id	sid	tid	token	lemma	pos
iajs-2809	1	1	70	70	NUM
iajs-2809	1	2	this	this	DET
iajs-2809	1	3	work	work	NOUN
iajs-2809	1	4	is	be	AUX
iajs-2809	1	5	licensed	license	VERB
iajs-2809	1	6	under	under	ADP
iajs-2809	1	7	a	a	DET
iajs-2809	1	8	creative	creative	ADJ
iajs-2809	1	9	commons	common	NOUN
iajs-2809	1	10	attribution	attribution	NOUN
iajs-2809	1	11	4.0	4.0	NUM
iajs-2809	1	12	international	international	ADJ
iajs-2809	1	13	license	license	NOUN
iajs-2809	1	14	.	.	PUNCT
iajs-2809	2	1	a	a	DET
iajs-2809	2	2	new	new	ADJ
iajs-2809	2	3	approach	approach	NOUN
iajs-2809	2	4	to	to	ADP
iajs-2809	2	5	solving	solve	VERB
iajs-2809	2	6	linear	linear	ADJ
iajs-2809	2	7	fractional	fractional	ADJ
iajs-2809	2	8	programming	programming	NOUN
iajs-2809	2	9	problem	problem	NOUN
iajs-2809	2	10	with	with	ADP
iajs-2809	2	11	rough	rough	ADJ
iajs-2809	2	12	interval	interval	NOUN
iajs-2809	2	13	coefficients	coefficient	NOUN
iajs-2809	2	14	in	in	ADP
iajs-2809	2	15	the	the	DET
iajs-2809	2	16	objective	objective	ADJ
iajs-2809	2	17	function	function	NOUN
iajs-2809	2	18	abstract	abstract	ADP
iajs-2809	2	19	this	this	DET
iajs-2809	2	20	paper	paper	NOUN
iajs-2809	2	21	presents	present	VERB
iajs-2809	2	22	a	a	DET
iajs-2809	2	23	linear	linear	ADJ
iajs-2809	2	24	fractional	fractional	ADJ
iajs-2809	2	25	programming	programming	NOUN
iajs-2809	2	26	problem	problem	NOUN
iajs-2809	2	27	(	(	PUNCT
iajs-2809	2	28	lfpp	lfpp	PROPN
iajs-2809	2	29	)	)	PUNCT
iajs-2809	2	30	with	with	ADP
iajs-2809	2	31	rough	rough	ADJ
iajs-2809	2	32	interval	interval	NOUN
iajs-2809	2	33	coefficients	coefficient	NOUN
iajs-2809	2	34	(	(	PUNCT
iajs-2809	2	35	rics	ric	NOUN
iajs-2809	2	36	)	)	PUNCT
iajs-2809	2	37	in	in	ADP
iajs-2809	2	38	the	the	DET
iajs-2809	2	39	objective	objective	ADJ
iajs-2809	2	40	function	function	NOUN
iajs-2809	2	41	.	.	PUNCT
iajs-2809	3	1	it	it	PRON
iajs-2809	3	2	shows	show	VERB
iajs-2809	3	3	that	that	SCONJ
iajs-2809	3	4	the	the	DET
iajs-2809	3	5	lfpp	lfpp	NOUN
iajs-2809	3	6	with	with	ADP
iajs-2809	3	7	rics	ric	NOUN
iajs-2809	3	8	in	in	ADP
iajs-2809	3	9	the	the	DET
iajs-2809	3	10	objective	objective	ADJ
iajs-2809	3	11	function	function	NOUN
iajs-2809	3	12	can	can	AUX
iajs-2809	3	13	be	be	AUX
iajs-2809	3	14	converted	convert	VERB
iajs-2809	3	15	into	into	ADP
iajs-2809	3	16	a	a	DET
iajs-2809	3	17	linear	linear	ADJ
iajs-2809	3	18	programming	programming	NOUN
iajs-2809	3	19	problem	problem	NOUN
iajs-2809	3	20	(	(	PUNCT
iajs-2809	3	21	lpp	lpp	PROPN
iajs-2809	3	22	)	)	PUNCT
iajs-2809	3	23	with	with	ADP
iajs-2809	3	24	rics	ric	NOUN
iajs-2809	3	25	by	by	ADP
iajs-2809	3	26	using	use	VERB
iajs-2809	3	27	the	the	DET
iajs-2809	3	28	variable	variable	ADJ
iajs-2809	3	29	transformations	transformation	NOUN
iajs-2809	3	30	.	.	PUNCT
iajs-2809	4	1	to	to	PART
iajs-2809	4	2	solve	solve	VERB
iajs-2809	4	3	this	this	DET
iajs-2809	4	4	problem	problem	NOUN
iajs-2809	4	5	,	,	PUNCT
iajs-2809	4	6	we	we	PRON
iajs-2809	4	7	will	will	AUX
iajs-2809	4	8	make	make	VERB
iajs-2809	4	9	two	two	NUM
iajs-2809	4	10	lpp	lpp	NOUN
iajs-2809	4	11	with	with	ADP
iajs-2809	4	12	interval	interval	NOUN
iajs-2809	4	13	coefficients	coefficient	NOUN
iajs-2809	4	14	(	(	PUNCT
iajs-2809	4	15	ics	ics	NOUN
iajs-2809	4	16	)	)	PUNCT
iajs-2809	4	17	.	.	PUNCT
iajs-2809	5	1	next	next	ADV
iajs-2809	5	2	,	,	PUNCT
iajs-2809	5	3	those	those	DET
iajs-2809	5	4	four	four	NUM
iajs-2809	5	5	lpps	lpp	NOUN
iajs-2809	5	6	can	can	AUX
iajs-2809	5	7	be	be	AUX
iajs-2809	5	8	constructed	construct	VERB
iajs-2809	5	9	under	under	ADP
iajs-2809	5	10	these	these	DET
iajs-2809	5	11	assumptions	assumption	NOUN
iajs-2809	5	12	;	;	PUNCT
iajs-2809	5	13	the	the	DET
iajs-2809	5	14	lpps	lpp	NOUN
iajs-2809	5	15	can	can	AUX
iajs-2809	5	16	be	be	AUX
iajs-2809	5	17	solved	solve	VERB
iajs-2809	5	18	by	by	ADP
iajs-2809	5	19	the	the	DET
iajs-2809	5	20	classical	classical	ADJ
iajs-2809	5	21	simplex	simplex	NOUN
iajs-2809	5	22	method	method	NOUN
iajs-2809	5	23	and	and	CCONJ
iajs-2809	5	24	used	use	VERB
iajs-2809	5	25	with	with	ADP
iajs-2809	5	26	ms	ms	PROPN
iajs-2809	5	27	excel	excel	PROPN
iajs-2809	5	28	solver	solver	NOUN
iajs-2809	5	29	.	.	PUNCT
iajs-2809	6	1	there	there	PRON
iajs-2809	6	2	is	be	VERB
iajs-2809	6	3	also	also	ADV
iajs-2809	6	4	argumentation	argumentation	NOUN
iajs-2809	6	5	about	about	ADP
iajs-2809	6	6	solving	solve	VERB
iajs-2809	6	7	this	this	DET
iajs-2809	6	8	type	type	NOUN
iajs-2809	6	9	of	of	ADP
iajs-2809	6	10	linear	linear	ADJ
iajs-2809	6	11	fractional	fractional	ADJ
iajs-2809	6	12	optimization	optimization	NOUN
iajs-2809	6	13	programming	programming	NOUN
iajs-2809	6	14	problem	problem	NOUN
iajs-2809	6	15	.	.	PUNCT
iajs-2809	7	1	the	the	DET
iajs-2809	7	2	derived	derive	VERB
iajs-2809	7	3	theory	theory	NOUN
iajs-2809	7	4	can	can	AUX
iajs-2809	7	5	be	be	AUX
iajs-2809	7	6	applied	apply	VERB
iajs-2809	7	7	to	to	ADP
iajs-2809	7	8	several	several	ADJ
iajs-2809	7	9	numerical	numerical	ADJ
iajs-2809	7	10	examples	example	NOUN
iajs-2809	7	11	with	with	ADP
iajs-2809	7	12	its	its	PRON
iajs-2809	7	13	details	detail	NOUN
iajs-2809	7	14	,	,	PUNCT
iajs-2809	7	15	but	but	CCONJ
iajs-2809	7	16	we	we	PRON
iajs-2809	7	17	show	show	VERB
iajs-2809	7	18	only	only	ADV
iajs-2809	7	19	two	two	NUM
iajs-2809	7	20	examples	example	NOUN
iajs-2809	7	21	for	for	ADP
iajs-2809	7	22	promising	promise	VERB
iajs-2809	7	23	.	.	PUNCT
iajs-2809	8	1	keywords	keyword	NOUN
iajs-2809	8	2	:	:	PUNCT
iajs-2809	8	3	linear	linear	ADJ
iajs-2809	8	4	fractional	fractional	ADJ
iajs-2809	8	5	programming	programming	NOUN
iajs-2809	8	6	,	,	PUNCT
iajs-2809	8	7	linear	linear	ADJ
iajs-2809	8	8	programming	programming	NOUN
iajs-2809	8	9	,	,	PUNCT
iajs-2809	8	10	rough	rough	ADJ
iajs-2809	8	11	interval	interval	NOUN
iajs-2809	8	12	function	function	NOUN
iajs-2809	8	13	,	,	PUNCT
iajs-2809	8	14	rough	rough	ADJ
iajs-2809	8	15	interval	interval	NOUN
iajs-2809	8	16	coefficients	coefficient	NOUN
iajs-2809	8	17	,	,	PUNCT
iajs-2809	8	18	interval	interval	NOUN
iajs-2809	8	19	coefficients	coefficient	NOUN
iajs-2809	8	20	.	.	PUNCT
iajs-2809	9	1	1	1	X
iajs-2809	9	2	.	.	X
iajs-2809	9	3	introduction	introduction	NOUN
iajs-2809	9	4	the	the	DET
iajs-2809	9	5	essential	essential	ADJ
iajs-2809	9	6	optimization	optimization	NOUN
iajs-2809	9	7	is	be	AUX
iajs-2809	9	8	nonlinear	nonlinear	ADJ
iajs-2809	9	9	programming	programming	NOUN
iajs-2809	9	10	;	;	PUNCT
iajs-2809	9	11	the	the	DET
iajs-2809	9	12	linear	linear	ADJ
iajs-2809	9	13	fractional	fractional	ADJ
iajs-2809	9	14	programming	programming	NOUN
iajs-2809	9	15	problem	problem	NOUN
iajs-2809	9	16	is	be	AUX
iajs-2809	9	17	one	one	NUM
iajs-2809	9	18	of	of	ADP
iajs-2809	9	19	them	they	PRON
iajs-2809	9	20	.	.	PUNCT
iajs-2809	10	1	the	the	DET
iajs-2809	10	2	coefficient	coefficient	NOUN
iajs-2809	10	3	on	on	ADP
iajs-2809	10	4	the	the	DET
iajs-2809	10	5	system	system	NOUN
iajs-2809	10	6	can	can	AUX
iajs-2809	10	7	commonly	commonly	ADV
iajs-2809	10	8	not	not	PART
iajs-2809	10	9	be	be	AUX
iajs-2809	10	10	established	establish	VERB
iajs-2809	10	11	exactly	exactly	ADV
iajs-2809	10	12	in	in	ADP
iajs-2809	10	13	various	various	ADJ
iajs-2809	10	14	linear	linear	ADJ
iajs-2809	10	15	programming	programming	NOUN
iajs-2809	10	16	problems	problem	NOUN
iajs-2809	10	17	.	.	PUNCT
iajs-2809	11	1	the	the	DET
iajs-2809	11	2	interval	interval	NOUN
iajs-2809	11	3	technique	technique	NOUN
iajs-2809	11	4	,	,	PUNCT
iajs-2809	11	5	in	in	ADP
iajs-2809	11	6	which	which	PRON
iajs-2809	11	7	unknown	unknown	ADJ
iajs-2809	11	8	coefficients	coefficient	NOUN
iajs-2809	11	9	are	be	AUX
iajs-2809	11	10	converted	convert	VERB
iajs-2809	11	11	into	into	ADP
iajs-2809	11	12	intervals	interval	NOUN
iajs-2809	11	13	,	,	PUNCT
iajs-2809	11	14	is	be	AUX
iajs-2809	11	15	one	one	NUM
iajs-2809	11	16	way	way	NOUN
iajs-2809	11	17	to	to	PART
iajs-2809	11	18	address	address	VERB
iajs-2809	11	19	this	this	DET
iajs-2809	11	20	programming	programming	NOUN
iajs-2809	11	21	problem	problem	NOUN
iajs-2809	11	22	.	.	PUNCT
iajs-2809	12	1	because	because	SCONJ
iajs-2809	12	2	many	many	ADJ
iajs-2809	12	3	real	real	ADJ
iajs-2809	12	4	-	-	PUNCT
iajs-2809	12	5	world	world	NOUN
iajs-2809	12	6	problems	problem	NOUN
iajs-2809	12	7	are	be	AUX
iajs-2809	12	8	expressed	express	VERB
iajs-2809	12	9	as	as	ADP
iajs-2809	12	10	fractional	fractional	ADJ
iajs-2809	12	11	functions	function	NOUN
iajs-2809	12	12	,	,	PUNCT
iajs-2809	12	13	fractional	fractional	ADJ
iajs-2809	12	14	programming	programming	NOUN
iajs-2809	12	15	has	have	AUX
iajs-2809	12	16	gained	gain	VERB
iajs-2809	12	17	much	much	ADJ
iajs-2809	12	18	control	control	NOUN
iajs-2809	12	19	.	.	PUNCT
iajs-2809	13	1	these	these	DET
iajs-2809	13	2	issues	issue	NOUN
iajs-2809	13	3	frequently	frequently	ADV
iajs-2809	13	4	arise	arise	VERB
iajs-2809	13	5	in	in	ADP
iajs-2809	13	6	situations	situation	NOUN
iajs-2809	13	7	involving	involve	VERB
iajs-2809	13	8	actual	actual	ADJ
iajs-2809	13	9	capital	capital	NOUN
iajs-2809	13	10	return	return	NOUN
iajs-2809	13	11	on	on	ADP
iajs-2809	13	12	investment	investment	NOUN
iajs-2809	13	13	versus	versus	ADP
iajs-2809	13	14	required	require	VERB
iajs-2809	13	15	capital	capital	NOUN
iajs-2809	13	16	.	.	PUNCT
iajs-2809	14	1	the	the	DET
iajs-2809	14	2	objective	objective	ADJ
iajs-2809	14	3	function	function	NOUN
iajs-2809	14	4	of	of	ADP
iajs-2809	14	5	a	a	DET
iajs-2809	14	6	linear	linear	ADJ
iajs-2809	14	7	fractional	fractional	ADJ
iajs-2809	14	8	programming	programming	NOUN
iajs-2809	14	9	problem	problem	NOUN
iajs-2809	14	10	is	be	AUX
iajs-2809	14	11	highly	highly	ADV
iajs-2809	14	12	valuable	valuable	ADJ
iajs-2809	14	13	in	in	ADP
iajs-2809	14	14	construction	construction	NOUN
iajs-2809	14	15	,	,	PUNCT
iajs-2809	14	16	such	such	ADJ
iajs-2809	14	17	as	as	ADP
iajs-2809	14	18	economic	economic	ADJ
iajs-2809	14	19	and	and	CCONJ
iajs-2809	14	20	company	company	NOUN
iajs-2809	14	21	organization	organization	NOUN
iajs-2809	14	22	.	.	PUNCT
iajs-2809	15	1	charnes	charne	NOUN
iajs-2809	15	2	and	and	CCONJ
iajs-2809	15	3	cooper	cooper	NOUN
iajs-2809	15	4	have	have	AUX
iajs-2809	15	5	presented	present	VERB
iajs-2809	15	6	several	several	ADJ
iajs-2809	15	7	solutions	solution	NOUN
iajs-2809	15	8	to	to	ADP
iajs-2809	15	9	this	this	DET
iajs-2809	15	10	problem	problem	NOUN
iajs-2809	15	11	.	.	PUNCT
iajs-2809	16	1	ibn	ibn	PROPN
iajs-2809	16	2	al	al	PROPN
iajs-2809	16	3	haitham	haitham	PROPN
iajs-2809	16	4	journal	journal	PROPN
iajs-2809	16	5	for	for	ADP
iajs-2809	16	6	pure	pure	ADJ
iajs-2809	16	7	and	and	CCONJ
iajs-2809	16	8	applied	apply	VERB
iajs-2809	16	9	science	science	NOUN
iajs-2809	16	10	journal	journal	PROPN
iajs-2809	16	11	homepage	homepage	NOUN
iajs-2809	16	12	:	:	PUNCT
iajs-2809	16	13	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2809	16	14	doi	doi	NOUN
iajs-2809	16	15	:	:	PUNCT
iajs-2809	16	16	10.30526/35.2.2809	10.30526/35.2.2809	PROPN
iajs-2809	16	17	article	article	NOUN
iajs-2809	16	18	history	history	NOUN
iajs-2809	16	19	:	:	PUNCT
iajs-2809	16	20	received	receive	VERB
iajs-2809	16	21	1	1	NUM
iajs-2809	16	22	,	,	PUNCT
iajs-2809	16	23	februry	februry	NOUN
iajs-2809	16	24	,	,	PUNCT
iajs-2809	16	25	2022	2022	NUM
iajs-2809	16	26	,	,	PUNCT
iajs-2809	16	27	accepted,7	accepted,7	NOUN
iajs-2809	16	28	,	,	PUNCT
iajs-2809	16	29	march	march	NOUN
iajs-2809	16	30	,	,	PUNCT
iajs-2809	16	31	2022	2022	NUM
iajs-2809	16	32	,	,	PUNCT
iajs-2809	16	33	published	publish	VERB
iajs-2809	16	34	in	in	ADP
iajs-2809	16	35	april	april	PROPN
iajs-2809	16	36	2022	2022	NUM
iajs-2809	16	37	.	.	PUNCT
iajs-2809	17	1	rebaz	rebaz	PROPN
iajs-2809	17	2	b.	b.	PROPN
iajs-2809	17	3	mustafa	mustafa	PROPN
iajs-2809	17	4	department	department	PROPN
iajs-2809	17	5	of	of	ADP
iajs-2809	17	6	mathematics	mathematics	PROPN
iajs-2809	17	7	,	,	PUNCT
iajs-2809	17	8	college	college	NOUN
iajs-2809	17	9	of	of	ADP
iajs-2809	17	10	education	education	NOUN
iajs-2809	17	11	,	,	PUNCT
iajs-2809	17	12	salahaddin	salahaddin	VERB
iajs-2809	17	13	university	university	NOUN
iajs-2809	17	14	,	,	PUNCT
iajs-2809	17	15	erbil	erbil	PROPN
iajs-2809	17	16	,	,	PUNCT
iajs-2809	17	17	iraq	iraq	PROPN
iajs-2809	17	18	.	.	PUNCT
iajs-2809	18	1	rebaz.mustafa@epu.edu.iq	rebaz.mustafa@epu.edu.iq	PROPN
iajs-2809	18	2	nejmaddin	nejmaddin	PROPN
iajs-2809	18	3	a.	a.	NOUN
iajs-2809	18	4	sulaiman	sulaiman	PROPN
iajs-2809	18	5	department	department	PROPN
iajs-2809	18	6	of	of	ADP
iajs-2809	18	7	mathematics	mathematics	PROPN
iajs-2809	18	8	,	,	PUNCT
iajs-2809	18	9	college	college	NOUN
iajs-2809	18	10	of	of	ADP
iajs-2809	18	11	education	education	NOUN
iajs-2809	18	12	,	,	PUNCT
iajs-2809	18	13	salahaddin	salahaddin	VERB
iajs-2809	18	14	university	university	NOUN
iajs-2809	18	15	,	,	PUNCT
iajs-2809	18	16	erbil	erbil	PROPN
iajs-2809	18	17	,	,	PUNCT
iajs-2809	18	18	iraq	iraq	PROPN
iajs-2809	18	19	.	.	PUNCT
iajs-2809	19	1	nejmadin.sulaiman@su.edu.krd	nejmadin.sulaiman@su.edu.krd	PROPN
iajs-2809	19	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2809	19	3	mailto:rebaz.mustafa@epu.edu.iq	mailto:rebaz.mustafa@epu.edu.iq	PROPN
iajs-2809	19	4	ibn	ibn	PROPN
iajs-2809	19	5	al	al	PROPN
iajs-2809	19	6	-	-	PUNCT
iajs-2809	19	7	haitham	haitham	PROPN
iajs-2809	19	8	jour	jour	X
iajs-2809	19	9	.	.	PROPN
iajs-2809	19	10	for	for	ADP
iajs-2809	19	11	pure	pure	ADJ
iajs-2809	19	12	&	&	CCONJ
iajs-2809	19	13	appl	appl	PROPN
iajs-2809	19	14	.	.	PUNCT
iajs-2809	20	1	sci	sci	PROPN
iajs-2809	20	2	.	.	PROPN
iajs-2809	21	1	53	53	NUM
iajs-2809	21	2	(	(	PUNCT
iajs-2809	21	3	2)2022	2)2022	VERB
iajs-2809	21	4	71	71	NUM
iajs-2809	21	5	charnes	charne	NOUN
iajs-2809	21	6	and	and	CCONJ
iajs-2809	21	7	cooper	cooper	NOUN
iajs-2809	21	8	’s	’s	PART
iajs-2809	21	9	method	method	NOUN
iajs-2809	22	1	[	[	X
iajs-2809	22	2	1]can	1]can	NUM
iajs-2809	22	3	convert	convert	VERB
iajs-2809	22	4	a	a	DET
iajs-2809	22	5	fractional	fractional	ADJ
iajs-2809	22	6	programming	programming	NOUN
iajs-2809	22	7	problem	problem	NOUN
iajs-2809	22	8	into	into	ADP
iajs-2809	22	9	a	a	DET
iajs-2809	22	10	linear	linear	ADJ
iajs-2809	22	11	programming	programming	NOUN
iajs-2809	22	12	problem	problem	NOUN
iajs-2809	22	13	,	,	PUNCT
iajs-2809	22	14	which	which	PRON
iajs-2809	22	15	is	be	AUX
iajs-2809	22	16	known	know	VERB
iajs-2809	22	17	as	as	ADP
iajs-2809	22	18	linear	linear	ADJ
iajs-2809	22	19	fractional	fractional	ADJ
iajs-2809	22	20	programming	programming	NOUN
iajs-2809	22	21	.	.	PUNCT
iajs-2809	23	1	[	[	X
iajs-2809	23	2	2	2	X
iajs-2809	23	3	]	]	PUNCT
iajs-2809	23	4	the	the	DET
iajs-2809	23	5	notion	notion	NOUN
iajs-2809	23	6	of	of	ADP
iajs-2809	23	7	a	a	DET
iajs-2809	23	8	rough	rough	ADJ
iajs-2809	23	9	interval	interval	NOUN
iajs-2809	23	10	will	will	AUX
iajs-2809	23	11	be	be	AUX
iajs-2809	23	12	applied	apply	VERB
iajs-2809	23	13	to	to	PART
iajs-2809	23	14	model	model	VERB
iajs-2809	23	15	dual	dual	ADJ
iajs-2809	23	16	uncertain	uncertain	ADJ
iajs-2809	23	17	information	information	NOUN
iajs-2809	23	18	of	of	ADP
iajs-2809	23	19	different	different	ADJ
iajs-2809	23	20	parameters	parameter	NOUN
iajs-2809	23	21	.	.	PUNCT
iajs-2809	24	1	the	the	DET
iajs-2809	24	2	related	relate	VERB
iajs-2809	24	3	solution	solution	NOUN
iajs-2809	24	4	method	method	NOUN
iajs-2809	24	5	for	for	ADP
iajs-2809	24	6	rough	rough	ADJ
iajs-2809	24	7	interval	interval	NOUN
iajs-2809	24	8	fuzzy	fuzzy	ADJ
iajs-2809	24	9	linear	linear	ADJ
iajs-2809	24	10	programming	programming	NOUN
iajs-2809	24	11	problems	problem	NOUN
iajs-2809	24	12	with	with	ADP
iajs-2809	24	13	dual	dual	ADJ
iajs-2809	24	14	uncertain	uncertain	ADJ
iajs-2809	24	15	solutions	solution	NOUN
iajs-2809	24	16	will	will	AUX
iajs-2809	24	17	be	be	AUX
iajs-2809	24	18	described	describe	VERB
iajs-2809	24	19	.	.	PUNCT
iajs-2809	25	1	the	the	DET
iajs-2809	25	2	rough	rough	ADJ
iajs-2809	25	3	set	set	NOUN
iajs-2809	25	4	(	(	PUNCT
iajs-2809	25	5	rs	rs	NOUN
iajs-2809	25	6	)	)	PUNCT
iajs-2809	25	7	theory	theory	NOUN
iajs-2809	25	8	's	's	PART
iajs-2809	25	9	main	main	ADJ
iajs-2809	25	10	principle	principle	NOUN
iajs-2809	25	11	is	be	AUX
iajs-2809	25	12	that	that	SCONJ
iajs-2809	25	13	each	each	DET
iajs-2809	25	14	ambiguous	ambiguous	ADJ
iajs-2809	25	15	notion	notion	NOUN
iajs-2809	25	16	is	be	AUX
iajs-2809	25	17	substituted	substitute	VERB
iajs-2809	25	18	by	by	ADP
iajs-2809	25	19	a	a	DET
iajs-2809	25	20	couple	couple	NOUN
iajs-2809	25	21	of	of	ADP
iajs-2809	25	22	exact	exact	ADJ
iajs-2809	25	23	conceptions	conception	NOUN
iajs-2809	25	24	known	know	VERB
iajs-2809	25	25	as	as	ADP
iajs-2809	25	26	the	the	DET
iajs-2809	25	27	lower	low	ADJ
iajs-2809	25	28	and	and	CCONJ
iajs-2809	25	29	upper	upper	ADJ
iajs-2809	25	30	approximations	approximation	NOUN
iajs-2809	25	31	of	of	ADP
iajs-2809	25	32	the	the	DET
iajs-2809	25	33	vague	vague	ADJ
iajs-2809	25	34	idea	idea	NOUN
iajs-2809	25	35	.	.	PUNCT
iajs-2809	26	1	for	for	ADP
iajs-2809	26	2	an	an	DET
iajs-2809	26	3	ambiguous	ambiguous	ADJ
iajs-2809	26	4	idea	idea	NOUN
iajs-2809	26	5	rs	rs	NOUN
iajs-2809	26	6	,	,	PUNCT
iajs-2809	26	7	a	a	DET
iajs-2809	26	8	lower	low	ADJ
iajs-2809	26	9	approximation	approximation	NOUN
iajs-2809	26	10	includes	include	VERB
iajs-2809	26	11	complete	complete	ADJ
iajs-2809	26	12	items	item	NOUN
iajs-2809	26	13	that	that	PRON
iajs-2809	26	14	surely	surely	ADV
iajs-2809	26	15	belong	belong	VERB
iajs-2809	26	16	to	to	ADP
iajs-2809	26	17	the	the	DET
iajs-2809	26	18	model	model	NOUN
iajs-2809	26	19	rs	rs	NOUN
iajs-2809	26	20	,	,	PUNCT
iajs-2809	26	21	and	and	CCONJ
iajs-2809	26	22	an	an	DET
iajs-2809	26	23	upper	upper	ADJ
iajs-2809	26	24	approximation	approximation	NOUN
iajs-2809	26	25	comprises	comprise	VERB
iajs-2809	26	26	all	all	DET
iajs-2809	26	27	items	item	NOUN
iajs-2809	26	28	that	that	PRON
iajs-2809	26	29	possibly	possibly	ADV
iajs-2809	26	30	belong	belong	VERB
iajs-2809	26	31	to	to	ADP
iajs-2809	26	32	the	the	DET
iajs-2809	26	33	model	model	NOUN
iajs-2809	26	34	rs	rs	NOUN
iajs-2809	26	35	.	.	PUNCT
iajs-2809	27	1	in	in	ADP
iajs-2809	27	2	other	other	ADJ
iajs-2809	27	3	words	word	NOUN
iajs-2809	27	4	,	,	PUNCT
iajs-2809	27	5	the	the	DET
iajs-2809	27	6	concept	concept	NOUN
iajs-2809	27	7	's	's	PART
iajs-2809	27	8	lower	low	ADJ
iajs-2809	27	9	approximation	approximation	NOUN
iajs-2809	27	10	is	be	AUX
iajs-2809	27	11	the	the	DET
iajs-2809	27	12	union	union	NOUN
iajs-2809	27	13	of	of	ADP
iajs-2809	27	14	all	all	DET
iajs-2809	27	15	fundamental	fundamental	ADJ
iajs-2809	27	16	concepts	concept	NOUN
iajs-2809	27	17	contained	contain	VERB
iajs-2809	27	18	in	in	ADP
iajs-2809	27	19	it	it	PRON
iajs-2809	27	20	.	.	PUNCT
iajs-2809	28	1	the	the	DET
iajs-2809	28	2	concept	concept	NOUN
iajs-2809	28	3	's	's	PART
iajs-2809	28	4	upper	upper	ADJ
iajs-2809	28	5	approximation	approximation	NOUN
iajs-2809	28	6	is	be	AUX
iajs-2809	28	7	the	the	DET
iajs-2809	28	8	union	union	NOUN
iajs-2809	28	9	of	of	ADP
iajs-2809	28	10	all	all	DET
iajs-2809	28	11	fundamental	fundamental	ADJ
iajs-2809	28	12	concepts	concept	NOUN
iajs-2809	28	13	with	with	ADP
iajs-2809	28	14	a	a	DET
iajs-2809	28	15	non	non	ADJ
iajs-2809	28	16	-	-	ADJ
iajs-2809	28	17	empty	empty	ADJ
iajs-2809	28	18	set	set	NOUN
iajs-2809	28	19	intersection	intersection	NOUN
iajs-2809	28	20	.	.	PUNCT
iajs-2809	29	1	pawlak[3	pawlak[3	PROPN
iajs-2809	29	2	]	]	PUNCT
iajs-2809	29	3	introduced	introduce	VERB
iajs-2809	29	4	the	the	DET
iajs-2809	29	5	rough	rough	ADJ
iajs-2809	29	6	set	set	NOUN
iajs-2809	29	7	theory	theory	NOUN
iajs-2809	29	8	in	in	ADP
iajs-2809	29	9	1982	1982	NUM
iajs-2809	29	10	as	as	ADP
iajs-2809	29	11	a	a	DET
iajs-2809	29	12	strategy	strategy	NOUN
iajs-2809	29	13	for	for	ADP
iajs-2809	29	14	managing	manage	VERB
iajs-2809	29	15	ambiguity	ambiguity	NOUN
iajs-2809	29	16	and	and	CCONJ
iajs-2809	29	17	uncertainty	uncertainty	NOUN
iajs-2809	29	18	together	together	ADV
iajs-2809	29	19	.	.	PUNCT
iajs-2809	30	1	many	many	ADJ
iajs-2809	30	2	techniques	technique	NOUN
iajs-2809	30	3	for	for	ADP
iajs-2809	30	4	solving	solve	VERB
iajs-2809	30	5	fractional	fractional	ADJ
iajs-2809	30	6	programming	programming	NOUN
iajs-2809	30	7	difficulties	difficulty	NOUN
iajs-2809	30	8	have	have	AUX
iajs-2809	30	9	emerged	emerge	VERB
iajs-2809	30	10	in	in	ADP
iajs-2809	30	11	recent	recent	ADJ
iajs-2809	30	12	decades	decade	NOUN
iajs-2809	30	13	.	.	PUNCT
iajs-2809	31	1	some	some	PRON
iajs-2809	31	2	works	work	VERB
iajs-2809	31	3	on	on	ADP
iajs-2809	31	4	rough	rough	ADJ
iajs-2809	31	5	programming	programming	NOUN
iajs-2809	31	6	have	have	AUX
iajs-2809	31	7	been	be	AUX
iajs-2809	31	8	developed	develop	VERB
iajs-2809	31	9	in	in	ADP
iajs-2809	31	10	the	the	DET
iajs-2809	31	11	last	last	ADJ
iajs-2809	31	12	decade	decade	NOUN
iajs-2809	31	13	.	.	PUNCT
iajs-2809	32	1	[	[	X
iajs-2809	32	2	4	4	X
iajs-2809	32	3	]	]	PUNCT
iajs-2809	32	4	and	and	CCONJ
iajs-2809	32	5	[	[	X
iajs-2809	32	6	5	5	NUM
iajs-2809	32	7	]	]	PUNCT
iajs-2809	32	8	developed	develop	VERB
iajs-2809	32	9	a	a	DET
iajs-2809	32	10	new	new	ADJ
iajs-2809	32	11	sort	sort	NOUN
iajs-2809	32	12	of	of	ADP
iajs-2809	32	13	rough	rough	ADJ
iajs-2809	32	14	programming	programming	NOUN
iajs-2809	32	15	recently	recently	ADV
iajs-2809	32	16	,	,	PUNCT
iajs-2809	32	17	where	where	SCONJ
iajs-2809	32	18	they	they	PRON
iajs-2809	32	19	established	establish	VERB
iajs-2809	32	20	two	two	NUM
iajs-2809	32	21	solution	solution	NOUN
iajs-2809	32	22	concepts	concept	NOUN
iajs-2809	32	23	:	:	PUNCT
iajs-2809	32	24	surely	surely	ADV
iajs-2809	32	25	optimum	optimum	ADJ
iajs-2809	32	26	solution	solution	NOUN
iajs-2809	32	27	and	and	CCONJ
iajs-2809	32	28	possibly	possibly	ADV
iajs-2809	32	29	optimal	optimal	ADJ
iajs-2809	32	30	solution	solution	NOUN
iajs-2809	32	31	.	.	PUNCT
iajs-2809	33	1	[	[	X
iajs-2809	33	2	6	6	NUM
iajs-2809	33	3	]	]	PUNCT
iajs-2809	33	4	proposed	propose	VERB
iajs-2809	33	5	rough	rough	ADJ
iajs-2809	33	6	intervals	interval	NOUN
iajs-2809	33	7	(	(	PUNCT
iajs-2809	33	8	ri	ri	NOUN
iajs-2809	33	9	)	)	PUNCT
iajs-2809	33	10	to	to	PART
iajs-2809	33	11	transact	transact	VERB
iajs-2809	33	12	with	with	ADP
iajs-2809	33	13	partly	partly	ADV
iajs-2809	33	14	uncertain	uncertain	ADJ
iajs-2809	33	15	or	or	CCONJ
iajs-2809	33	16	ill	ill	ADV
iajs-2809	33	17	-	-	PUNCT
iajs-2809	33	18	defined	define	VERB
iajs-2809	33	19	relevant	relevant	ADJ
iajs-2809	33	20	variables	variable	NOUN
iajs-2809	33	21	.	.	PUNCT
iajs-2809	34	1	the	the	DET
iajs-2809	34	2	rough	rough	ADJ
iajs-2809	34	3	set	set	NOUN
iajs-2809	34	4	ideas	idea	NOUN
iajs-2809	34	5	are	be	AUX
iajs-2809	34	6	adapted	adapt	VERB
iajs-2809	34	7	to	to	PART
iajs-2809	34	8	represent	represent	VERB
iajs-2809	34	9	constant	constant	ADJ
iajs-2809	34	10	variables	variable	NOUN
iajs-2809	34	11	.	.	PUNCT
iajs-2809	35	1	it	it	PRON
iajs-2809	35	2	is	be	AUX
iajs-2809	35	3	worth	worth	ADJ
iajs-2809	35	4	noting	note	VERB
iajs-2809	35	5	that	that	SCONJ
iajs-2809	35	6	,	,	PUNCT
iajs-2809	35	7	at	at	ADP
iajs-2809	35	8	first	first	ADV
iajs-2809	35	9	,	,	PUNCT
iajs-2809	35	10	rough	rough	ADJ
iajs-2809	35	11	sets	set	NOUN
iajs-2809	35	12	could	could	AUX
iajs-2809	35	13	only	only	ADV
iajs-2809	35	14	handle	handle	VERB
iajs-2809	35	15	discrete	discrete	ADJ
iajs-2809	35	16	objects	object	NOUN
iajs-2809	35	17	and	and	CCONJ
iajs-2809	35	18	could	could	AUX
iajs-2809	35	19	n't	not	PART
iajs-2809	35	20	express	express	VERB
iajs-2809	35	21	continuous	continuous	ADJ
iajs-2809	35	22	values	value	NOUN
iajs-2809	35	23	.	.	PUNCT
iajs-2809	36	1	rough	rough	ADJ
iajs-2809	36	2	intervals	interval	NOUN
iajs-2809	36	3	are	be	AUX
iajs-2809	36	4	a	a	DET
iajs-2809	36	5	subset	subset	NOUN
iajs-2809	36	6	of	of	ADP
iajs-2809	36	7	rss	rss	NOUN
iajs-2809	36	8	.	.	PUNCT
iajs-2809	37	1	it	it	PRON
iajs-2809	37	2	satisfies	satisfy	VERB
iajs-2809	37	3	all	all	PRON
iajs-2809	37	4	of	of	ADP
iajs-2809	37	5	the	the	DET
iajs-2809	37	6	rough	rough	ADJ
iajs-2809	37	7	set	set	NOUN
iajs-2809	37	8	's	's	PART
iajs-2809	37	9	attributes	attribute	NOUN
iajs-2809	37	10	and	and	CCONJ
iajs-2809	37	11	key	key	ADJ
iajs-2809	37	12	ideas	idea	NOUN
iajs-2809	37	13	,	,	PUNCT
iajs-2809	37	14	such	such	ADJ
iajs-2809	37	15	as	as	ADP
iajs-2809	37	16	the	the	DET
iajs-2809	37	17	notions	notion	NOUN
iajs-2809	37	18	of	of	ADP
iajs-2809	37	19	upper	upper	ADJ
iajs-2809	37	20	and	and	CCONJ
iajs-2809	37	21	lower	low	ADJ
iajs-2809	37	22	approximation[7	approximation[7	NOUN
iajs-2809	37	23	]	]	PUNCT
iajs-2809	37	24	.	.	PUNCT
iajs-2809	38	1	[	[	X
iajs-2809	38	2	8	8	NUM
iajs-2809	38	3	]	]	PUNCT
iajs-2809	38	4	introduced	introduce	VERB
iajs-2809	38	5	a	a	DET
iajs-2809	38	6	method	method	NOUN
iajs-2809	38	7	to	to	PART
iajs-2809	38	8	solve	solve	VERB
iajs-2809	38	9	lfpp	lfpp	NOUN
iajs-2809	38	10	with	with	ADP
iajs-2809	38	11	ics	ic	NOUN
iajs-2809	38	12	in	in	ADP
iajs-2809	38	13	the	the	DET
iajs-2809	38	14	objective	objective	ADJ
iajs-2809	38	15	function	function	NOUN
iajs-2809	38	16	by	by	ADP
iajs-2809	38	17	variable	variable	ADJ
iajs-2809	38	18	transformation	transformation	NOUN
iajs-2809	38	19	.	.	PUNCT
iajs-2809	39	1	the	the	DET
iajs-2809	39	2	initial	initial	ADJ
iajs-2809	39	3	problem	problem	NOUN
iajs-2809	39	4	is	be	AUX
iajs-2809	39	5	converted	convert	VERB
iajs-2809	39	6	into	into	ADP
iajs-2809	39	7	a	a	DET
iajs-2809	39	8	nonlinear	nonlinear	ADJ
iajs-2809	39	9	programming	programming	NOUN
iajs-2809	39	10	problem	problem	NOUN
iajs-2809	39	11	,	,	PUNCT
iajs-2809	39	12	which	which	PRON
iajs-2809	39	13	is	be	AUX
iajs-2809	39	14	finally	finally	ADV
iajs-2809	39	15	transformed	transform	VERB
iajs-2809	39	16	into	into	ADP
iajs-2809	39	17	an	an	DET
iajs-2809	39	18	lpp	lpp	NOUN
iajs-2809	39	19	.	.	PUNCT
iajs-2809	40	1	[	[	X
iajs-2809	40	2	7	7	X
iajs-2809	40	3	]	]	PUNCT
iajs-2809	40	4	presented	present	VERB
iajs-2809	40	5	a	a	DET
iajs-2809	40	6	solution	solution	NOUN
iajs-2809	40	7	of	of	ADP
iajs-2809	40	8	the	the	DET
iajs-2809	40	9	fully	fully	ADV
iajs-2809	40	10	rough	rough	ADJ
iajs-2809	40	11	interval	interval	NOUN
iajs-2809	40	12	coefficients	coefficient	NOUN
iajs-2809	40	13	of	of	ADP
iajs-2809	40	14	lpp	lpp	PROPN
iajs-2809	40	15	.	.	PUNCT
iajs-2809	41	1	they	they	PRON
iajs-2809	41	2	constructed	construct	VERB
iajs-2809	41	3	two	two	NUM
iajs-2809	41	4	-	-	PUNCT
iajs-2809	41	5	interval	interval	NOUN
iajs-2809	41	6	coefficients	coefficient	NOUN
iajs-2809	41	7	of	of	ADP
iajs-2809	41	8	lpp	lpp	PROPN
iajs-2809	41	9	and	and	CCONJ
iajs-2809	41	10	found	find	VERB
iajs-2809	41	11	some	some	DET
iajs-2809	41	12	new	new	ADJ
iajs-2809	41	13	solutions	solution	NOUN
iajs-2809	41	14	.	.	PUNCT
iajs-2809	42	1	[	[	X
iajs-2809	42	2	9	9	NUM
iajs-2809	42	3	]	]	PUNCT
iajs-2809	42	4	suggested	suggest	VERB
iajs-2809	42	5	a	a	DET
iajs-2809	42	6	new	new	ADJ
iajs-2809	42	7	method	method	NOUN
iajs-2809	42	8	to	to	PART
iajs-2809	42	9	solve	solve	VERB
iajs-2809	42	10	the	the	DET
iajs-2809	42	11	rough	rough	ADJ
iajs-2809	42	12	interval	interval	NOUN
iajs-2809	42	13	linear	linear	NOUN
iajs-2809	42	14	fractional	fractional	ADJ
iajs-2809	42	15	programming	programming	NOUN
iajs-2809	42	16	problem	problem	NOUN
iajs-2809	42	17	and	and	CCONJ
iajs-2809	42	18	introduced	introduce	VERB
iajs-2809	42	19	two	two	NUM
iajs-2809	42	20	possible	possible	ADJ
iajs-2809	42	21	kinds	kind	NOUN
iajs-2809	42	22	of	of	ADP
iajs-2809	42	23	formula	formula	NOUN
iajs-2809	42	24	transformation	transformation	NOUN
iajs-2809	42	25	with	with	ADP
iajs-2809	42	26	its	its	PRON
iajs-2809	42	27	proof	proof	NOUN
iajs-2809	42	28	.	.	PUNCT
iajs-2809	43	1	they	they	PRON
iajs-2809	43	2	obtained	obtain	VERB
iajs-2809	43	3	optimal	optimal	ADJ
iajs-2809	43	4	values	value	NOUN
iajs-2809	43	5	and	and	CCONJ
iajs-2809	43	6	solutions	solution	NOUN
iajs-2809	43	7	for	for	ADP
iajs-2809	43	8	the	the	DET
iajs-2809	43	9	initial	initial	ADJ
iajs-2809	43	10	lfpp	lfpp	NOUN
iajs-2809	43	11	with	with	ADP
iajs-2809	43	12	rics	ric	NOUN
iajs-2809	43	13	.	.	PUNCT
iajs-2809	44	1	khalifa[10	khalifa[10	X
iajs-2809	44	2	]	]	PUNCT
iajs-2809	44	3	defined	define	VERB
iajs-2809	44	4	a	a	DET
iajs-2809	44	5	new	new	ADJ
iajs-2809	44	6	approach	approach	NOUN
iajs-2809	44	7	to	to	ADP
iajs-2809	44	8	solving	solve	VERB
iajs-2809	44	9	lfpp	lfpp	NOUN
iajs-2809	44	10	with	with	ADP
iajs-2809	44	11	rics	ric	NOUN
iajs-2809	44	12	in	in	ADP
iajs-2809	44	13	the	the	DET
iajs-2809	44	14	objective	objective	ADJ
iajs-2809	44	15	function	function	NOUN
iajs-2809	44	16	when	when	SCONJ
iajs-2809	44	17	the	the	DET
iajs-2809	44	18	problem	problem	NOUN
iajs-2809	44	19	can	can	AUX
iajs-2809	44	20	be	be	AUX
iajs-2809	44	21	transformed	transform	VERB
iajs-2809	44	22	into	into	ADP
iajs-2809	44	23	a	a	DET
iajs-2809	44	24	series	series	NOUN
iajs-2809	44	25	of	of	ADP
iajs-2809	44	26	lpps	lpp	NOUN
iajs-2809	44	27	with	with	ADP
iajs-2809	44	28	ri	ri	PROPN
iajs-2809	44	29	under	under	ADP
iajs-2809	44	30	some	some	DET
iajs-2809	44	31	assumptions	assumption	NOUN
iajs-2809	44	32	.	.	PUNCT
iajs-2809	45	1	we	we	PRON
iajs-2809	45	2	can	can	AUX
iajs-2809	45	3	be	be	AUX
iajs-2809	45	4	used	use	VERB
iajs-2809	45	5	our	our	PRON
iajs-2809	45	6	approach	approach	NOUN
iajs-2809	45	7	of	of	ADP
iajs-2809	45	8	multi	multi	ADJ
iajs-2809	45	9	-	-	ADJ
iajs-2809	45	10	objective	objective	ADJ
iajs-2809	45	11	linear	linear	ADJ
iajs-2809	45	12	fractional	fractional	ADJ
iajs-2809	45	13	programming	programming	NOUN
iajs-2809	45	14	problem	problem	NOUN
iajs-2809	45	15	or	or	CCONJ
iajs-2809	45	16	multiobjective	multiobjective	ADJ
iajs-2809	45	17	linear	linear	ADJ
iajs-2809	45	18	programming	programming	NOUN
iajs-2809	45	19	problem	problem	NOUN
iajs-2809	45	20	with	with	ADP
iajs-2809	45	21	rics	ric	NOUN
iajs-2809	45	22	,	,	PUNCT
iajs-2809	45	23	when	when	SCONJ
iajs-2809	45	24	maybe	maybe	ADV
iajs-2809	45	25	changed	change	VERB
iajs-2809	45	26	the	the	DET
iajs-2809	45	27	coefficients	coefficient	NOUN
iajs-2809	45	28	of	of	ADP
iajs-2809	45	29	examples	example	NOUN
iajs-2809	45	30	from[11],[12	from[11],[12	X
iajs-2809	45	31	]	]	PUNCT
iajs-2809	45	32	,	,	PUNCT
iajs-2809	45	33	and	and	CCONJ
iajs-2809	45	34	[	[	X
iajs-2809	45	35	13	13	NUM
iajs-2809	45	36	]	]	PUNCT
iajs-2809	45	37	into	into	ADP
iajs-2809	45	38	rics	ric	NOUN
iajs-2809	45	39	.	.	PUNCT
iajs-2809	46	1	in	in	ADP
iajs-2809	46	2	this	this	DET
iajs-2809	46	3	paper	paper	NOUN
iajs-2809	46	4	,	,	PUNCT
iajs-2809	46	5	the	the	DET
iajs-2809	46	6	concentration	concentration	NOUN
iajs-2809	46	7	of	of	ADP
iajs-2809	46	8	our	our	PRON
iajs-2809	46	9	argumentation	argumentation	NOUN
iajs-2809	46	10	and	and	CCONJ
iajs-2809	46	11	investigation	investigation	NOUN
iajs-2809	46	12	promotes	promote	VERB
iajs-2809	46	13	an	an	DET
iajs-2809	46	14	approach	approach	NOUN
iajs-2809	46	15	to	to	ADP
iajs-2809	46	16	finding	find	VERB
iajs-2809	46	17	surely	surely	ADV
iajs-2809	46	18	and	and	CCONJ
iajs-2809	46	19	possibly	possibly	ADV
iajs-2809	46	20	optimal	optimal	ADJ
iajs-2809	46	21	solutions	solution	NOUN
iajs-2809	46	22	to	to	ADP
iajs-2809	46	23	a	a	DET
iajs-2809	46	24	linear	linear	ADJ
iajs-2809	46	25	fractional	fractional	ADJ
iajs-2809	46	26	programming	programming	NOUN
iajs-2809	46	27	problem	problem	NOUN
iajs-2809	46	28	with	with	ADP
iajs-2809	46	29	rough	rough	ADJ
iajs-2809	46	30	interval	interval	NOUN
iajs-2809	46	31	coefficients	coefficient	NOUN
iajs-2809	46	32	in	in	ADP
iajs-2809	46	33	the	the	DET
iajs-2809	46	34	objective	objective	ADJ
iajs-2809	46	35	function	function	NOUN
iajs-2809	46	36	.	.	PUNCT
iajs-2809	47	1	the	the	DET
iajs-2809	47	2	remainder	remainder	NOUN
iajs-2809	47	3	of	of	ADP
iajs-2809	47	4	the	the	DET
iajs-2809	47	5	paper	paper	NOUN
iajs-2809	47	6	is	be	AUX
iajs-2809	47	7	laid	lay	VERB
iajs-2809	47	8	out	out	ADP
iajs-2809	47	9	as	as	SCONJ
iajs-2809	47	10	follows	follow	VERB
iajs-2809	47	11	.	.	PUNCT
iajs-2809	48	1	section	section	NOUN
iajs-2809	48	2	2	2	NUM
iajs-2809	48	3	presents	present	VERB
iajs-2809	48	4	some	some	DET
iajs-2809	48	5	fundamental	fundamental	ADJ
iajs-2809	48	6	understanding	understanding	NOUN
iajs-2809	48	7	of	of	ADP
iajs-2809	48	8	rough	rough	ADJ
iajs-2809	48	9	intervals	interval	NOUN
iajs-2809	48	10	.	.	PUNCT
iajs-2809	49	1	the	the	DET
iajs-2809	49	2	formulation	formulation	NOUN
iajs-2809	49	3	and	and	CCONJ
iajs-2809	49	4	derivation	derivation	NOUN
iajs-2809	49	5	of	of	ADP
iajs-2809	49	6	lfpp	lfpp	NOUN
iajs-2809	49	7	with	with	ADP
iajs-2809	49	8	rics	ric	NOUN
iajs-2809	49	9	are	be	AUX
iajs-2809	49	10	described	describe	VERB
iajs-2809	49	11	in	in	ADP
iajs-2809	49	12	section	section	NOUN
iajs-2809	49	13	3	3	NUM
iajs-2809	49	14	.	.	PUNCT
iajs-2809	50	1	two	two	NUM
iajs-2809	50	2	numerical	numerical	ADJ
iajs-2809	50	3	examples	example	NOUN
iajs-2809	50	4	are	be	AUX
iajs-2809	50	5	put	put	VERB
iajs-2809	50	6	to	to	ADP
iajs-2809	50	7	the	the	DET
iajs-2809	50	8	test	test	NOUN
iajs-2809	50	9	in	in	ADP
iajs-2809	50	10	section	section	NOUN
iajs-2809	50	11	4	4	NUM
iajs-2809	50	12	.	.	PUNCT
iajs-2809	51	1	finally	finally	ADV
iajs-2809	51	2	,	,	PUNCT
iajs-2809	51	3	in	in	ADP
iajs-2809	51	4	section	section	NOUN
iajs-2809	51	5	5	5	NUM
iajs-2809	51	6	,	,	PUNCT
iajs-2809	51	7	there	there	PRON
iajs-2809	51	8	are	be	VERB
iajs-2809	51	9	some	some	DET
iajs-2809	51	10	concluding	conclude	VERB
iajs-2809	51	11	observations	observation	NOUN
iajs-2809	51	12	.	.	PUNCT
iajs-2809	52	1	ibn	ibn	PROPN
iajs-2809	52	2	al	al	PROPN
iajs-2809	52	3	-	-	PUNCT
iajs-2809	52	4	haitham	haitham	PROPN
iajs-2809	52	5	jour	jour	X
iajs-2809	52	6	.	.	PROPN
iajs-2809	52	7	for	for	ADP
iajs-2809	52	8	pure	pure	ADJ
iajs-2809	52	9	&	&	CCONJ
iajs-2809	52	10	appl	appl	PROPN
iajs-2809	52	11	.	.	PUNCT
iajs-2809	53	1	sci	sci	PROPN
iajs-2809	53	2	.	.	PROPN
iajs-2809	54	1	53	53	NUM
iajs-2809	54	2	(	(	PUNCT
iajs-2809	54	3	2)2022	2)2022	VERB
iajs-2809	54	4	72	72	NUM
iajs-2809	54	5	2	2	NUM
iajs-2809	54	6	.	.	PUNCT
iajs-2809	55	1	definitions	definition	NOUN
iajs-2809	55	2	definition	definition	NOUN
iajs-2809	55	3	:	:	PUNCT
iajs-2809	55	4	2.1	2.1	NUM
iajs-2809	55	5	.	.	PUNCT
iajs-2809	56	1	let	let	VERB
iajs-2809	56	2	𝐸	𝐸	PROPN
iajs-2809	56	3	denotes	denote	NOUN
iajs-2809	56	4	a	a	DET
iajs-2809	56	5	compact	compact	ADJ
iajs-2809	56	6	set	set	NOUN
iajs-2809	56	7	of	of	ADP
iajs-2809	56	8	real	real	ADJ
iajs-2809	56	9	numbers	number	NOUN
iajs-2809	56	10	.	.	PUNCT
iajs-2809	57	1	a	a	DET
iajs-2809	57	2	rough	rough	ADJ
iajs-2809	57	3	interval	interval	NOUN
iajs-2809	57	4	𝐴	𝐴	PROPN
iajs-2809	57	5	⊆	⊆	NUM
iajs-2809	57	6	𝐸	𝐸	PROPN
iajs-2809	57	7	is	be	AUX
iajs-2809	57	8	defined	define	VERB
iajs-2809	57	9	𝐴	𝐴	NOUN
iajs-2809	57	10	=	=	SYM
iajs-2809	57	11	(	(	PUNCT
iajs-2809	57	12	𝐴∗	𝐴∗	PROPN
iajs-2809	57	13	,	,	PUNCT
iajs-2809	57	14	𝐴∗	𝐴∗	NUM
iajs-2809	57	15	)	)	PUNCT
iajs-2809	57	16	,	,	PUNCT
iajs-2809	57	17	where	where	SCONJ
iajs-2809	57	18	𝐴∗	𝐴∗	NOUN
iajs-2809	57	19	⊆	⊆	NUM
iajs-2809	57	20	𝐴∗	𝐴∗	NUM
iajs-2809	57	21	,	,	PUNCT
iajs-2809	57	22	𝐴∗	𝐴∗	NUM
iajs-2809	57	23	and	and	CCONJ
iajs-2809	57	24	𝐴∗are	𝐴∗are	VERB
iajs-2809	57	25	conventional	conventional	ADJ
iajs-2809	57	26	lower	low	ADJ
iajs-2809	57	27	and	and	CCONJ
iajs-2809	57	28	upper	upper	ADJ
iajs-2809	57	29	approximation	approximation	NOUN
iajs-2809	57	30	intervals	interval	NOUN
iajs-2809	57	31	of𝐴	of𝐴	PROPN
iajs-2809	57	32	,	,	PUNCT
iajs-2809	57	33	respectively	respectively	ADV
iajs-2809	57	34	.	.	PUNCT
iajs-2809	58	1	𝐴∗	𝐴∗	X
iajs-2809	59	1	=	=	PUNCT
iajs-2809	60	1	[	[	X
iajs-2809	60	2	𝑎∗	𝑎∗	PROPN
iajs-2809	60	3	𝑙	𝑙	X
iajs-2809	60	4	,	,	PUNCT
iajs-2809	60	5	𝑎∗	𝑎∗	PROPN
iajs-2809	60	6	𝑢]and𝐴∗	𝑢]and𝐴∗	NOUN
iajs-2809	60	7	=	=	PUNCT
iajs-2809	61	1	[	[	X
iajs-2809	61	2	𝑎∗𝑙	𝑎∗𝑙	NUM
iajs-2809	61	3	,	,	PUNCT
iajs-2809	61	4	𝑎∗𝑢	𝑎∗𝑢	NUM
iajs-2809	61	5	]	]	PUNCT
iajs-2809	61	6	.	.	PUNCT
iajs-2809	62	1	where	where	SCONJ
iajs-2809	62	2	𝑎∗	𝑎∗	PROPN
iajs-2809	62	3	𝑙	𝑙	X
iajs-2809	62	4	,	,	PUNCT
iajs-2809	62	5	𝑎∗	𝑎∗	PROPN
iajs-2809	62	6	𝑢	𝑢	NOUN
iajs-2809	62	7	,	,	PUNCT
iajs-2809	62	8	𝑎∗𝑙	𝑎∗𝑙	PUNCT
iajs-2809	62	9	and	and	CCONJ
iajs-2809	62	10	𝑎∗𝑢	𝑎∗𝑢	NOUN
iajs-2809	62	11	are	be	AUX
iajs-2809	62	12	real	real	ADJ
iajs-2809	62	13	numbers	number	NOUN
iajs-2809	62	14	.	.	PUNCT
iajs-2809	63	1	note	note	VERB
iajs-2809	63	2	that	that	SCONJ
iajs-2809	63	3	the	the	DET
iajs-2809	63	4	intervals	interval	NOUN
iajs-2809	63	5	𝐴∗	𝐴∗	NOUN
iajs-2809	63	6	and	and	CCONJ
iajs-2809	63	7	𝐴∗	𝐴∗	NUM
iajs-2809	63	8	are	be	AUX
iajs-2809	63	9	not	not	PART
iajs-2809	63	10	the	the	DET
iajs-2809	63	11	complement	complement	NOUN
iajs-2809	63	12	each	each	DET
iajs-2809	63	13	other	other	ADJ
iajs-2809	63	14	.	.	PUNCT
iajs-2809	64	1	the	the	DET
iajs-2809	64	2	arithmetic	arithmetic	ADJ
iajs-2809	64	3	operations	operation	NOUN
iajs-2809	64	4	on	on	ADP
iajs-2809	64	5	rough	rough	ADJ
iajs-2809	64	6	intervals	interval	NOUN
iajs-2809	64	7	are	be	AUX
iajs-2809	64	8	based	base	VERB
iajs-2809	64	9	on	on	ADP
iajs-2809	64	10	interval	interval	NOUN
iajs-2809	64	11	arithmetic	arithmetic	ADJ
iajs-2809	64	12	.	.	PUNCT
iajs-2809	65	1	some	some	PRON
iajs-2809	65	2	of	of	ADP
iajs-2809	65	3	these	these	DET
iajs-2809	65	4	mathematical	mathematical	ADJ
iajs-2809	65	5	procedures	procedure	NOUN
iajs-2809	65	6	will	will	AUX
iajs-2809	65	7	be	be	AUX
iajs-2809	65	8	explained	explain	VERB
iajs-2809	65	9	as	as	SCONJ
iajs-2809	65	10	follows	follow	VERB
iajs-2809	65	11	:	:	PUNCT
iajs-2809	65	12	let	let	VERB
iajs-2809	65	13	𝐴	𝐴	PROPN
iajs-2809	65	14	=	=	SYM
iajs-2809	65	15	(	(	PUNCT
iajs-2809	65	16	[	[	X
iajs-2809	65	17	𝑎∗	𝑎∗	PROPN
iajs-2809	65	18	𝑙	𝑙	X
iajs-2809	65	19	,	,	PUNCT
iajs-2809	65	20	𝑎∗	𝑎∗	PROPN
iajs-2809	65	21	𝑢	𝑢	X
iajs-2809	65	22	]	]	X
iajs-2809	65	23	,	,	PUNCT
iajs-2809	65	24	[	[	PUNCT
iajs-2809	65	25	𝑎∗𝑙	𝑎∗𝑙	X
iajs-2809	65	26	,	,	PUNCT
iajs-2809	65	27	𝑎∗𝑢	𝑎∗𝑢	X
iajs-2809	65	28	]	]	PUNCT
iajs-2809	65	29	)	)	PUNCT
iajs-2809	65	30	and	and	CCONJ
iajs-2809	65	31	𝐵	𝐵	NOUN
iajs-2809	65	32	=	=	SYM
iajs-2809	65	33	(	(	PUNCT
iajs-2809	65	34	[	[	X
iajs-2809	65	35	𝑏∗	𝑏∗	PROPN
iajs-2809	65	36	𝑙	𝑙	X
iajs-2809	65	37	,	,	PUNCT
iajs-2809	65	38	𝑏∗	𝑏∗	PROPN
iajs-2809	65	39	𝑢	𝑢	X
iajs-2809	65	40	]	]	X
iajs-2809	65	41	,	,	PUNCT
iajs-2809	65	42	[	[	PUNCT
iajs-2809	65	43	𝑏∗𝑙	𝑏∗𝑙	X
iajs-2809	65	44	,	,	PUNCT
iajs-2809	65	45	𝑏∗𝑢	𝑏∗𝑢	X
iajs-2809	65	46	]	]	PUNCT
iajs-2809	65	47	)	)	PUNCT
iajs-2809	65	48	be	be	AUX
iajs-2809	65	49	two	two	NUM
iajs-2809	65	50	rough	rough	ADJ
iajs-2809	65	51	intervals	interval	NOUN
iajs-2809	65	52	.	.	PUNCT
iajs-2809	66	1	then	then	ADV
iajs-2809	66	2	we	we	PRON
iajs-2809	66	3	have	have	VERB
iajs-2809	66	4	:	:	PUNCT
iajs-2809	66	5	𝐴	𝐴	PROPN
iajs-2809	66	6	+	+	CCONJ
iajs-2809	66	7	𝐵	𝐵	NOUN
iajs-2809	66	8	=	=	SYM
iajs-2809	66	9	(	(	PUNCT
iajs-2809	66	10	[	[	X
iajs-2809	66	11	𝑎∗	𝑎∗	PROPN
iajs-2809	66	12	𝑙	𝑙	NOUN
iajs-2809	66	13	+	+	NUM
iajs-2809	66	14	𝑏∗	𝑏∗	PROPN
iajs-2809	67	1	𝑙	𝑙	X
iajs-2809	67	2	,	,	PUNCT
iajs-2809	67	3	𝑎∗	𝑎∗	PROPN
iajs-2809	67	4	𝑢	𝑢	PROPN
iajs-2809	67	5	+	+	CCONJ
iajs-2809	67	6	𝑏∗	𝑏∗	PROPN
iajs-2809	67	7	𝑢	𝑢	X
iajs-2809	67	8	]	]	X
iajs-2809	67	9	,	,	PUNCT
iajs-2809	67	10	[	[	PUNCT
iajs-2809	67	11	𝑎∗𝑙	𝑎∗𝑙	X
iajs-2809	67	12	+	+	NUM
iajs-2809	67	13	𝑏∗𝑙	𝑏∗𝑙	NOUN
iajs-2809	67	14	,	,	PUNCT
iajs-2809	67	15	𝑎∗𝑢	𝑎∗𝑢	X
iajs-2809	67	16	+	+	CCONJ
iajs-2809	67	17	𝑏∗𝑢	𝑏∗𝑢	X
iajs-2809	67	18	]	]	PUNCT
iajs-2809	67	19	)	)	PUNCT
iajs-2809	67	20	𝐴	𝐴	PROPN
iajs-2809	67	21	−	−	PROPN
iajs-2809	67	22	𝐵	𝐵	NOUN
iajs-2809	67	23	=	=	SYM
iajs-2809	67	24	(	(	PUNCT
iajs-2809	68	1	[	[	X
iajs-2809	68	2	𝑎∗	𝑎∗	PROPN
iajs-2809	68	3	𝑙	𝑙	X
iajs-2809	68	4	−	−	PROPN
iajs-2809	68	5	𝑏∗	𝑏∗	PROPN
iajs-2809	68	6	𝑢	𝑢	PROPN
iajs-2809	68	7	,	,	PUNCT
iajs-2809	68	8	𝑎∗	𝑎∗	PROPN
iajs-2809	68	9	𝑢	𝑢	ADV
iajs-2809	68	10	−	−	PROPN
iajs-2809	68	11	𝑏∗	𝑏∗	PROPN
iajs-2809	68	12	𝑙	𝑙	X
iajs-2809	68	13	]	]	X
iajs-2809	68	14	,	,	PUNCT
iajs-2809	68	15	[	[	PUNCT
iajs-2809	68	16	𝑎∗𝑙	𝑎∗𝑙	PUNCT
iajs-2809	68	17	−	−	X
iajs-2809	68	18	𝑏∗𝑢	𝑏∗𝑢	NOUN
iajs-2809	68	19	,	,	PUNCT
iajs-2809	68	20	𝑎∗𝑢	𝑎∗𝑢	NUM
iajs-2809	68	21	−	−	PROPN
iajs-2809	68	22	𝑏∗𝑙	𝑏∗𝑙	NOUN
iajs-2809	68	23	]	]	PUNCT
iajs-2809	68	24	)	)	PUNCT
iajs-2809	69	1	𝐴	𝐴	PROPN
iajs-2809	69	2	×	×	NOUN
iajs-2809	69	3	𝐵	𝐵	NOUN
iajs-2809	69	4	=	=	SYM
iajs-2809	69	5	(	(	PUNCT
iajs-2809	69	6	[	[	X
iajs-2809	69	7	𝑎∗	𝑎∗	PROPN
iajs-2809	69	8	𝑙	𝑙	X
iajs-2809	69	9	×	×	NOUN
iajs-2809	69	10	𝑏∗	𝑏∗	PROPN
iajs-2809	69	11	𝑙	𝑙	X
iajs-2809	69	12	,	,	PUNCT
iajs-2809	69	13	𝑎∗	𝑎∗	PROPN
iajs-2809	69	14	𝑢	𝑢	X
iajs-2809	69	15	×	×	NOUN
iajs-2809	69	16	𝑏∗	𝑏∗	PROPN
iajs-2809	69	17	𝑢	𝑢	X
iajs-2809	69	18	]	]	X
iajs-2809	69	19	,	,	PUNCT
iajs-2809	69	20	[	[	PUNCT
iajs-2809	69	21	𝑎∗𝑙	𝑎∗𝑙	NUM
iajs-2809	69	22	×	×	PROPN
iajs-2809	69	23	𝑏∗𝑙	𝑏∗𝑙	PROPN
iajs-2809	69	24	,	,	PUNCT
iajs-2809	69	25	𝑎∗𝑢	𝑎∗𝑢	NUM
iajs-2809	69	26	×	×	PROPN
iajs-2809	69	27	𝑏∗𝑢	𝑏∗𝑢	X
iajs-2809	69	28	]	]	PUNCT
iajs-2809	69	29	)	)	PUNCT
iajs-2809	69	30	𝐴	𝐴	PROPN
iajs-2809	69	31	÷	÷	NUM
iajs-2809	69	32	𝐵	𝐵	NOUN
iajs-2809	69	33	=	=	SYM
iajs-2809	69	34	(	(	PUNCT
iajs-2809	69	35	[	[	X
iajs-2809	69	36	𝑎∗	𝑎∗	PROPN
iajs-2809	69	37	𝑙	𝑙	X
iajs-2809	69	38	÷	÷	X
iajs-2809	69	39	𝑏∗	𝑏∗	PROPN
iajs-2809	69	40	𝑢	𝑢	PROPN
iajs-2809	69	41	,	,	PUNCT
iajs-2809	69	42	𝑎∗	𝑎∗	PROPN
iajs-2809	69	43	𝑢	𝑢	X
iajs-2809	69	44	÷	÷	X
iajs-2809	69	45	𝑏∗	𝑏∗	PROPN
iajs-2809	69	46	𝑙	𝑙	X
iajs-2809	69	47	]	]	X
iajs-2809	69	48	,	,	PUNCT
iajs-2809	69	49	[	[	PUNCT
iajs-2809	69	50	𝑎∗𝑙	𝑎∗𝑙	NUM
iajs-2809	69	51	÷	÷	X
iajs-2809	69	52	𝑏∗𝑢	𝑏∗𝑢	PROPN
iajs-2809	69	53	,	,	PUNCT
iajs-2809	69	54	𝑎∗𝑢	𝑎∗𝑢	NUM
iajs-2809	69	55	÷	÷	SYM
iajs-2809	69	56	𝑏∗𝑙	𝑏∗𝑙	NOUN
iajs-2809	69	57	]	]	PUNCT
iajs-2809	69	58	)	)	PUNCT
iajs-2809	69	59	definition	definition	NOUN
iajs-2809	69	60	:	:	PUNCT
iajs-2809	69	61	2.2	2.2	NUM
iajs-2809	69	62	.	.	PUNCT
iajs-2809	69	63	function	function	VERB
iajs-2809	69	64	𝑓	𝑓	NOUN
iajs-2809	69	65	:	:	PUNCT
iajs-2809	69	66	𝑅𝑛	𝑅𝑛	PROPN
iajs-2809	69	67	→	→	SYM
iajs-2809	69	68	𝐴	𝐴	PROPN
iajs-2809	69	69	is	be	AUX
iajs-2809	69	70	called	call	VERB
iajs-2809	69	71	a	a	DET
iajs-2809	69	72	rough	rough	ADJ
iajs-2809	69	73	interval	interval	NOUN
iajs-2809	69	74	function	function	NOUN
iajs-2809	69	75	with	with	ADP
iajs-2809	69	76	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2809	69	77	)	)	PUNCT
iajs-2809	69	78	=	=	SYM
iajs-2809	69	79	(	(	PUNCT
iajs-2809	69	80	𝑓∗(𝑥	𝑓∗(𝑥	PROPN
iajs-2809	69	81	)	)	PUNCT
iajs-2809	69	82	,	,	PUNCT
iajs-2809	69	83	𝑓∗(𝑥	𝑓∗(𝑥	PROPN
iajs-2809	69	84	)	)	PUNCT
iajs-2809	69	85	)	)	PUNCT
iajs-2809	69	86	,	,	PUNCT
iajs-2809	69	87	where	where	SCONJ
iajs-2809	69	88	for	for	ADP
iajs-2809	69	89	every𝑥	every𝑥	PROPN
iajs-2809	69	90	∈	∈	PROPN
iajs-2809	69	91	𝑅𝑛	𝑅𝑛	PROPN
iajs-2809	69	92	,	,	PUNCT
iajs-2809	69	93	𝑓∗(𝑥	𝑓∗(𝑥	PROPN
iajs-2809	69	94	)	)	PUNCT
iajs-2809	69	95	,	,	PUNCT
iajs-2809	69	96	𝑓∗(𝑥	𝑓∗(𝑥	PROPN
iajs-2809	69	97	)	)	PUNCT
iajs-2809	69	98	are	be	AUX
iajs-2809	69	99	lower	low	ADJ
iajs-2809	69	100	and	and	CCONJ
iajs-2809	69	101	upper	upper	ADJ
iajs-2809	69	102	approximation	approximation	NOUN
iajs-2809	69	103	interval	interval	NOUN
iajs-2809	69	104	valued	value	VERB
iajs-2809	69	105	functions	function	NOUN
iajs-2809	69	106	.	.	PUNCT
iajs-2809	70	1	definition	definition	NOUN
iajs-2809	70	2	:	:	PUNCT
iajs-2809	70	3	2.3	2.3	NUM
iajs-2809	70	4	.	.	PUNCT
iajs-2809	71	1	a	a	DET
iajs-2809	71	2	feasible	feasible	ADJ
iajs-2809	71	3	point	point	NOUN
iajs-2809	71	4	𝑥∗	𝑥∗	PROPN
iajs-2809	71	5	∈	∈	PROPN
iajs-2809	71	6	𝑆	𝑆	PROPN
iajs-2809	71	7	is	be	AUX
iajs-2809	71	8	said	say	VERB
iajs-2809	71	9	to	to	PART
iajs-2809	71	10	be	be	AUX
iajs-2809	71	11	an	an	DET
iajs-2809	71	12	optimal	optimal	ADJ
iajs-2809	71	13	solution	solution	NOUN
iajs-2809	71	14	of	of	ADP
iajs-2809	71	15	optimization	optimization	NOUN
iajs-2809	71	16	problem	problem	NOUN
iajs-2809	71	17	lfpp	lfpp	VERB
iajs-2809	71	18	with	with	ADP
iajs-2809	71	19	rics	ric	NOUN
iajs-2809	71	20	,	,	PUNCT
iajs-2809	71	21	if	if	SCONJ
iajs-2809	71	22	there	there	PRON
iajs-2809	71	23	does	do	AUX
iajs-2809	71	24	not	not	PART
iajs-2809	71	25	exist	exist	VERB
iajs-2809	71	26	𝑥	𝑥	DET
iajs-2809	71	27	∈	∈	PROPN
iajs-2809	71	28	𝑆	𝑆	PROPN
iajs-2809	71	29	,	,	PUNCT
iajs-2809	71	30	such	such	ADJ
iajs-2809	71	31	that𝑓(𝑥∗	that𝑓(𝑥∗	NOUN
iajs-2809	71	32	)	)	PUNCT
iajs-2809	71	33	≤	≤	NUM
iajs-2809	71	34	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2809	71	35	)	)	PUNCT
iajs-2809	71	36	.	.	PUNCT
iajs-2809	72	1	where	where	SCONJ
iajs-2809	72	2	𝑆	𝑆	PROPN
iajs-2809	72	3	=	=	SYM
iajs-2809	72	4	{	{	PUNCT
iajs-2809	72	5	𝑥	𝑥	PUNCT
iajs-2809	72	6	∈	∈	PROPN
iajs-2809	72	7	𝑅𝑛	𝑅𝑛	NOUN
iajs-2809	72	8	:	:	PUNCT
iajs-2809	72	9	𝐴𝑥	𝐴𝑥	NOUN
iajs-2809	72	10	≤	≤	NOUN
iajs-2809	72	11	𝑏	𝑏	NOUN
iajs-2809	72	12	,	,	PUNCT
iajs-2809	72	13	𝑥	𝑥	PRON
iajs-2809	72	14	≥	≥	NOUN
iajs-2809	72	15	0	0	NUM
iajs-2809	72	16	}	}	SYM
iajs-2809	72	17	3	3	NUM
iajs-2809	72	18	.	.	X
iajs-2809	72	19	formulation	formulation	NOUN
iajs-2809	72	20	of	of	ADP
iajs-2809	72	21	the	the	DET
iajs-2809	72	22	problem	problem	NOUN
iajs-2809	72	23	the	the	DET
iajs-2809	72	24	generally	generally	ADV
iajs-2809	72	25	extended	extended	ADJ
iajs-2809	72	26	form	form	NOUN
iajs-2809	72	27	of	of	ADP
iajs-2809	72	28	a	a	DET
iajs-2809	72	29	lfpp	lfpp	NOUN
iajs-2809	72	30	with	with	ADP
iajs-2809	72	31	rics	ric	NOUN
iajs-2809	72	32	in	in	ADP
iajs-2809	72	33	the	the	DET
iajs-2809	72	34	objective	objective	ADJ
iajs-2809	72	35	function	function	NOUN
iajs-2809	72	36	is	be	AUX
iajs-2809	72	37	as	as	SCONJ
iajs-2809	72	38	follows	follow	VERB
iajs-2809	72	39	:	:	PUNCT
iajs-2809	72	40	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	72	41	𝑍	𝑍	PROPN
iajs-2809	72	42	=	=	SYM
iajs-2809	72	43	∑	∑	PUNCT
iajs-2809	72	44	{	{	PUNCT
iajs-2809	72	45	(	(	PUNCT
iajs-2809	72	46	𝐴∗𝑖	𝐴∗𝑖	PROPN
iajs-2809	72	47	,	,	PUNCT
iajs-2809	72	48	𝐴𝑖	𝐴𝑖	PROPN
iajs-2809	72	49	∗)𝑥𝑖	∗)𝑥𝑖	PUNCT
iajs-2809	73	1	+	+	PUNCT
iajs-2809	73	2	𝑘	𝑘	PRON
iajs-2809	73	3	𝑖=1	𝑖=1	PROPN
iajs-2809	73	4	(	(	PUNCT
iajs-2809	73	5	𝐴∗𝑘+1	𝐴∗𝑘+1	NOUN
iajs-2809	73	6	,	,	PUNCT
iajs-2809	73	7	𝐴𝑘+1	𝐴𝑘+1	PART
iajs-2809	73	8	∗	∗	NOUN
iajs-2809	73	9	)	)	PUNCT
iajs-2809	73	10	}	}	PUNCT
iajs-2809	73	11	∑	∑	PUNCT
iajs-2809	73	12	{	{	PUNCT
iajs-2809	73	13	(	(	PUNCT
iajs-2809	73	14	𝐵∗𝑖	𝐵∗𝑖	PROPN
iajs-2809	73	15	,	,	PUNCT
iajs-2809	73	16	𝐵𝑖	𝐵𝑖	PROPN
iajs-2809	73	17	∗)𝑥𝑖	∗)𝑥𝑖	PUNCT
iajs-2809	74	1	+	+	PROPN
iajs-2809	74	2	𝑘	𝑘	PRON
iajs-2809	74	3	𝑖=1	𝑖=1	PROPN
iajs-2809	74	4	(	(	PUNCT
iajs-2809	74	5	𝐵∗𝑘+1	𝐵∗𝑘+1	INTJ
iajs-2809	74	6	,	,	PUNCT
iajs-2809	74	7	𝐵𝑘+1	𝐵𝑘+1	NOUN
iajs-2809	74	8	∗	∗	NOUN
iajs-2809	74	9	)	)	PUNCT
iajs-2809	74	10	}	}	PUNCT
iajs-2809	74	11	subject	subject	ADJ
iajs-2809	74	12	to	to	ADP
iajs-2809	74	13	:	:	PUNCT
iajs-2809	74	14	(	(	PUNCT
iajs-2809	74	15	1	1	X
iajs-2809	74	16	)	)	PUNCT
iajs-2809	74	17	∑	∑	PUNCT
iajs-2809	74	18	𝐴𝑖𝑥𝑖	𝐴𝑖𝑥𝑖	PROPN
iajs-2809	74	19	≤	≤	NOUN
iajs-2809	75	1	𝑏	𝑏	DET
iajs-2809	75	2	𝑘	𝑘	X
iajs-2809	75	3	𝑖=1	𝑖=1	PUNCT
iajs-2809	75	4	for	for	ADP
iajs-2809	75	5	each	each	DET
iajs-2809	75	6	𝑥𝑖	𝑥𝑖	PROPN
iajs-2809	75	7	≥	≥	NOUN
iajs-2809	75	8	0	0	NUM
iajs-2809	75	9	,	,	PUNCT
iajs-2809	75	10	𝑖	𝑖	NOUN
iajs-2809	75	11	=	=	SYM
iajs-2809	75	12	1,2	1,2	NUM
iajs-2809	75	13	,	,	PUNCT
iajs-2809	75	14	…	…	PUNCT
iajs-2809	75	15	,	,	PUNCT
iajs-2809	75	16	𝑘.	𝑘.	NOUN
iajs-2809	75	17	where	where	SCONJ
iajs-2809	75	18	𝐴𝑖	𝐴𝑖	PROPN
iajs-2809	75	19	,	,	PUNCT
iajs-2809	75	20	𝑖	𝑖	NOUN
iajs-2809	75	21	=	=	SYM
iajs-2809	75	22	1,2	1,2	NUM
iajs-2809	75	23	,	,	PUNCT
iajs-2809	75	24	…	…	PUNCT
iajs-2809	75	25	,	,	PUNCT
iajs-2809	75	26	𝑘	𝑘	NOUN
iajs-2809	75	27	,	,	PUNCT
iajs-2809	75	28	and	and	CCONJ
iajs-2809	75	29	𝑏	𝑏	PROPN
iajs-2809	75	30	both	both	PRON
iajs-2809	75	31	are	be	AUX
iajs-2809	75	32	m	m	ADJ
iajs-2809	75	33	-	-	ADJ
iajs-2809	75	34	dimensional	dimensional	ADJ
iajs-2809	75	35	constant	constant	ADJ
iajs-2809	75	36	column	column	NOUN
iajs-2809	75	37	vectors	vector	NOUN
iajs-2809	75	38	.	.	PUNCT
iajs-2809	76	1	ibn	ibn	PROPN
iajs-2809	76	2	al	al	PROPN
iajs-2809	76	3	-	-	PUNCT
iajs-2809	76	4	haitham	haitham	PROPN
iajs-2809	76	5	jour	jour	X
iajs-2809	76	6	.	.	PROPN
iajs-2809	76	7	for	for	ADP
iajs-2809	76	8	pure	pure	ADJ
iajs-2809	76	9	&	&	CCONJ
iajs-2809	76	10	appl	appl	PROPN
iajs-2809	76	11	.	.	PUNCT
iajs-2809	77	1	sci	sci	PROPN
iajs-2809	77	2	.	.	PROPN
iajs-2809	78	1	53	53	NUM
iajs-2809	78	2	(	(	PUNCT
iajs-2809	78	3	2)2022	2)2022	NUM
iajs-2809	78	4	73	73	NUM
iajs-2809	78	5	assume	assume	VERB
iajs-2809	78	6	that∑	that∑	PROPN
iajs-2809	78	7	(	(	PUNCT
iajs-2809	78	8	𝐵∗𝑖	𝐵∗𝑖	PROPN
iajs-2809	78	9	,	,	PUNCT
iajs-2809	78	10	𝐵𝑖	𝐵𝑖	PROPN
iajs-2809	78	11	∗)𝑥𝑖	∗)𝑥𝑖	PUNCT
iajs-2809	79	1	+	+	PROPN
iajs-2809	79	2	𝑘	𝑘	PRON
iajs-2809	79	3	𝑖=1	𝑖=1	PROPN
iajs-2809	79	4	(	(	PUNCT
iajs-2809	79	5	𝐵∗𝑘+1	𝐵∗𝑘+1	INTJ
iajs-2809	79	6	,	,	PUNCT
iajs-2809	79	7	𝐵𝑘+1	𝐵𝑘+1	NOUN
iajs-2809	79	8	∗	∗	NOUN
iajs-2809	79	9	)	)	PUNCT
iajs-2809	79	10	>	>	X
iajs-2809	79	11	0	0	NUM
iajs-2809	79	12	,	,	PUNCT
iajs-2809	79	13	for	for	ADP
iajs-2809	79	14	all𝑥𝑖	all𝑥𝑖	PROPN
iajs-2809	79	15	∈	∈	PROPN
iajs-2809	79	16	𝑆	𝑆	PROPN
iajs-2809	79	17	,	,	PUNCT
iajs-2809	79	18	𝑖	𝑖	NOUN
iajs-2809	79	19	=	=	SYM
iajs-2809	79	20	1,2	1,2	NUM
iajs-2809	79	21	,	,	PUNCT
iajs-2809	79	22	…	…	PUNCT
iajs-2809	79	23	,	,	PUNCT
iajs-2809	79	24	𝑘	𝑘	X
iajs-2809	79	25	,	,	PUNCT
iajs-2809	79	26	where	where	SCONJ
iajs-2809	79	27	𝑆	𝑆	PROPN
iajs-2809	79	28	is	be	AUX
iajs-2809	79	29	the	the	DET
iajs-2809	79	30	compact	compact	ADJ
iajs-2809	79	31	feasible	feasible	ADJ
iajs-2809	79	32	region	region	NOUN
iajs-2809	79	33	of	of	ADP
iajs-2809	79	34	the	the	DET
iajs-2809	79	35	problem	problem	NOUN
iajs-2809	79	36	(	(	PUNCT
iajs-2809	79	37	1	1	NUM
iajs-2809	79	38	)	)	PUNCT
iajs-2809	79	39	.	.	PUNCT
iajs-2809	80	1	in	in	ADP
iajs-2809	80	2	the	the	DET
iajs-2809	80	3	deriving	deriving	NOUN
iajs-2809	80	4	theory	theory	NOUN
iajs-2809	80	5	for	for	ADP
iajs-2809	80	6	solving	solve	VERB
iajs-2809	80	7	the	the	DET
iajs-2809	80	8	problem	problem	NOUN
iajs-2809	80	9	(	(	PUNCT
iajs-2809	80	10	1	1	NUM
iajs-2809	80	11	)	)	PUNCT
iajs-2809	80	12	,	,	PUNCT
iajs-2809	80	13	we	we	PRON
iajs-2809	80	14	introduce	introduce	VERB
iajs-2809	80	15	the	the	DET
iajs-2809	80	16	variable	variable	NOUN
iajs-2809	80	17	,	,	PUNCT
iajs-2809	80	18	t=	t=	NOUN
iajs-2809	80	19	1	1	NUM
iajs-2809	80	20	∑	∑	PROPN
iajs-2809	80	21	(	(	PUNCT
iajs-2809	80	22	b∗i	b∗i	ADJ
iajs-2809	80	23	,	,	PUNCT
iajs-2809	80	24	bi	bi	NOUN
iajs-2809	80	25	∗)xi+k	∗)xi+k	PROPN
iajs-2809	80	26	i=1	i=1	PROPN
iajs-2809	80	27	(	(	PUNCT
iajs-2809	80	28	b∗k+1,bk+1	b∗k+1,bk+1	ADJ
iajs-2809	80	29	∗	∗	NOUN
iajs-2809	80	30	)	)	PUNCT
iajs-2809	80	31	,	,	PUNCT
iajs-2809	80	32	to	to	PART
iajs-2809	80	33	transform	transform	VERB
iajs-2809	80	34	lfp	lfp	PROPN
iajs-2809	80	35	problem	problem	NOUN
iajs-2809	80	36	with	with	ADP
iajs-2809	80	37	rics	ric	NOUN
iajs-2809	80	38	in	in	ADP
iajs-2809	80	39	the	the	DET
iajs-2809	80	40	objective	objective	ADJ
iajs-2809	80	41	function	function	NOUN
iajs-2809	80	42	into	into	ADP
iajs-2809	80	43	a	a	DET
iajs-2809	80	44	rough	rough	ADJ
iajs-2809	80	45	interval	interval	NOUN
iajs-2809	80	46	linear	linear	NOUN
iajs-2809	80	47	programming	programming	NOUN
iajs-2809	80	48	problem	problem	NOUN
iajs-2809	80	49	by	by	ADP
iajs-2809	80	50	the	the	DET
iajs-2809	80	51	method	method	NOUN
iajs-2809	80	52	of	of	ADP
iajs-2809	80	53	charnes	charne	NOUN
iajs-2809	80	54	and	and	CCONJ
iajs-2809	80	55	cooper[1	cooper[1	VERB
iajs-2809	80	56	]	]	PUNCT
iajs-2809	80	57	,	,	PUNCT
iajs-2809	80	58	and	and	CCONJ
iajs-2809	80	59	then	then	ADV
iajs-2809	80	60	we	we	PRON
iajs-2809	80	61	have	have	VERB
iajs-2809	80	62	:	:	PUNCT
iajs-2809	80	63	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	80	64	𝑍	𝑍	PROPN
iajs-2809	80	65	=	=	SYM
iajs-2809	80	66	∑	∑	PUNCT
iajs-2809	80	67	(	(	PUNCT
iajs-2809	80	68	𝐴∗𝑖	𝐴∗𝑖	PROPN
iajs-2809	80	69	,	,	PUNCT
iajs-2809	80	70	𝐴𝑖	𝐴𝑖	PROPN
iajs-2809	80	71	∗)𝑥𝑖𝑡	∗)𝑥𝑖𝑡	NOUN
iajs-2809	81	1	+	+	PROPN
iajs-2809	81	2	𝑘	𝑘	X
iajs-2809	81	3	𝑖=1	𝑖=1	PROPN
iajs-2809	81	4	(	(	PUNCT
iajs-2809	81	5	𝐴∗𝑘+1	𝐴∗𝑘+1	NOUN
iajs-2809	81	6	,	,	PUNCT
iajs-2809	81	7	𝐴𝑘+1	𝐴𝑘+1	PART
iajs-2809	81	8	∗	∗	NOUN
iajs-2809	81	9	)	)	PUNCT
iajs-2809	82	1	𝑡	𝑡	PROPN
iajs-2809	82	2	subject	subject	ADJ
iajs-2809	82	3	to	to	ADP
iajs-2809	82	4	:	:	PUNCT
iajs-2809	82	5	∑	∑	PROPN
iajs-2809	82	6	(	(	PUNCT
iajs-2809	82	7	𝐵∗𝑖	𝐵∗𝑖	PROPN
iajs-2809	82	8	,	,	PUNCT
iajs-2809	82	9	𝐵𝑖	𝐵𝑖	PROPN
iajs-2809	82	10	∗)𝑥𝑖𝑡	∗)𝑥𝑖𝑡	PROPN
iajs-2809	83	1	+	+	PROPN
iajs-2809	83	2	𝑘	𝑘	X
iajs-2809	83	3	𝑖=1	𝑖=1	PROPN
iajs-2809	83	4	(	(	PUNCT
iajs-2809	83	5	𝐵∗𝑘+1	𝐵∗𝑘+1	INTJ
iajs-2809	83	6	,	,	PUNCT
iajs-2809	83	7	𝐵𝑘+1	𝐵𝑘+1	NOUN
iajs-2809	83	8	∗	∗	NOUN
iajs-2809	83	9	)	)	PUNCT
iajs-2809	83	10	𝑡	𝑡	NOUN
iajs-2809	83	11	=	=	SYM
iajs-2809	83	12	1	1	NUM
iajs-2809	83	13	(	(	PUNCT
iajs-2809	83	14	2	2	NUM
iajs-2809	83	15	)	)	PUNCT
iajs-2809	83	16	∑	∑	PUNCT
iajs-2809	84	1	𝐴𝑖𝑥𝑖𝑡	𝐴𝑖𝑥𝑖𝑡	PROPN
iajs-2809	84	2	−	−	PROPN
iajs-2809	84	3	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	84	4	≤	≤	ADJ
iajs-2809	84	5	0𝑘	0𝑘	NOUN
iajs-2809	84	6	𝑖=1	𝑖=1	PUNCT
iajs-2809	85	1	𝑥𝑖	𝑥𝑖	X
iajs-2809	85	2	,	,	PUNCT
iajs-2809	85	3	𝑡	𝑡	X
iajs-2809	85	4	≥	≥	NOUN
iajs-2809	85	5	0	0	NUM
iajs-2809	85	6	,	,	PUNCT
iajs-2809	85	7	𝑖	𝑖	NOUN
iajs-2809	85	8	=	=	SYM
iajs-2809	85	9	1,2	1,2	NUM
iajs-2809	85	10	,	,	PUNCT
iajs-2809	85	11	…	…	PUNCT
iajs-2809	85	12	,	,	PUNCT
iajs-2809	85	13	𝑘.	𝑘.	NOUN
iajs-2809	85	14	by	by	ADP
iajs-2809	85	15	introducing	introduce	VERB
iajs-2809	85	16	variable	variable	ADJ
iajs-2809	85	17	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	85	18	=	=	SYM
iajs-2809	85	19	𝑥𝑖𝑡	𝑥𝑖𝑡	PROPN
iajs-2809	85	20	,	,	PUNCT
iajs-2809	85	21	𝑖	𝑖	X
iajs-2809	85	22	=	=	SYM
iajs-2809	85	23	1,2	1,2	NUM
iajs-2809	85	24	,	,	PUNCT
iajs-2809	85	25	…	…	PUNCT
iajs-2809	85	26	,	,	PUNCT
iajs-2809	85	27	𝑘	𝑘	X
iajs-2809	85	28	,	,	PUNCT
iajs-2809	85	29	problem	problem	NOUN
iajs-2809	85	30	(	(	PUNCT
iajs-2809	85	31	2	2	X
iajs-2809	85	32	)	)	PUNCT
iajs-2809	85	33	is	be	AUX
iajs-2809	85	34	transformed	transform	VERB
iajs-2809	85	35	into	into	ADP
iajs-2809	85	36	the	the	DET
iajs-2809	85	37	following	follow	VERB
iajs-2809	85	38	equivalent	equivalent	ADJ
iajs-2809	85	39	problem	problem	NOUN
iajs-2809	85	40	:	:	PUNCT
iajs-2809	85	41	.	.	PUNCT
iajs-2809	86	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	86	2	𝑍	𝑍	PROPN
iajs-2809	86	3	=	=	SYM
iajs-2809	86	4	∑	∑	PUNCT
iajs-2809	86	5	(	(	PUNCT
iajs-2809	86	6	𝐴∗𝑖	𝐴∗𝑖	PROPN
iajs-2809	86	7	,	,	PUNCT
iajs-2809	86	8	𝐴𝑖	𝐴𝑖	PROPN
iajs-2809	86	9	∗)𝑢𝑖	∗)𝑢𝑖	PUNCT
iajs-2809	87	1	+	+	NOUN
iajs-2809	87	2	𝑘	𝑘	X
iajs-2809	87	3	𝑖=1	𝑖=1	PROPN
iajs-2809	87	4	(	(	PUNCT
iajs-2809	87	5	𝐴∗𝑘+1	𝐴∗𝑘+1	NOUN
iajs-2809	87	6	,	,	PUNCT
iajs-2809	87	7	𝐴𝑘+1	𝐴𝑘+1	PART
iajs-2809	87	8	∗	∗	NOUN
iajs-2809	87	9	)	)	PUNCT
iajs-2809	88	1	𝑡	𝑡	PROPN
iajs-2809	88	2	subject	subject	ADJ
iajs-2809	88	3	to	to	ADP
iajs-2809	88	4	:	:	PUNCT
iajs-2809	88	5	∑	∑	PROPN
iajs-2809	88	6	(	(	PUNCT
iajs-2809	88	7	𝐵∗𝑖	𝐵∗𝑖	PROPN
iajs-2809	88	8	,	,	PUNCT
iajs-2809	88	9	𝐵𝑖	𝐵𝑖	PROPN
iajs-2809	88	10	∗)𝑢𝑖	∗)𝑢𝑖	PUNCT
iajs-2809	89	1	+	+	NOUN
iajs-2809	89	2	𝑘	𝑘	X
iajs-2809	89	3	𝑖=1	𝑖=1	PROPN
iajs-2809	89	4	(	(	PUNCT
iajs-2809	89	5	𝐵∗𝑘+1	𝐵∗𝑘+1	INTJ
iajs-2809	89	6	,	,	PUNCT
iajs-2809	89	7	𝐵𝑘+1	𝐵𝑘+1	NOUN
iajs-2809	89	8	∗	∗	NOUN
iajs-2809	89	9	)	)	PUNCT
iajs-2809	89	10	𝑡	𝑡	NOUN
iajs-2809	89	11	=	=	SYM
iajs-2809	89	12	1	1	NUM
iajs-2809	89	13	(	(	PUNCT
iajs-2809	89	14	3	3	NUM
iajs-2809	89	15	)	)	PUNCT
iajs-2809	89	16	∑	∑	PUNCT
iajs-2809	90	1	𝐴𝑖𝑢𝑖	𝐴𝑖𝑢𝑖	VERB
iajs-2809	90	2	−	−	NOUN
iajs-2809	90	3	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	90	4	≤	≤	NOUN
iajs-2809	90	5	0𝑘	0𝑘	NOUN
iajs-2809	90	6	𝑖=1	𝑖=1	PUNCT
iajs-2809	91	1	𝑢𝑖	𝑢𝑖	INTJ
iajs-2809	91	2	,	,	PUNCT
iajs-2809	91	3	𝑡	𝑡	X
iajs-2809	91	4	≥	≥	NOUN
iajs-2809	91	5	0	0	NUM
iajs-2809	91	6	,	,	PUNCT
iajs-2809	91	7	𝑖	𝑖	NOUN
iajs-2809	91	8	=	=	SYM
iajs-2809	91	9	1,2	1,2	NUM
iajs-2809	91	10	,	,	PUNCT
iajs-2809	91	11	…	…	PUNCT
iajs-2809	91	12	,	,	PUNCT
iajs-2809	91	13	𝑘.	𝑘.	NOUN
iajs-2809	91	14	by	by	ADP
iajs-2809	91	15	hamzehee	hamzehee	PROPN
iajs-2809	91	16	et	et	PROPN
iajs-2809	91	17	al	al	PROPN
iajs-2809	91	18	.	.	PUNCT
iajs-2809	92	1	[	[	X
iajs-2809	92	2	7	7	NUM
iajs-2809	92	3	]	]	PUNCT
iajs-2809	92	4	,	,	PUNCT
iajs-2809	92	5	we	we	PRON
iajs-2809	92	6	will	will	AUX
iajs-2809	92	7	build	build	VERB
iajs-2809	92	8	two	two	NUM
iajs-2809	92	9	linear	linear	ADJ
iajs-2809	92	10	programming	programming	NOUN
iajs-2809	92	11	with	with	ADP
iajs-2809	92	12	interval	interval	NOUN
iajs-2809	92	13	coefficients	coefficient	NOUN
iajs-2809	92	14	.	.	PUNCT
iajs-2809	93	1	one	one	NUM
iajs-2809	93	2	of	of	ADP
iajs-2809	93	3	these	these	DET
iajs-2809	93	4	problems	problem	NOUN
iajs-2809	93	5	is	be	AUX
iajs-2809	93	6	linear	linear	PROPN
iajs-2809	93	7	programming	programming	NOUN
iajs-2809	93	8	,	,	PUNCT
iajs-2809	93	9	where	where	SCONJ
iajs-2809	93	10	all	all	PRON
iajs-2809	93	11	of	of	ADP
iajs-2809	93	12	it	it	PRON
iajs-2809	93	13	is	be	AUX
iajs-2809	93	14	coefficients	coefficient	NOUN
iajs-2809	93	15	are	be	AUX
iajs-2809	93	16	lower	low	ADJ
iajs-2809	93	17	approximate	approximate	ADJ
iajs-2809	93	18	to	to	ADP
iajs-2809	93	19	rough	rough	ADJ
iajs-2809	93	20	intervals	interval	NOUN
iajs-2809	93	21	.	.	PUNCT
iajs-2809	94	1	the	the	DET
iajs-2809	94	2	other	other	ADJ
iajs-2809	94	3	is	be	AUX
iajs-2809	94	4	linear	linear	NOUN
iajs-2809	94	5	programming	programming	NOUN
iajs-2809	94	6	where	where	SCONJ
iajs-2809	94	7	all	all	PRON
iajs-2809	94	8	of	of	ADP
iajs-2809	94	9	its	its	PRON
iajs-2809	94	10	coefficients	coefficient	NOUN
iajs-2809	94	11	are	be	AUX
iajs-2809	94	12	upper	upper	ADJ
iajs-2809	94	13	approximate	approximate	NOUN
iajs-2809	94	14	of	of	ADP
iajs-2809	94	15	rough	rough	ADJ
iajs-2809	94	16	intervals	interval	NOUN
iajs-2809	94	17	.	.	PUNCT
iajs-2809	95	1	to	to	PART
iajs-2809	95	2	solve	solve	VERB
iajs-2809	95	3	problem	problem	NOUN
iajs-2809	95	4	(	(	PUNCT
iajs-2809	95	5	3	3	NUM
iajs-2809	95	6	)	)	PUNCT
iajs-2809	95	7	,	,	PUNCT
iajs-2809	95	8	the	the	DET
iajs-2809	95	9	following	follow	VERB
iajs-2809	95	10	two	two	NUM
iajs-2809	95	11	linear	linear	ADJ
iajs-2809	95	12	programming	programming	NOUN
iajs-2809	95	13	with	with	ADP
iajs-2809	95	14	interval	interval	NOUN
iajs-2809	95	15	coefficients	coefficient	NOUN
iajs-2809	95	16	are	be	AUX
iajs-2809	95	17	discussed	discuss	VERB
iajs-2809	95	18	in	in	ADP
iajs-2809	95	19	the	the	DET
iajs-2809	95	20	sequel	sequel	NOUN
iajs-2809	95	21	.	.	PUNCT
iajs-2809	96	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	96	2	𝑍	𝑍	PROPN
iajs-2809	96	3	=	=	SYM
iajs-2809	96	4	∑	∑	PUNCT
iajs-2809	96	5	𝐴∗𝑖𝑢𝑖	𝐴∗𝑖𝑢𝑖	PROPN
iajs-2809	96	6	+	+	PROPN
iajs-2809	96	7	𝑘	𝑘	NOUN
iajs-2809	96	8	𝑖=1	𝑖=1	PUNCT
iajs-2809	96	9	𝐴∗𝑖+𝑘𝑡	𝐴∗𝑖+𝑘𝑡	PROPN
iajs-2809	96	10	subject	subject	ADJ
iajs-2809	96	11	to	to	ADP
iajs-2809	96	12	:	:	PUNCT
iajs-2809	96	13	∑	∑	PUNCT
iajs-2809	96	14	𝐵∗𝑖𝑢𝑖	𝐵∗𝑖𝑢𝑖	PROPN
iajs-2809	96	15	+	+	PROPN
iajs-2809	96	16	𝑘	𝑘	PRON
iajs-2809	96	17	𝑖=1	𝑖=1	NOUN
iajs-2809	96	18	𝐵∗𝑖+𝑘𝑡	𝐵∗𝑖+𝑘𝑡	NOUN
iajs-2809	96	19	=	=	SYM
iajs-2809	96	20	1	1	NUM
iajs-2809	96	21	(	(	PUNCT
iajs-2809	96	22	4	4	NUM
iajs-2809	96	23	)	)	PUNCT
iajs-2809	96	24	∑	∑	PUNCT
iajs-2809	96	25	𝐴𝑖𝑢𝑖	𝐴𝑖𝑢𝑖	VERB
iajs-2809	96	26	−	−	NOUN
iajs-2809	96	27	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	96	28	≤	≤	NOUN
iajs-2809	96	29	0𝑘	0𝑘	NOUN
iajs-2809	96	30	𝑖=1	𝑖=1	PUNCT
iajs-2809	97	1	𝑢𝑖	𝑢𝑖	INTJ
iajs-2809	97	2	,	,	PUNCT
iajs-2809	97	3	𝑡	𝑡	X
iajs-2809	97	4	≥	≥	NOUN
iajs-2809	97	5	0	0	NUM
iajs-2809	97	6	,	,	PUNCT
iajs-2809	97	7	𝑖	𝑖	NOUN
iajs-2809	97	8	=	=	SYM
iajs-2809	97	9	1,2	1,2	NUM
iajs-2809	97	10	,	,	PUNCT
iajs-2809	97	11	…	…	PUNCT
iajs-2809	97	12	,	,	PUNCT
iajs-2809	97	13	𝑘.	𝑘.	PROPN
iajs-2809	97	14	ibn	ibn	PROPN
iajs-2809	97	15	al	al	PROPN
iajs-2809	97	16	-	-	PUNCT
iajs-2809	97	17	haitham	haitham	PROPN
iajs-2809	97	18	jour	jour	X
iajs-2809	97	19	.	.	PROPN
iajs-2809	98	1	for	for	ADP
iajs-2809	98	2	pure	pure	ADJ
iajs-2809	98	3	&	&	CCONJ
iajs-2809	98	4	appl	appl	PROPN
iajs-2809	98	5	.	.	PUNCT
iajs-2809	99	1	sci	sci	PROPN
iajs-2809	99	2	.	.	PROPN
iajs-2809	100	1	53	53	NUM
iajs-2809	100	2	(	(	PUNCT
iajs-2809	100	3	2)2022	2)2022	NOUN
iajs-2809	100	4	74	74	NUM
iajs-2809	100	5	and	and	CCONJ
iajs-2809	100	6	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	100	7	𝑍	𝑍	PROPN
iajs-2809	100	8	=	=	SYM
iajs-2809	100	9	∑	∑	PUNCT
iajs-2809	100	10	𝐴𝑖	𝐴𝑖	PROPN
iajs-2809	100	11	∗𝑢𝑖	∗𝑢𝑖	PROPN
iajs-2809	101	1	+	+	PROPN
iajs-2809	101	2	𝑘	𝑘	PRON
iajs-2809	101	3	𝑖=1	𝑖=1	PROPN
iajs-2809	101	4	𝐴𝑖+𝑘	𝐴𝑖+𝑘	PROPN
iajs-2809	101	5	∗	∗	NOUN
iajs-2809	101	6	𝑡	𝑡	NOUN
iajs-2809	101	7	subject	subject	ADJ
iajs-2809	101	8	to	to	ADP
iajs-2809	101	9	:	:	PUNCT
iajs-2809	101	10	∑	∑	PROPN
iajs-2809	101	11	𝐵𝑖	𝐵𝑖	PROPN
iajs-2809	101	12	∗𝑢𝑖	∗𝑢𝑖	PROPN
iajs-2809	102	1	+	+	PROPN
iajs-2809	103	1	𝑘	𝑘	PRON
iajs-2809	103	2	𝑖=1	𝑖=1	PROPN
iajs-2809	103	3	𝐵𝑖+𝑘	𝐵𝑖+𝑘	NUM
iajs-2809	103	4	∗	∗	NOUN
iajs-2809	103	5	𝑡	𝑡	X
iajs-2809	103	6	=	=	NOUN
iajs-2809	103	7	1	1	NUM
iajs-2809	103	8	(	(	PUNCT
iajs-2809	103	9	5	5	NUM
iajs-2809	103	10	)	)	PUNCT
iajs-2809	103	11	∑	∑	PUNCT
iajs-2809	103	12	𝐴𝑖𝑢𝑖	𝐴𝑖𝑢𝑖	VERB
iajs-2809	103	13	−	−	NOUN
iajs-2809	103	14	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	103	15	≤	≤	NOUN
iajs-2809	103	16	0𝑘	0𝑘	NOUN
iajs-2809	103	17	𝑖=1	𝑖=1	PUNCT
iajs-2809	104	1	𝑢𝑖	𝑢𝑖	INTJ
iajs-2809	104	2	,	,	PUNCT
iajs-2809	104	3	𝑡	𝑡	X
iajs-2809	104	4	≥	≥	NOUN
iajs-2809	104	5	0	0	NUM
iajs-2809	104	6	,	,	PUNCT
iajs-2809	104	7	𝑖	𝑖	NOUN
iajs-2809	104	8	=	=	SYM
iajs-2809	104	9	1,2	1,2	NUM
iajs-2809	104	10	,	,	PUNCT
iajs-2809	104	11	…	…	PUNCT
iajs-2809	104	12	,	,	PUNCT
iajs-2809	104	13	𝑘.	𝑘.	NOUN
iajs-2809	104	14	in	in	ADP
iajs-2809	104	15	the	the	DET
iajs-2809	104	16	beginning	beginning	NOUN
iajs-2809	105	1	,	,	PUNCT
iajs-2809	105	2	we	we	PRON
iajs-2809	105	3	can	can	AUX
iajs-2809	105	4	derive	derive	VERB
iajs-2809	105	5	problem	problem	NOUN
iajs-2809	105	6	(	(	PUNCT
iajs-2809	105	7	4	4	NUM
iajs-2809	105	8	)	)	PUNCT
iajs-2809	105	9	.	.	PUNCT
iajs-2809	106	1	to	to	PART
iajs-2809	106	2	solve	solve	VERB
iajs-2809	106	3	the	the	DET
iajs-2809	106	4	problem	problem	NOUN
iajs-2809	106	5	(	(	PUNCT
iajs-2809	106	6	4	4	X
iajs-2809	106	7	)	)	PUNCT
iajs-2809	106	8	an	an	DET
iajs-2809	106	9	equivalent	equivalent	ADJ
iajs-2809	106	10	problem	problem	NOUN
iajs-2809	106	11	can	can	AUX
iajs-2809	106	12	be	be	AUX
iajs-2809	106	13	written	write	VERB
iajs-2809	106	14	:	:	PUNCT
iajs-2809	106	15	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	106	16	𝑍	𝑍	PROPN
iajs-2809	106	17	=	=	SYM
iajs-2809	106	18	[	[	X
iajs-2809	106	19	𝑎∗1	𝑎∗1	ADJ
iajs-2809	106	20	𝑙	𝑙	X
iajs-2809	106	21	,	,	PUNCT
iajs-2809	106	22	𝑎∗1	𝑎∗1	ADJ
iajs-2809	106	23	𝑢	𝑢	X
iajs-2809	106	24	]	]	X
iajs-2809	106	25	𝑢1	𝑢1	PROPN
iajs-2809	106	26	+	+	CCONJ
iajs-2809	107	1	[	[	X
iajs-2809	107	2	𝑎∗2	𝑎∗2	NOUN
iajs-2809	107	3	𝑙	𝑙	X
iajs-2809	107	4	,	,	PUNCT
iajs-2809	107	5	𝑎∗2	𝑎∗2	NOUN
iajs-2809	107	6	𝑢	𝑢	X
iajs-2809	107	7	]	]	X
iajs-2809	107	8	𝑢2	𝑢2	PROPN
iajs-2809	107	9	+	+	SYM
iajs-2809	107	10	⋯	⋯	PROPN
iajs-2809	107	11	+	+	CCONJ
iajs-2809	107	12	[	[	X
iajs-2809	107	13	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	107	14	𝑙	𝑙	X
iajs-2809	107	15	,	,	PUNCT
iajs-2809	107	16	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	107	17	𝑢	𝑢	X
iajs-2809	107	18	]	]	X
iajs-2809	107	19	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	107	20	+	+	X
iajs-2809	107	21	[	[	X
iajs-2809	107	22	𝑎∗𝑘+1	𝑎∗𝑘+1	X
iajs-2809	107	23	𝑙	𝑙	NUM
iajs-2809	107	24	,	,	PUNCT
iajs-2809	107	25	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	107	26	𝑢	𝑢	X
iajs-2809	107	27	]	]	X
iajs-2809	107	28	𝑡	𝑡	X
iajs-2809	107	29	subject	subject	ADJ
iajs-2809	107	30	to	to	ADP
iajs-2809	107	31	:	:	PUNCT
iajs-2809	108	1	[	[	X
iajs-2809	108	2	𝑏∗1	𝑏∗1	NOUN
iajs-2809	108	3	𝑙	𝑙	X
iajs-2809	108	4	,	,	PUNCT
iajs-2809	108	5	𝑏∗1	𝑏∗1	ADJ
iajs-2809	108	6	𝑢	𝑢	X
iajs-2809	108	7	]	]	X
iajs-2809	108	8	𝑢1	𝑢1	NOUN
iajs-2809	108	9	+	+	CCONJ
iajs-2809	109	1	[	[	X
iajs-2809	109	2	𝑏∗2	𝑏∗2	NOUN
iajs-2809	109	3	𝑙	𝑙	X
iajs-2809	109	4	,	,	PUNCT
iajs-2809	109	5	𝑏∗2	𝑏∗2	ADJ
iajs-2809	109	6	𝑢	𝑢	X
iajs-2809	109	7	]	]	X
iajs-2809	109	8	𝑢2	𝑢2	PROPN
iajs-2809	109	9	+	+	SYM
iajs-2809	109	10	⋯	⋯	PROPN
iajs-2809	109	11	+	+	CCONJ
iajs-2809	109	12	[	[	X
iajs-2809	109	13	𝑏∗𝑘	𝑏∗𝑘	X
iajs-2809	109	14	𝑙	𝑙	NUM
iajs-2809	109	15	,	,	PUNCT
iajs-2809	109	16	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	109	17	𝑢	𝑢	X
iajs-2809	109	18	]	]	X
iajs-2809	109	19	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	109	20	+	+	X
iajs-2809	110	1	[	[	PUNCT
iajs-2809	110	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	110	3	𝑙	𝑙	X
iajs-2809	110	4	,	,	PUNCT
iajs-2809	110	5	𝑏∗𝑘+1	𝑏∗𝑘+1	X
iajs-2809	110	6	𝑢	𝑢	X
iajs-2809	110	7	]	]	X
iajs-2809	110	8	𝑡	𝑡	X
iajs-2809	110	9	=	=	SYM
iajs-2809	110	10	1	1	NUM
iajs-2809	110	11	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	110	12	+	+	CCONJ
iajs-2809	110	13	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	110	14	+	+	SYM
iajs-2809	110	15	⋯	⋯	PROPN
iajs-2809	111	1	+	+	CCONJ
iajs-2809	111	2	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	111	3	−	−	NOUN
iajs-2809	111	4	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	111	5	≤	≤	NOUN
iajs-2809	111	6	0	0	NUM
iajs-2809	111	7	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	111	8	,	,	PUNCT
iajs-2809	111	9	𝑡	𝑡	X
iajs-2809	111	10	≥	≥	NOUN
iajs-2809	111	11	0	0	NUM
iajs-2809	111	12	,	,	PUNCT
iajs-2809	111	13	𝑖	𝑖	NOUN
iajs-2809	111	14	=	=	SYM
iajs-2809	111	15	1,2	1,2	NUM
iajs-2809	111	16	,	,	PUNCT
iajs-2809	111	17	…	…	PUNCT
iajs-2809	111	18	,	,	PUNCT
iajs-2809	111	19	𝑘.	𝑘.	VERB
iajs-2809	111	20	the	the	DET
iajs-2809	111	21	linear	linear	ADJ
iajs-2809	111	22	combination	combination	NOUN
iajs-2809	111	23	of	of	ADP
iajs-2809	111	24	each	each	DET
iajs-2809	111	25	region	region	NOUN
iajs-2809	111	26	interval	interval	NOUN
iajs-2809	111	27	of	of	ADP
iajs-2809	111	28	the	the	DET
iajs-2809	111	29	problem	problem	NOUN
iajs-2809	111	30	(	(	PUNCT
iajs-2809	111	31	4	4	X
iajs-2809	111	32	)	)	PUNCT
iajs-2809	111	33	yields	yield	VERB
iajs-2809	111	34	the	the	DET
iajs-2809	111	35	following	follow	VERB
iajs-2809	111	36	problem	problem	NOUN
iajs-2809	111	37	:	:	PUNCT
iajs-2809	111	38	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	111	39	𝑍	𝑍	PROPN
iajs-2809	111	40	=	=	SYM
iajs-2809	111	41	[	[	X
iajs-2809	111	42	𝛾1𝑎∗1	𝛾1𝑎∗1	NOUN
iajs-2809	111	43	𝑙	𝑙	X
iajs-2809	111	44	+	+	CCONJ
iajs-2809	111	45	(	(	PUNCT
iajs-2809	111	46	1	1	NUM
iajs-2809	111	47	−	−	PROPN
iajs-2809	111	48	𝛾1)𝑎∗1	𝛾1)𝑎∗1	NOUN
iajs-2809	111	49	𝑢	𝑢	X
iajs-2809	111	50	]	]	X
iajs-2809	111	51	𝑢1	𝑢1	NOUN
iajs-2809	111	52	+	+	CCONJ
iajs-2809	111	53	⋯	⋯	VERB
iajs-2809	111	54	+	+	CCONJ
iajs-2809	112	1	[	[	X
iajs-2809	112	2	𝛾𝑘𝑎∗𝑘	𝛾𝑘𝑎∗𝑘	X
iajs-2809	112	3	𝑙	𝑙	X
iajs-2809	112	4	+	+	CCONJ
iajs-2809	112	5	(	(	PUNCT
iajs-2809	112	6	1	1	NUM
iajs-2809	112	7	−	−	NOUN
iajs-2809	112	8	𝛾𝑘)𝑎∗𝑘	𝛾𝑘)𝑎∗𝑘	NOUN
iajs-2809	112	9	𝑢	𝑢	X
iajs-2809	112	10	]	]	PUNCT
iajs-2809	112	11	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	112	12	+	+	X
iajs-2809	113	1	[	[	X
iajs-2809	113	2	𝛾𝑘+1𝑎∗𝑘+1	𝛾𝑘+1𝑎∗𝑘+1	NOUN
iajs-2809	113	3	𝑙	𝑙	X
iajs-2809	113	4	+	+	CCONJ
iajs-2809	113	5	(	(	PUNCT
iajs-2809	113	6	1	1	NUM
iajs-2809	113	7	−	−	PROPN
iajs-2809	113	8	𝛾𝑘+1)𝑎∗𝑘+1	𝛾𝑘+1)𝑎∗𝑘+1	NUM
iajs-2809	113	9	𝑢	𝑢	X
iajs-2809	113	10	]	]	X
iajs-2809	113	11	𝑡	𝑡	X
iajs-2809	113	12	subject	subject	ADJ
iajs-2809	113	13	to	to	ADP
iajs-2809	113	14	:	:	PUNCT
iajs-2809	114	1	[	[	X
iajs-2809	114	2	𝛿1𝑏∗1	𝛿1𝑏∗1	X
iajs-2809	114	3	𝑙	𝑙	X
iajs-2809	114	4	+	+	CCONJ
iajs-2809	114	5	(	(	PUNCT
iajs-2809	114	6	1	1	NUM
iajs-2809	114	7	−	−	PROPN
iajs-2809	114	8	𝛿1)𝑏∗1	𝛿1)𝑏∗1	NUM
iajs-2809	114	9	𝑢	𝑢	X
iajs-2809	114	10	]	]	X
iajs-2809	114	11	𝑢1	𝑢1	NOUN
iajs-2809	114	12	+	+	CCONJ
iajs-2809	114	13	⋯	⋯	VERB
iajs-2809	114	14	+	+	CCONJ
iajs-2809	115	1	[	[	X
iajs-2809	115	2	𝛿𝑘𝑏∗𝑘	𝛿𝑘𝑏∗𝑘	NOUN
iajs-2809	115	3	𝑙	𝑙	X
iajs-2809	115	4	+	+	CCONJ
iajs-2809	115	5	(	(	PUNCT
iajs-2809	115	6	1	1	NUM
iajs-2809	115	7	−	−	NOUN
iajs-2809	115	8	𝛿𝑘)𝑏∗𝑘	𝛿𝑘)𝑏∗𝑘	NOUN
iajs-2809	115	9	𝑢	𝑢	X
iajs-2809	115	10	]	]	X
iajs-2809	115	11	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	115	12	+	+	NOUN
iajs-2809	116	1	[	[	X
iajs-2809	116	2	𝛿𝑘+1𝑏∗𝑘+1	𝛿𝑘+1𝑏∗𝑘+1	NOUN
iajs-2809	116	3	𝑙	𝑙	X
iajs-2809	116	4	+	+	CCONJ
iajs-2809	116	5	(	(	PUNCT
iajs-2809	116	6	1	1	NUM
iajs-2809	116	7	−	−	NOUN
iajs-2809	116	8	𝛿𝑘+1)𝑏∗𝑘+1	𝛿𝑘+1)𝑏∗𝑘+1	X
iajs-2809	116	9	𝑢	𝑢	X
iajs-2809	116	10	]	]	X
iajs-2809	116	11	𝑡	𝑡	X
iajs-2809	116	12	=	=	SYM
iajs-2809	116	13	1	1	NUM
iajs-2809	116	14	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	116	15	+	+	CCONJ
iajs-2809	116	16	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	116	17	+	+	SYM
iajs-2809	116	18	⋯	⋯	PROPN
iajs-2809	116	19	+	+	CCONJ
iajs-2809	116	20	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	116	21	−	−	NOUN
iajs-2809	116	22	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	116	23	≤	≤	NOUN
iajs-2809	116	24	0	0	NUM
iajs-2809	116	25	(	(	PUNCT
iajs-2809	116	26	6	6	NUM
iajs-2809	116	27	)	)	PUNCT
iajs-2809	116	28	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	116	29	,	,	PUNCT
iajs-2809	116	30	𝑡	𝑡	X
iajs-2809	116	31	≥	≥	NOUN
iajs-2809	116	32	0	0	NUM
iajs-2809	116	33	,	,	PUNCT
iajs-2809	116	34	𝑖	𝑖	NOUN
iajs-2809	116	35	=	=	SYM
iajs-2809	116	36	1,2	1,2	NUM
iajs-2809	116	37	,	,	PUNCT
iajs-2809	116	38	…	…	PUNCT
iajs-2809	116	39	,	,	PUNCT
iajs-2809	116	40	𝑘	𝑘	X
iajs-2809	116	41	,	,	PUNCT
iajs-2809	116	42	𝛾𝑖	𝛾𝑖	INTJ
iajs-2809	116	43	,	,	PUNCT
iajs-2809	116	44	𝛿𝑖	𝛿𝑖	X
iajs-2809	116	45	∈	∈	PROPN
iajs-2809	117	1	[	[	X
iajs-2809	117	2	0,1	0,1	NUM
iajs-2809	117	3	]	]	PUNCT
iajs-2809	117	4	,	,	PUNCT
iajs-2809	117	5	𝑖	𝑖	SYM
iajs-2809	117	6	=	=	SYM
iajs-2809	117	7	1,2	1,2	NUM
iajs-2809	117	8	,	,	PUNCT
iajs-2809	117	9	…	…	PUNCT
iajs-2809	117	10	,	,	PUNCT
iajs-2809	117	11	𝑘	𝑘	X
iajs-2809	118	1	+	+	NOUN
iajs-2809	118	2	1	1	NUM
iajs-2809	118	3	.	.	PUNCT
iajs-2809	119	1	the	the	DET
iajs-2809	119	2	equality	equality	NOUN
iajs-2809	119	3	constraint	constraint	NOUN
iajs-2809	119	4	in	in	ADP
iajs-2809	119	5	problem	problem	NOUN
iajs-2809	119	6	(	(	PUNCT
iajs-2809	119	7	6	6	NUM
iajs-2809	119	8	)	)	PUNCT
iajs-2809	119	9	can	can	AUX
iajs-2809	119	10	be	be	AUX
iajs-2809	119	11	further	far	ADV
iajs-2809	119	12	reduced	reduce	VERB
iajs-2809	119	13	to	to	ADP
iajs-2809	119	14	:	:	PUNCT
iajs-2809	119	15	[	[	X
iajs-2809	119	16	𝛿1𝑢1(𝑏∗1	𝛿1𝑢1(𝑏∗1	X
iajs-2809	119	17	𝑙	𝑙	X
iajs-2809	119	18	−	−	NOUN
iajs-2809	119	19	𝑏∗1	𝑏∗1	ADJ
iajs-2809	119	20	𝑢	𝑢	NOUN
iajs-2809	119	21	)	)	PUNCT
iajs-2809	120	1	+	+	CCONJ
iajs-2809	120	2	⋯	⋯	X
iajs-2809	120	3	+	+	CCONJ
iajs-2809	120	4	𝛿𝑘𝑢𝑘(𝑏∗𝑘	𝛿𝑘𝑢𝑘(𝑏∗𝑘	PROPN
iajs-2809	120	5	𝑙	𝑙	PRON
iajs-2809	120	6	−	−	NOUN
iajs-2809	120	7	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	120	8	𝑢	𝑢	NOUN
iajs-2809	120	9	)	)	PUNCT
iajs-2809	120	10	+	+	CCONJ
iajs-2809	120	11	𝛿𝑘+1𝑡(𝑏∗𝑘+1	𝛿𝑘+1𝑡(𝑏∗𝑘+1	NOUN
iajs-2809	120	12	𝑙	𝑙	PRON
iajs-2809	120	13	−	−	PROPN
iajs-2809	120	14	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	120	15	𝑢	𝑢	PROPN
iajs-2809	120	16	)	)	PUNCT
iajs-2809	120	17	]	]	PUNCT
iajs-2809	121	1	+	+	CCONJ
iajs-2809	121	2	𝑏∗1	𝑏∗1	ADJ
iajs-2809	121	3	𝑢	𝑢	PRON
iajs-2809	121	4	𝑢1	𝑢1	PROPN
iajs-2809	121	5	+	+	CCONJ
iajs-2809	121	6	⋯	⋯	PROPN
iajs-2809	121	7	+	+	CCONJ
iajs-2809	121	8	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	121	9	𝑢	𝑢	X
iajs-2809	121	10	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	121	11	+	+	CCONJ
iajs-2809	121	12	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	121	13	𝑢	𝑢	X
iajs-2809	121	14	𝑡	𝑡	PROPN
iajs-2809	121	15	=	=	SYM
iajs-2809	121	16	1	1	NUM
iajs-2809	121	17	(	(	PUNCT
iajs-2809	121	18	7	7	NUM
iajs-2809	121	19	)	)	PUNCT
iajs-2809	121	20	since	since	SCONJ
iajs-2809	121	21	𝑢𝑖	𝑢𝑖	INTJ
iajs-2809	121	22	,	,	PUNCT
iajs-2809	121	23	𝑡	𝑡	X
iajs-2809	121	24	≥	≥	NOUN
iajs-2809	121	25	0	0	NUM
iajs-2809	121	26	,	,	PUNCT
iajs-2809	121	27	𝑖	𝑖	NOUN
iajs-2809	121	28	=	=	SYM
iajs-2809	121	29	1,2	1,2	NUM
iajs-2809	121	30	,	,	PUNCT
iajs-2809	121	31	…	…	PUNCT
iajs-2809	121	32	,	,	PUNCT
iajs-2809	121	33	𝑘	𝑘	X
iajs-2809	121	34	,	,	PUNCT
iajs-2809	121	35	𝛿𝑖	𝛿𝑖	X
iajs-2809	121	36	∈	∈	PROPN
iajs-2809	122	1	[	[	X
iajs-2809	122	2	0,1	0,1	NUM
iajs-2809	122	3	]	]	PUNCT
iajs-2809	122	4	,	,	PUNCT
iajs-2809	122	5	(	(	PUNCT
iajs-2809	122	6	𝑏∗𝑖	𝑏∗𝑖	PUNCT
iajs-2809	122	7	𝑢	𝑢	X
iajs-2809	122	8	−	−	PROPN
iajs-2809	122	9	𝑏∗𝑖	𝑏∗𝑖	NOUN
iajs-2809	122	10	𝑙	𝑙	NUM
iajs-2809	122	11	)	)	PUNCT
iajs-2809	122	12	≥	≥	NOUN
iajs-2809	122	13	0	0	NUM
iajs-2809	122	14	,	,	PUNCT
iajs-2809	122	15	𝑖	𝑖	NOUN
iajs-2809	122	16	=	=	SYM
iajs-2809	122	17	1,2	1,2	NUM
iajs-2809	122	18	,	,	PUNCT
iajs-2809	122	19	…	…	PUNCT
iajs-2809	122	20	,	,	PUNCT
iajs-2809	122	21	𝑘	𝑘	X
iajs-2809	122	22	+	+	NOUN
iajs-2809	122	23	1	1	NUM
iajs-2809	122	24	.	.	PUNCT
iajs-2809	122	25	therefore	therefore	ADV
iajs-2809	122	26	,	,	PUNCT
iajs-2809	122	27	(	(	PUNCT
iajs-2809	122	28	7	7	X
iajs-2809	122	29	)	)	PUNCT
iajs-2809	122	30	can	can	AUX
iajs-2809	122	31	be	be	AUX
iajs-2809	122	32	written	write	VERB
iajs-2809	122	33	as	as	ADP
iajs-2809	122	34	:	:	PUNCT
iajs-2809	122	35	ibn	ibn	PROPN
iajs-2809	122	36	al	al	PROPN
iajs-2809	122	37	-	-	PUNCT
iajs-2809	122	38	haitham	haitham	PROPN
iajs-2809	122	39	jour	jour	X
iajs-2809	122	40	.	.	PROPN
iajs-2809	122	41	for	for	ADP
iajs-2809	122	42	pure	pure	ADJ
iajs-2809	122	43	&	&	CCONJ
iajs-2809	122	44	appl	appl	PROPN
iajs-2809	122	45	.	.	PUNCT
iajs-2809	123	1	sci	sci	PROPN
iajs-2809	123	2	.	.	PROPN
iajs-2809	124	1	53	53	NUM
iajs-2809	124	2	(	(	PUNCT
iajs-2809	124	3	2)2022	2)2022	VERB
iajs-2809	124	4	75	75	NUM
iajs-2809	124	5	1	1	NUM
iajs-2809	124	6	≤	≤	NUM
iajs-2809	124	7	1	1	NUM
iajs-2809	124	8	+	+	CCONJ
iajs-2809	125	1	[	[	X
iajs-2809	125	2	𝛿1𝑢1(𝑏∗1	𝛿1𝑢1(𝑏∗1	X
iajs-2809	125	3	𝑢	𝑢	PRON
iajs-2809	125	4	−	−	NOUN
iajs-2809	125	5	𝑏∗1	𝑏∗1	ADJ
iajs-2809	125	6	𝑙	𝑙	X
iajs-2809	125	7	)	)	PUNCT
iajs-2809	126	1	+	+	CCONJ
iajs-2809	126	2	⋯	⋯	VERB
iajs-2809	126	3	+	+	CCONJ
iajs-2809	126	4	𝛿𝑘𝑢𝑘(𝑏∗𝑘	𝛿𝑘𝑢𝑘(𝑏∗𝑘	PROPN
iajs-2809	126	5	𝑢	𝑢	NOUN
iajs-2809	126	6	−	−	PROPN
iajs-2809	126	7	𝑏∗𝑘	𝑏∗𝑘	NOUN
iajs-2809	126	8	𝑙	𝑙	X
iajs-2809	126	9	)	)	PUNCT
iajs-2809	127	1	+	+	CCONJ
iajs-2809	127	2	𝛿𝑘+1𝑡(𝑏∗𝑘+1	𝛿𝑘+1𝑡(𝑏∗𝑘+1	NOUN
iajs-2809	127	3	𝑢	𝑢	NOUN
iajs-2809	127	4	−	−	NOUN
iajs-2809	127	5	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	127	6	𝑙	𝑙	PROPN
iajs-2809	127	7	)	)	PUNCT
iajs-2809	127	8	]	]	PUNCT
iajs-2809	128	1	≤	≤	NUM
iajs-2809	128	2	1	1	NUM
iajs-2809	128	3	+	+	CCONJ
iajs-2809	129	1	[	[	X
iajs-2809	129	2	𝑢1(𝑏∗1	𝑢1(𝑏∗1	PROPN
iajs-2809	129	3	𝑢	𝑢	ADP
iajs-2809	129	4	−	−	NOUN
iajs-2809	129	5	𝑏∗1	𝑏∗1	ADJ
iajs-2809	129	6	𝑙	𝑙	X
iajs-2809	129	7	)	)	PUNCT
iajs-2809	130	1	+	+	CCONJ
iajs-2809	130	2	⋯	⋯	VERB
iajs-2809	130	3	+	+	CCONJ
iajs-2809	130	4	𝑢𝑘(𝑏∗𝑘	𝑢𝑘(𝑏∗𝑘	X
iajs-2809	130	5	𝑢	𝑢	PROPN
iajs-2809	130	6	−	−	PROPN
iajs-2809	130	7	𝑏∗𝑘	𝑏∗𝑘	NOUN
iajs-2809	130	8	𝑙	𝑙	X
iajs-2809	130	9	)	)	PUNCT
iajs-2809	130	10	+	+	CCONJ
iajs-2809	130	11	𝑡(𝑏∗𝑘+1	𝑡(𝑏∗𝑘+1	NUM
iajs-2809	130	12	𝑢	𝑢	NOUN
iajs-2809	130	13	−	−	NOUN
iajs-2809	130	14	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	130	15	𝑙	𝑙	PROPN
iajs-2809	130	16	)	)	PUNCT
iajs-2809	130	17	]	]	PUNCT
iajs-2809	131	1	(	(	PUNCT
iajs-2809	131	2	8)	8)	NUM
iajs-2809	131	3	by	by	ADP
iajs-2809	131	4	combining	combine	VERB
iajs-2809	131	5	(	(	PUNCT
iajs-2809	131	6	7	7	NUM
iajs-2809	131	7	)	)	PUNCT
iajs-2809	131	8	and	and	CCONJ
iajs-2809	131	9	(	(	PUNCT
iajs-2809	131	10	8)	8)	NUM
iajs-2809	131	11	,	,	PUNCT
iajs-2809	131	12	the	the	DET
iajs-2809	131	13	result	result	NOUN
iajs-2809	131	14	is	be	AUX
iajs-2809	131	15	1	1	NUM
iajs-2809	131	16	≤	≤	NUM
iajs-2809	131	17	𝑏∗1	𝑏∗1	ADJ
iajs-2809	131	18	𝑢	𝑢	PRON
iajs-2809	131	19	𝑢1	𝑢1	PROPN
iajs-2809	131	20	+	+	CCONJ
iajs-2809	131	21	⋯	⋯	PROPN
iajs-2809	131	22	+	+	CCONJ
iajs-2809	131	23	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	131	24	𝑢	𝑢	X
iajs-2809	131	25	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	131	26	+	+	CCONJ
iajs-2809	131	27	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	131	28	𝑢	𝑢	X
iajs-2809	131	29	𝑡	𝑡	PROPN
iajs-2809	131	30	≤	≤	ADV
iajs-2809	131	31	1	1	NUM
iajs-2809	131	32	+	+	CCONJ
iajs-2809	131	33	𝑢1(𝑏∗1	𝑢1(𝑏∗1	PROPN
iajs-2809	131	34	𝑢	𝑢	PART
iajs-2809	131	35	−	−	NOUN
iajs-2809	131	36	𝑏∗1	𝑏∗1	ADJ
iajs-2809	131	37	𝑙	𝑙	X
iajs-2809	131	38	)	)	PUNCT
iajs-2809	132	1	+	+	CCONJ
iajs-2809	132	2	⋯	⋯	VERB
iajs-2809	132	3	+	+	CCONJ
iajs-2809	132	4	𝑢𝑘(𝑏∗𝑘	𝑢𝑘(𝑏∗𝑘	X
iajs-2809	132	5	𝑢	𝑢	PROPN
iajs-2809	132	6	−	−	PROPN
iajs-2809	132	7	𝑏∗𝑘	𝑏∗𝑘	NOUN
iajs-2809	132	8	𝑙	𝑙	X
iajs-2809	132	9	)	)	PUNCT
iajs-2809	132	10	+	+	CCONJ
iajs-2809	132	11	𝑡(𝑏∗𝑘+1	𝑡(𝑏∗𝑘+1	NUM
iajs-2809	132	12	𝑢	𝑢	NOUN
iajs-2809	132	13	−	−	NOUN
iajs-2809	132	14	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	132	15	𝑙	𝑙	NOUN
iajs-2809	132	16	)	)	PUNCT
iajs-2809	132	17	(	(	PUNCT
iajs-2809	132	18	9	9	X
iajs-2809	132	19	)	)	PUNCT
iajs-2809	132	20	this	this	PRON
iajs-2809	132	21	is	be	AUX
iajs-2809	132	22	then	then	ADV
iajs-2809	132	23	reduced	reduce	VERB
iajs-2809	132	24	to	to	ADP
iajs-2809	132	25	𝑏∗1	𝑏∗1	PROPN
iajs-2809	132	26	𝑢	𝑢	PART
iajs-2809	132	27	𝑢1	𝑢1	PROPN
iajs-2809	132	28	+	+	CCONJ
iajs-2809	132	29	⋯	⋯	PROPN
iajs-2809	132	30	+	+	CCONJ
iajs-2809	132	31	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	132	32	𝑢	𝑢	X
iajs-2809	132	33	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	132	34	+	+	CCONJ
iajs-2809	132	35	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	132	36	𝑢	𝑢	X
iajs-2809	132	37	𝑡	𝑡	PROPN
iajs-2809	132	38	≥	≥	NUM
iajs-2809	132	39	1	1	NUM
iajs-2809	132	40	(	(	PUNCT
iajs-2809	132	41	10	10	NUM
iajs-2809	132	42	)	)	PUNCT
iajs-2809	132	43	and	and	CCONJ
iajs-2809	132	44	𝑏∗1	𝑏∗1	ADJ
iajs-2809	132	45	𝑙	𝑙	PRON
iajs-2809	132	46	𝑢1	𝑢1	PROPN
iajs-2809	132	47	+	+	CCONJ
iajs-2809	132	48	⋯	⋯	PROPN
iajs-2809	132	49	+	+	CCONJ
iajs-2809	132	50	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	132	51	𝑙	𝑙	X
iajs-2809	132	52	𝑢𝑘	𝑢𝑘	NOUN
iajs-2809	133	1	+	+	CCONJ
iajs-2809	133	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	133	3	𝑙	𝑙	X
iajs-2809	133	4	𝑡	𝑡	NOUN
iajs-2809	133	5	≤	≤	ADV
iajs-2809	133	6	1	1	NUM
iajs-2809	133	7	(	(	PUNCT
iajs-2809	133	8	11	11	NUM
iajs-2809	133	9	)	)	PUNCT
iajs-2809	133	10	therefore	therefore	ADV
iajs-2809	133	11	,	,	PUNCT
iajs-2809	133	12	using	use	VERB
iajs-2809	133	13	(	(	PUNCT
iajs-2809	133	14	10	10	NUM
iajs-2809	133	15	)	)	PUNCT
iajs-2809	133	16	and	and	CCONJ
iajs-2809	133	17	(	(	PUNCT
iajs-2809	133	18	11	11	NUM
iajs-2809	133	19	)	)	PUNCT
iajs-2809	133	20	,	,	PUNCT
iajs-2809	133	21	the	the	DET
iajs-2809	133	22	equation	equation	NOUN
iajs-2809	133	23	(	(	PUNCT
iajs-2809	133	24	6	6	NUM
iajs-2809	133	25	)	)	PUNCT
iajs-2809	133	26	is	be	AUX
iajs-2809	133	27	converted	convert	VERB
iajs-2809	133	28	into	into	ADP
iajs-2809	133	29	the	the	DET
iajs-2809	133	30	following	follow	VERB
iajs-2809	133	31	equation	equation	NOUN
iajs-2809	133	32	:	:	PUNCT
iajs-2809	133	33	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	133	34	𝑍	𝑍	PROPN
iajs-2809	133	35	=	=	SYM
iajs-2809	134	1	[	[	X
iajs-2809	134	2	𝛾1𝑎∗1	𝛾1𝑎∗1	NOUN
iajs-2809	134	3	𝑙	𝑙	X
iajs-2809	134	4	+	+	CCONJ
iajs-2809	134	5	(	(	PUNCT
iajs-2809	134	6	1	1	NUM
iajs-2809	134	7	−	−	PROPN
iajs-2809	134	8	𝛾1)𝑎∗1	𝛾1)𝑎∗1	NOUN
iajs-2809	134	9	𝑢	𝑢	X
iajs-2809	134	10	]	]	X
iajs-2809	134	11	𝑢1	𝑢1	NOUN
iajs-2809	134	12	+	+	CCONJ
iajs-2809	134	13	⋯	⋯	VERB
iajs-2809	134	14	+	+	CCONJ
iajs-2809	135	1	[	[	X
iajs-2809	135	2	𝛾𝑘𝑎∗𝑘	𝛾𝑘𝑎∗𝑘	X
iajs-2809	135	3	𝑙	𝑙	X
iajs-2809	135	4	+	+	CCONJ
iajs-2809	135	5	(	(	PUNCT
iajs-2809	135	6	1	1	NUM
iajs-2809	135	7	−	−	NOUN
iajs-2809	135	8	𝛾𝑘)𝑎∗𝑘	𝛾𝑘)𝑎∗𝑘	NOUN
iajs-2809	135	9	𝑢	𝑢	X
iajs-2809	135	10	]	]	PUNCT
iajs-2809	135	11	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	135	12	+	+	X
iajs-2809	136	1	[	[	X
iajs-2809	136	2	𝛾𝑘+1𝑎∗𝑘+1	𝛾𝑘+1𝑎∗𝑘+1	NOUN
iajs-2809	136	3	𝑙	𝑙	X
iajs-2809	136	4	+	+	CCONJ
iajs-2809	136	5	(	(	PUNCT
iajs-2809	136	6	1	1	NUM
iajs-2809	136	7	−	−	PROPN
iajs-2809	136	8	𝛾𝑘+1)𝑎∗𝑘+1	𝛾𝑘+1)𝑎∗𝑘+1	NUM
iajs-2809	136	9	𝑢	𝑢	X
iajs-2809	136	10	]	]	X
iajs-2809	136	11	𝑡	𝑡	X
iajs-2809	136	12	subject	subject	ADJ
iajs-2809	136	13	to	to	ADP
iajs-2809	136	14	:	:	PUNCT
iajs-2809	136	15	𝑏∗1	𝑏∗1	PROPN
iajs-2809	136	16	𝑢	𝑢	PRON
iajs-2809	136	17	𝑢1	𝑢1	PROPN
iajs-2809	136	18	+	+	CCONJ
iajs-2809	136	19	⋯	⋯	PROPN
iajs-2809	136	20	+	+	CCONJ
iajs-2809	136	21	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	136	22	𝑢	𝑢	X
iajs-2809	136	23	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	136	24	+	+	CCONJ
iajs-2809	136	25	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	136	26	𝑢	𝑢	X
iajs-2809	136	27	𝑡	𝑡	PROPN
iajs-2809	136	28	≥	≥	NUM
iajs-2809	136	29	1	1	NUM
iajs-2809	136	30	(	(	PUNCT
iajs-2809	136	31	12	12	NUM
iajs-2809	136	32	)	)	PUNCT
iajs-2809	136	33	𝑏∗1	𝑏∗1	NOUN
iajs-2809	136	34	𝑙	𝑙	DET
iajs-2809	136	35	𝑢1	𝑢1	PROPN
iajs-2809	136	36	+	+	CCONJ
iajs-2809	136	37	⋯	⋯	PROPN
iajs-2809	136	38	+	+	CCONJ
iajs-2809	136	39	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	136	40	𝑙	𝑙	X
iajs-2809	136	41	𝑢𝑘	𝑢𝑘	NOUN
iajs-2809	137	1	+	+	CCONJ
iajs-2809	137	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	137	3	𝑙	𝑙	X
iajs-2809	137	4	𝑡	𝑡	NOUN
iajs-2809	137	5	≤	≤	ADV
iajs-2809	137	6	1	1	NUM
iajs-2809	137	7	(	(	PUNCT
iajs-2809	137	8	12	12	NUM
iajs-2809	137	9	)	)	PUNCT
iajs-2809	138	1	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	138	2	+	+	CCONJ
iajs-2809	138	3	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	138	4	+	+	SYM
iajs-2809	138	5	⋯	⋯	PROPN
iajs-2809	138	6	+	+	CCONJ
iajs-2809	138	7	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	138	8	−	−	NOUN
iajs-2809	138	9	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	138	10	≤	≤	NOUN
iajs-2809	138	11	0	0	NUM
iajs-2809	138	12	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	138	13	,	,	PUNCT
iajs-2809	138	14	𝑡	𝑡	X
iajs-2809	138	15	≥	≥	NOUN
iajs-2809	138	16	0	0	NUM
iajs-2809	138	17	,	,	PUNCT
iajs-2809	138	18	𝑖	𝑖	NOUN
iajs-2809	138	19	=	=	SYM
iajs-2809	138	20	1,2	1,2	NUM
iajs-2809	138	21	,	,	PUNCT
iajs-2809	138	22	…	…	PUNCT
iajs-2809	138	23	,	,	PUNCT
iajs-2809	138	24	𝑘	𝑘	X
iajs-2809	138	25	,	,	PUNCT
iajs-2809	138	26	𝛾𝑖	𝛾𝑖	ADV
iajs-2809	138	27	∈	∈	PROPN
iajs-2809	139	1	[	[	X
iajs-2809	139	2	0,1	0,1	NUM
iajs-2809	139	3	]	]	PUNCT
iajs-2809	139	4	,	,	PUNCT
iajs-2809	139	5	𝑖	𝑖	SYM
iajs-2809	139	6	=	=	SYM
iajs-2809	139	7	1,2	1,2	NUM
iajs-2809	139	8	,	,	PUNCT
iajs-2809	139	9	…	…	PUNCT
iajs-2809	139	10	,	,	PUNCT
iajs-2809	139	11	𝑘	𝑘	X
iajs-2809	139	12	+	+	NOUN
iajs-2809	139	13	1	1	X
iajs-2809	139	14	.	.	PUNCT
iajs-2809	140	1	in	in	ADP
iajs-2809	140	2	addition	addition	NOUN
iajs-2809	140	3	,	,	PUNCT
iajs-2809	140	4	if	if	SCONJ
iajs-2809	140	5	we	we	PRON
iajs-2809	140	6	let	let	VERB
iajs-2809	140	7	the	the	DET
iajs-2809	140	8	point	point	NOUN
iajs-2809	140	9	(	(	PUNCT
iajs-2809	140	10	�	�	NOUN
iajs-2809	140	11	̅	̅	NOUN
iajs-2809	140	12	�	�	NOUN
iajs-2809	140	13	1	1	NUM
iajs-2809	140	14	,	,	PUNCT
iajs-2809	140	15	�	�	NOUN
iajs-2809	140	16	̅	̅	NOUN
iajs-2809	140	17	�	�	NOUN
iajs-2809	140	18	2	2	NUM
iajs-2809	140	19	,	,	PUNCT
iajs-2809	140	20	…	…	PUNCT
iajs-2809	140	21	,	,	PUNCT
iajs-2809	140	22	�	�	NOUN
iajs-2809	140	23	̅	̅	NOUN
iajs-2809	140	24	�	�	X
iajs-2809	140	25	𝑘	𝑘	NOUN
iajs-2809	140	26	,	,	PUNCT
iajs-2809	140	27	𝑡̅	𝑡̅	PROPN
iajs-2809	140	28	)	)	PUNCT
iajs-2809	140	29	of	of	ADP
iajs-2809	140	30	the	the	DET
iajs-2809	140	31	feasible	feasible	ADJ
iajs-2809	140	32	region	region	NOUN
iajs-2809	140	33	from	from	ADP
iajs-2809	140	34	the	the	DET
iajs-2809	140	35	problem	problem	NOUN
iajs-2809	140	36	(	(	PUNCT
iajs-2809	140	37	12	12	NUM
iajs-2809	140	38	)	)	PUNCT
iajs-2809	140	39	,	,	PUNCT
iajs-2809	140	40	with	with	ADP
iajs-2809	140	41	𝛾𝑖	𝛾𝑖	ADV
iajs-2809	140	42	∈	∈	PROPN
iajs-2809	140	43	[	[	X
iajs-2809	140	44	0,1	0,1	NUM
iajs-2809	140	45	]	]	PUNCT
iajs-2809	140	46	,	,	PUNCT
iajs-2809	140	47	(	(	PUNCT
iajs-2809	140	48	𝑎∗𝑖	𝑎∗𝑖	NOUN
iajs-2809	140	49	𝑙	𝑙	PRON
iajs-2809	140	50	−	−	PROPN
iajs-2809	140	51	𝑎∗𝑖	𝑎∗𝑖	NUM
iajs-2809	140	52	𝑢	𝑢	X
iajs-2809	140	53	)	)	PUNCT
iajs-2809	140	54	≤	≤	NOUN
iajs-2809	140	55	0	0	NUM
iajs-2809	140	56	for𝑖	for𝑖	NOUN
iajs-2809	140	57	=	=	SYM
iajs-2809	140	58	1,2	1,2	NUM
iajs-2809	140	59	,	,	PUNCT
iajs-2809	140	60	…	…	PUNCT
iajs-2809	140	61	,	,	PUNCT
iajs-2809	140	62	𝑘	𝑘	X
iajs-2809	140	63	+	+	NOUN
iajs-2809	140	64	1	1	NUM
iajs-2809	140	65	.	.	PUNCT
iajs-2809	140	66	before	before	SCONJ
iajs-2809	140	67	the	the	DET
iajs-2809	140	68	objective	objective	NOUN
iajs-2809	140	69	of	of	ADP
iajs-2809	140	70	the	the	DET
iajs-2809	140	71	problem	problem	NOUN
iajs-2809	140	72	(	(	PUNCT
iajs-2809	140	73	12	12	NUM
iajs-2809	140	74	)	)	PUNCT
iajs-2809	140	75	can	can	AUX
iajs-2809	140	76	be	be	AUX
iajs-2809	140	77	written	write	VERB
iajs-2809	140	78	as	as	ADP
iajs-2809	140	79	:	:	PUNCT
iajs-2809	140	80	[	[	X
iajs-2809	140	81	𝛾1(𝑎∗1	𝛾1(𝑎∗1	PROPN
iajs-2809	140	82	𝑙	𝑙	PROPN
iajs-2809	140	83	−	−	PROPN
iajs-2809	140	84	𝑎∗1	𝑎∗1	ADJ
iajs-2809	140	85	𝑢	𝑢	NOUN
iajs-2809	140	86	)	)	PUNCT
iajs-2809	140	87	]	]	PUNCT
iajs-2809	140	88	�	�	SYM
iajs-2809	140	89	̅	̅	NOUN
iajs-2809	140	90	�	�	NOUN
iajs-2809	140	91	1	1	NUM
iajs-2809	140	92	+	+	SYM
iajs-2809	140	93	⋯	⋯	VERB
iajs-2809	140	94	+	+	CCONJ
iajs-2809	141	1	[	[	X
iajs-2809	141	2	𝛾𝑘(𝑎∗𝑘	𝛾𝑘(𝑎∗𝑘	X
iajs-2809	141	3	𝑙	𝑙	X
iajs-2809	141	4	−	−	PROPN
iajs-2809	141	5	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	141	6	𝑢	𝑢	NOUN
iajs-2809	141	7	)	)	PUNCT
iajs-2809	141	8	]	]	PUNCT
iajs-2809	141	9	�	�	SYM
iajs-2809	141	10	̅	̅	NOUN
iajs-2809	141	11	�	�	NOUN
iajs-2809	141	12	𝑘	𝑘	NOUN
iajs-2809	141	13	+	+	PROPN
iajs-2809	142	1	[	[	X
iajs-2809	142	2	𝛾𝑘+1(𝑎∗𝑘+1	𝛾𝑘+1(𝑎∗𝑘+1	NOUN
iajs-2809	142	3	𝑙	𝑙	PRON
iajs-2809	142	4	−	−	PROPN
iajs-2809	142	5	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	142	6	𝑢	𝑢	NOUN
iajs-2809	142	7	)	)	PUNCT
iajs-2809	142	8	]	]	PUNCT
iajs-2809	142	9	𝑡̅	𝑡̅	X
iajs-2809	142	10	+	+	CCONJ
iajs-2809	142	11	𝑎∗1	𝑎∗1	ADJ
iajs-2809	142	12	𝑢	𝑢	X
iajs-2809	142	13	�	�	PROPN
iajs-2809	142	14	̅	̅	NOUN
iajs-2809	142	15	�	�	NOUN
iajs-2809	142	16	1	1	NUM
iajs-2809	142	17	+	+	SYM
iajs-2809	142	18	⋯	⋯	VERB
iajs-2809	142	19	+	+	CCONJ
iajs-2809	142	20	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	142	21	𝑢	𝑢	PRON
iajs-2809	142	22	�	�	PROPN
iajs-2809	142	23	̅	̅	NOUN
iajs-2809	142	24	�	�	NOUN
iajs-2809	142	25	𝑘	𝑘	DET
iajs-2809	142	26	+	+	NOUN
iajs-2809	142	27	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	142	28	𝑢	𝑢	PROPN
iajs-2809	142	29	𝑡̅	𝑡̅	PROPN
iajs-2809	142	30	≥	≥	NOUN
iajs-2809	142	31	[	[	X
iajs-2809	142	32	(	(	PUNCT
iajs-2809	142	33	𝑎∗1	𝑎∗1	ADJ
iajs-2809	142	34	𝑙	𝑙	PRON
iajs-2809	142	35	−	−	PROPN
iajs-2809	142	36	𝑎∗1	𝑎∗1	ADJ
iajs-2809	142	37	𝑢	𝑢	NOUN
iajs-2809	142	38	)	)	PUNCT
iajs-2809	142	39	]	]	PUNCT
iajs-2809	142	40	�	�	SYM
iajs-2809	142	41	̅	̅	NOUN
iajs-2809	142	42	�	�	NOUN
iajs-2809	142	43	1	1	NUM
iajs-2809	142	44	+	+	SYM
iajs-2809	142	45	⋯	⋯	VERB
iajs-2809	142	46	+	+	CCONJ
iajs-2809	143	1	[	[	X
iajs-2809	143	2	(	(	PUNCT
iajs-2809	143	3	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	143	4	𝑙	𝑙	X
iajs-2809	143	5	−	−	DET
iajs-2809	143	6	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	143	7	𝑢	𝑢	NOUN
iajs-2809	143	8	)	)	PUNCT
iajs-2809	143	9	]	]	PUNCT
iajs-2809	143	10	�	�	SYM
iajs-2809	143	11	̅	̅	NOUN
iajs-2809	143	12	�	�	NOUN
iajs-2809	143	13	𝑘	𝑘	NOUN
iajs-2809	143	14	+	+	X
iajs-2809	144	1	[	[	X
iajs-2809	144	2	(	(	PUNCT
iajs-2809	144	3	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	144	4	𝑙	𝑙	PRON
iajs-2809	144	5	−	−	PROPN
iajs-2809	144	6	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	144	7	𝑢	𝑢	NOUN
iajs-2809	144	8	)	)	PUNCT
iajs-2809	144	9	]	]	PUNCT
iajs-2809	145	1	𝑡̅	𝑡̅	X
iajs-2809	145	2	+	+	CCONJ
iajs-2809	145	3	𝑎∗1	𝑎∗1	ADJ
iajs-2809	145	4	𝑢	𝑢	X
iajs-2809	145	5	�	�	PROPN
iajs-2809	145	6	̅	̅	NOUN
iajs-2809	145	7	�	�	NOUN
iajs-2809	145	8	1	1	NUM
iajs-2809	145	9	+	+	SYM
iajs-2809	145	10	⋯	⋯	VERB
iajs-2809	145	11	+	+	CCONJ
iajs-2809	145	12	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	145	13	𝑢	𝑢	PRON
iajs-2809	145	14	�	�	PROPN
iajs-2809	145	15	̅	̅	NOUN
iajs-2809	145	16	�	�	NOUN
iajs-2809	145	17	𝑘	𝑘	PRON
iajs-2809	145	18	+	+	NOUN
iajs-2809	145	19	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	145	20	𝑢	𝑢	ADJ
iajs-2809	145	21	𝑡̅	𝑡̅	PROPN
iajs-2809	145	22	=	=	SYM
iajs-2809	145	23	𝑎∗1	𝑎∗1	ADJ
iajs-2809	145	24	𝑙	𝑙	PRON
iajs-2809	145	25	�	�	NOUN
iajs-2809	145	26	̅	̅	NOUN
iajs-2809	145	27	�	�	NOUN
iajs-2809	145	28	1	1	NUM
iajs-2809	145	29	+	+	SYM
iajs-2809	145	30	⋯	⋯	VERB
iajs-2809	145	31	+	+	CCONJ
iajs-2809	145	32	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	145	33	𝑙	𝑙	PRON
iajs-2809	145	34	�	�	PROPN
iajs-2809	145	35	̅	̅	NOUN
iajs-2809	145	36	�	�	NOUN
iajs-2809	145	37	𝑘	𝑘	PRON
iajs-2809	145	38	+	+	NOUN
iajs-2809	145	39	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	145	40	𝑙	𝑙	PROPN
iajs-2809	145	41	𝑡	𝑡	PROPN
iajs-2809	145	42	̅	̅	NOUN
iajs-2809	145	43	the	the	DET
iajs-2809	145	44	right	right	ADJ
iajs-2809	145	45	-	-	PUNCT
iajs-2809	145	46	hand	hand	NOUN
iajs-2809	145	47	side	side	NOUN
iajs-2809	145	48	of	of	ADP
iajs-2809	145	49	the	the	DET
iajs-2809	145	50	above	above	ADJ
iajs-2809	145	51	equality	equality	NOUN
iajs-2809	145	52	proves	prove	VERB
iajs-2809	145	53	that	that	DET
iajs-2809	145	54	𝑎∗1	𝑎∗1	NOUN
iajs-2809	145	55	𝑙	𝑙	X
iajs-2809	145	56	,	,	PUNCT
iajs-2809	145	57	…	…	PUNCT
iajs-2809	145	58	,	,	PUNCT
iajs-2809	145	59	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	145	60	𝑙	𝑙	X
iajs-2809	145	61	,	,	PUNCT
iajs-2809	145	62	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	145	63	𝑙	𝑙	PRON
iajs-2809	145	64	is	be	AUX
iajs-2809	145	65	the	the	DET
iajs-2809	145	66	surely	surely	ADV
iajs-2809	145	67	lower	low	ADJ
iajs-2809	145	68	limit	limit	NOUN
iajs-2809	145	69	of	of	ADP
iajs-2809	145	70	the	the	DET
iajs-2809	145	71	interval	interval	NOUN
iajs-2809	145	72	coefficients	coefficient	NOUN
iajs-2809	145	73	in	in	ADP
iajs-2809	145	74	the	the	DET
iajs-2809	145	75	objective	objective	ADJ
iajs-2809	145	76	function	function	NOUN
iajs-2809	145	77	,	,	PUNCT
iajs-2809	145	78	and	and	CCONJ
iajs-2809	145	79	the	the	DET
iajs-2809	145	80	same	same	ADJ
iajs-2809	145	81	situation	situation	NOUN
iajs-2809	145	82	of	of	ADP
iajs-2809	145	83	the	the	DET
iajs-2809	145	84	problem	problem	NOUN
iajs-2809	145	85	(	(	PUNCT
iajs-2809	145	86	12	12	NUM
iajs-2809	145	87	)	)	PUNCT
iajs-2809	145	88	,	,	PUNCT
iajs-2809	145	89	with	with	ADP
iajs-2809	145	90	𝛾𝑖	𝛾𝑖	ADV
iajs-2809	145	91	∈	∈	PROPN
iajs-2809	146	1	[	[	X
iajs-2809	146	2	0,1	0,1	NUM
iajs-2809	146	3	]	]	PUNCT
iajs-2809	146	4	,	,	PUNCT
iajs-2809	146	5	(	(	PUNCT
iajs-2809	146	6	𝑎∗𝑖	𝑎∗𝑖	VERB
iajs-2809	146	7	𝑢	𝑢	DET
iajs-2809	146	8	−	−	PROPN
iajs-2809	146	9	𝑎∗𝑖	𝑎∗𝑖	SYM
iajs-2809	146	10	𝑙	𝑙	X
iajs-2809	146	11	)	)	PUNCT
iajs-2809	146	12	≥	≥	NOUN
iajs-2809	146	13	0	0	NUM
iajs-2809	146	14	for𝑖	for𝑖	NOUN
iajs-2809	146	15	=	=	SYM
iajs-2809	146	16	1,2	1,2	NUM
iajs-2809	146	17	,	,	PUNCT
iajs-2809	146	18	…	…	PUNCT
iajs-2809	146	19	,	,	PUNCT
iajs-2809	146	20	𝑘	𝑘	X
iajs-2809	146	21	+	+	NOUN
iajs-2809	146	22	1	1	NUM
iajs-2809	146	23	.	.	PUNCT
iajs-2809	146	24	before	before	ADP
iajs-2809	146	25	the	the	DET
iajs-2809	146	26	objective	objective	NOUN
iajs-2809	146	27	of	of	ADP
iajs-2809	146	28	the	the	DET
iajs-2809	146	29	problem	problem	NOUN
iajs-2809	146	30	(	(	PUNCT
iajs-2809	146	31	12	12	NUM
iajs-2809	146	32	)	)	PUNCT
iajs-2809	146	33	can	can	AUX
iajs-2809	146	34	be	be	AUX
iajs-2809	146	35	written	write	VERB
iajs-2809	146	36	as	as	ADP
iajs-2809	146	37	:	:	PUNCT
iajs-2809	146	38	ibn	ibn	PROPN
iajs-2809	146	39	al	al	PROPN
iajs-2809	146	40	-	-	PUNCT
iajs-2809	146	41	haitham	haitham	PROPN
iajs-2809	146	42	jour	jour	X
iajs-2809	146	43	.	.	PROPN
iajs-2809	146	44	for	for	ADP
iajs-2809	146	45	pure	pure	ADJ
iajs-2809	146	46	&	&	CCONJ
iajs-2809	146	47	appl	appl	PROPN
iajs-2809	146	48	.	.	PUNCT
iajs-2809	147	1	sci	sci	PROPN
iajs-2809	147	2	.	.	PROPN
iajs-2809	148	1	53	53	NUM
iajs-2809	148	2	(	(	PUNCT
iajs-2809	148	3	2)2022	2)2022	NUM
iajs-2809	148	4	76	76	NUM
iajs-2809	149	1	[	[	X
iajs-2809	149	2	𝛾1(𝑎∗1	𝛾1(𝑎∗1	PROPN
iajs-2809	149	3	𝑢	𝑢	NOUN
iajs-2809	149	4	−	−	PROPN
iajs-2809	149	5	𝑎∗1	𝑎∗1	NOUN
iajs-2809	149	6	𝑙	𝑙	PROPN
iajs-2809	149	7	)	)	PUNCT
iajs-2809	149	8	]	]	PUNCT
iajs-2809	149	9	�	�	SYM
iajs-2809	149	10	̅	̅	NOUN
iajs-2809	149	11	�	�	NOUN
iajs-2809	149	12	1	1	NUM
iajs-2809	149	13	+	+	SYM
iajs-2809	149	14	⋯	⋯	VERB
iajs-2809	149	15	+	+	CCONJ
iajs-2809	150	1	[	[	X
iajs-2809	150	2	𝛾𝑘(𝑎∗𝑘	𝛾𝑘(𝑎∗𝑘	X
iajs-2809	150	3	𝑢	𝑢	X
iajs-2809	150	4	−	−	PROPN
iajs-2809	150	5	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	150	6	𝑙	𝑙	PROPN
iajs-2809	150	7	)	)	PUNCT
iajs-2809	150	8	]	]	PUNCT
iajs-2809	150	9	�	�	SYM
iajs-2809	150	10	̅	̅	NOUN
iajs-2809	150	11	�	�	NOUN
iajs-2809	150	12	𝑘	𝑘	NOUN
iajs-2809	150	13	+	+	PROPN
iajs-2809	150	14	[	[	X
iajs-2809	150	15	𝛾𝑘+1(𝑎∗𝑘+1	𝛾𝑘+1(𝑎∗𝑘+1	NOUN
iajs-2809	150	16	𝑢	𝑢	PART
iajs-2809	150	17	−	−	PROPN
iajs-2809	150	18	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	150	19	𝑙	𝑙	PROPN
iajs-2809	150	20	)	)	PUNCT
iajs-2809	150	21	]	]	PUNCT
iajs-2809	150	22	𝑡̅	𝑡̅	ADP
iajs-2809	150	23	+	+	CCONJ
iajs-2809	150	24	𝑎∗1	𝑎∗1	ADJ
iajs-2809	150	25	𝑙	𝑙	PRON
iajs-2809	150	26	�	�	NOUN
iajs-2809	150	27	̅	̅	NOUN
iajs-2809	150	28	�	�	NOUN
iajs-2809	150	29	1	1	NUM
iajs-2809	150	30	+	+	SYM
iajs-2809	150	31	⋯	⋯	VERB
iajs-2809	150	32	+	+	CCONJ
iajs-2809	150	33	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	150	34	𝑙	𝑙	PRON
iajs-2809	150	35	�	�	PROPN
iajs-2809	150	36	̅	̅	NOUN
iajs-2809	150	37	�	�	NOUN
iajs-2809	150	38	𝑘	𝑘	PRON
iajs-2809	150	39	+	+	NOUN
iajs-2809	150	40	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	150	41	𝑙	𝑙	PROPN
iajs-2809	150	42	𝑡̅	𝑡̅	PROPN
iajs-2809	150	43	≥	≥	NOUN
iajs-2809	151	1	[	[	X
iajs-2809	151	2	(	(	PUNCT
iajs-2809	151	3	𝑎∗1	𝑎∗1	ADJ
iajs-2809	151	4	𝑢	𝑢	PRON
iajs-2809	151	5	−	−	PROPN
iajs-2809	151	6	𝑎∗1	𝑎∗1	NOUN
iajs-2809	151	7	𝑙	𝑙	PROPN
iajs-2809	151	8	)	)	PUNCT
iajs-2809	151	9	]	]	PUNCT
iajs-2809	151	10	�	�	SYM
iajs-2809	151	11	̅	̅	NOUN
iajs-2809	151	12	�	�	NOUN
iajs-2809	151	13	1	1	NUM
iajs-2809	151	14	+	+	SYM
iajs-2809	151	15	⋯	⋯	VERB
iajs-2809	151	16	+	+	CCONJ
iajs-2809	151	17	[	[	X
iajs-2809	151	18	(	(	PUNCT
iajs-2809	151	19	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	151	20	𝑢	𝑢	X
iajs-2809	151	21	−	−	PROPN
iajs-2809	151	22	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	151	23	𝑙	𝑙	PROPN
iajs-2809	151	24	)	)	PUNCT
iajs-2809	151	25	]	]	PUNCT
iajs-2809	151	26	�	�	SYM
iajs-2809	151	27	̅	̅	NOUN
iajs-2809	151	28	�	�	NOUN
iajs-2809	151	29	𝑘	𝑘	NOUN
iajs-2809	151	30	+	+	X
iajs-2809	151	31	[	[	X
iajs-2809	151	32	(	(	PUNCT
iajs-2809	151	33	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	151	34	𝑢	𝑢	PROPN
iajs-2809	151	35	−	−	PROPN
iajs-2809	151	36	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	151	37	𝑙	𝑙	PROPN
iajs-2809	151	38	)	)	PUNCT
iajs-2809	152	1	]	]	PUNCT
iajs-2809	152	2	𝑡̅	𝑡̅	ADP
iajs-2809	152	3	+	+	CCONJ
iajs-2809	152	4	𝑎∗1	𝑎∗1	ADJ
iajs-2809	152	5	𝑙	𝑙	PRON
iajs-2809	152	6	�	�	NOUN
iajs-2809	152	7	̅	̅	NOUN
iajs-2809	152	8	�	�	NOUN
iajs-2809	152	9	1	1	NUM
iajs-2809	152	10	+	+	SYM
iajs-2809	152	11	⋯	⋯	VERB
iajs-2809	152	12	+	+	CCONJ
iajs-2809	152	13	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	152	14	𝑙	𝑙	PRON
iajs-2809	152	15	�	�	PROPN
iajs-2809	152	16	̅	̅	NOUN
iajs-2809	152	17	�	�	NOUN
iajs-2809	152	18	𝑘	𝑘	PRON
iajs-2809	152	19	+	+	NOUN
iajs-2809	152	20	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	152	21	𝑙	𝑙	PRON
iajs-2809	152	22	𝑡̅	𝑡̅	PROPN
iajs-2809	152	23	=	=	SYM
iajs-2809	152	24	𝑎∗1	𝑎∗1	PROPN
iajs-2809	152	25	𝑢	𝑢	X
iajs-2809	152	26	�	�	PROPN
iajs-2809	152	27	̅	̅	NOUN
iajs-2809	152	28	�	�	NOUN
iajs-2809	152	29	1	1	NUM
iajs-2809	152	30	+	+	SYM
iajs-2809	152	31	⋯	⋯	VERB
iajs-2809	152	32	+	+	CCONJ
iajs-2809	152	33	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	152	34	𝑢	𝑢	PRON
iajs-2809	152	35	�	�	PROPN
iajs-2809	152	36	̅	̅	NOUN
iajs-2809	152	37	�	�	NOUN
iajs-2809	152	38	𝑘	𝑘	DET
iajs-2809	152	39	+	+	NOUN
iajs-2809	152	40	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	152	41	𝑢	𝑢	PROPN
iajs-2809	152	42	𝑡	𝑡	PROPN
iajs-2809	152	43	̅	̅	NOUN
iajs-2809	152	44	the	the	DET
iajs-2809	152	45	right	right	ADJ
iajs-2809	152	46	hand	hand	NOUN
iajs-2809	152	47	side	side	NOUN
iajs-2809	152	48	of	of	ADP
iajs-2809	152	49	the	the	DET
iajs-2809	152	50	above	above	ADJ
iajs-2809	152	51	equality	equality	NOUN
iajs-2809	152	52	proves	prove	VERB
iajs-2809	152	53	that	that	DET
iajs-2809	152	54	𝑎∗1	𝑎∗1	ADJ
iajs-2809	152	55	𝑢	𝑢	NOUN
iajs-2809	152	56	,	,	PUNCT
iajs-2809	152	57	…	…	PUNCT
iajs-2809	152	58	,	,	PUNCT
iajs-2809	152	59	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	152	60	𝑢	𝑢	NOUN
iajs-2809	152	61	,	,	PUNCT
iajs-2809	152	62	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	152	63	𝑢	𝑢	NOUN
iajs-2809	152	64	is	be	AUX
iajs-2809	152	65	the	the	DET
iajs-2809	152	66	surely	surely	ADV
iajs-2809	152	67	upper	upper	ADJ
iajs-2809	152	68	limit	limit	NOUN
iajs-2809	152	69	of	of	ADP
iajs-2809	152	70	the	the	DET
iajs-2809	152	71	interval	interval	NOUN
iajs-2809	152	72	coefficient	coefficient	NOUN
iajs-2809	152	73	in	in	ADP
iajs-2809	152	74	the	the	DET
iajs-2809	152	75	objective	objective	ADJ
iajs-2809	152	76	function	function	NOUN
iajs-2809	152	77	.	.	PUNCT
iajs-2809	153	1	now	now	ADV
iajs-2809	153	2	,	,	PUNCT
iajs-2809	153	3	the	the	DET
iajs-2809	153	4	surely	surely	ADV
iajs-2809	153	5	best	good	ADJ
iajs-2809	153	6	optimum	optimum	ADJ
iajs-2809	153	7	objective	objective	ADJ
iajs-2809	153	8	function	function	NOUN
iajs-2809	153	9	is	be	AUX
iajs-2809	153	10	𝑍	𝑍	NOUN
iajs-2809	153	11	=	=	SYM
iajs-2809	153	12	𝑎∗1	𝑎∗1	ADJ
iajs-2809	153	13	𝑢	𝑢	PROPN
iajs-2809	153	14	𝑢1	𝑢1	PROPN
iajs-2809	153	15	+	+	CCONJ
iajs-2809	153	16	⋯	⋯	PROPN
iajs-2809	153	17	+	+	CCONJ
iajs-2809	153	18	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	153	19	𝑢	𝑢	X
iajs-2809	153	20	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	153	21	+	+	NOUN
iajs-2809	153	22	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	153	23	𝑢	𝑢	PROPN
iajs-2809	153	24	𝑡	𝑡	PROPN
iajs-2809	153	25	and	and	CCONJ
iajs-2809	153	26	the	the	DET
iajs-2809	153	27	surely	surely	ADV
iajs-2809	153	28	worst	bad	ADJ
iajs-2809	153	29	optimum	optimum	ADJ
iajs-2809	153	30	objective	objective	ADJ
iajs-2809	153	31	function	function	NOUN
iajs-2809	153	32	is𝑍	is𝑍	NOUN
iajs-2809	153	33	=	=	SYM
iajs-2809	153	34	𝑎∗1	𝑎∗1	ADJ
iajs-2809	153	35	𝑙	𝑙	DET
iajs-2809	153	36	𝑢1	𝑢1	PROPN
iajs-2809	153	37	+	+	CCONJ
iajs-2809	153	38	⋯	⋯	PROPN
iajs-2809	153	39	+	+	NOUN
iajs-2809	153	40	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	153	41	𝑙	𝑙	X
iajs-2809	153	42	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	153	43	+	+	PROPN
iajs-2809	153	44	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	153	45	𝑙	𝑙	DET
iajs-2809	153	46	𝑡.	𝑡.	NOUN
iajs-2809	153	47	while	while	NOUN
iajs-2809	153	48	,	,	PUNCT
iajs-2809	153	49	to	to	PART
iajs-2809	153	50	get	get	VERB
iajs-2809	153	51	the	the	DET
iajs-2809	153	52	surely	surely	ADV
iajs-2809	153	53	worst	bad	ADJ
iajs-2809	153	54	optimum	optimum	ADJ
iajs-2809	153	55	taken	take	VERB
iajs-2809	153	56	of	of	ADP
iajs-2809	153	57	the	the	DET
iajs-2809	153	58	surely	surely	ADV
iajs-2809	153	59	lower	low	ADJ
iajs-2809	153	60	limit	limit	NOUN
iajs-2809	153	61	of	of	ADP
iajs-2809	153	62	the	the	DET
iajs-2809	153	63	interval	interval	NOUN
iajs-2809	153	64	coefficients	coefficient	NOUN
iajs-2809	153	65	in	in	ADP
iajs-2809	153	66	the	the	DET
iajs-2809	153	67	objective	objective	ADJ
iajs-2809	153	68	function	function	NOUN
iajs-2809	153	69	,	,	PUNCT
iajs-2809	153	70	thus	thus	ADV
iajs-2809	153	71	the	the	DET
iajs-2809	153	72	conversions	conversion	NOUN
iajs-2809	153	73	formula	formula	NOUN
iajs-2809	153	74	of	of	ADP
iajs-2809	153	75	lfp	lfp	NOUN
iajs-2809	153	76	problems	problem	NOUN
iajs-2809	153	77	with	with	ADP
iajs-2809	153	78	rough	rough	ADJ
iajs-2809	153	79	interval	interval	NOUN
iajs-2809	153	80	coefficients	coefficient	NOUN
iajs-2809	153	81	in	in	ADP
iajs-2809	153	82	the	the	DET
iajs-2809	153	83	objective	objective	ADJ
iajs-2809	153	84	function	function	NOUN
iajs-2809	153	85	of	of	ADP
iajs-2809	153	86	the	the	DET
iajs-2809	153	87	surely	surely	ADV
iajs-2809	153	88	lower	low	ADJ
iajs-2809	153	89	limit	limit	NOUN
iajs-2809	153	90	,	,	PUNCT
iajs-2809	153	91	can	can	AUX
iajs-2809	153	92	be	be	AUX
iajs-2809	153	93	written	write	VERB
iajs-2809	153	94	as	as	SCONJ
iajs-2809	153	95	follows	follow	VERB
iajs-2809	153	96	:	:	PUNCT
iajs-2809	153	97	but	but	CCONJ
iajs-2809	153	98	in	in	ADP
iajs-2809	153	99	this	this	DET
iajs-2809	153	100	case	case	NOUN
iajs-2809	153	101	,	,	PUNCT
iajs-2809	153	102	assume	assume	VERB
iajs-2809	153	103	that	that	SCONJ
iajs-2809	153	104	every	every	DET
iajs-2809	153	105	constraint	constraint	NOUN
iajs-2809	153	106	is	be	AUX
iajs-2809	153	107	smaller	small	ADJ
iajs-2809	153	108	than	than	ADP
iajs-2809	153	109	or	or	CCONJ
iajs-2809	153	110	equal	equal	ADJ
iajs-2809	153	111	.	.	PUNCT
iajs-2809	154	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	154	2	𝑍	𝑍	PROPN
iajs-2809	154	3	=	=	SYM
iajs-2809	154	4	𝑎∗1	𝑎∗1	NOUN
iajs-2809	154	5	𝑙	𝑙	DET
iajs-2809	154	6	𝑢1	𝑢1	PROPN
iajs-2809	154	7	+	+	CCONJ
iajs-2809	154	8	⋯	⋯	PROPN
iajs-2809	154	9	+	+	NOUN
iajs-2809	154	10	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	154	11	𝑙	𝑙	X
iajs-2809	154	12	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	154	13	+	+	PROPN
iajs-2809	154	14	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	154	15	𝑙	𝑙	X
iajs-2809	154	16	𝑡	𝑡	PROPN
iajs-2809	154	17	subject	subject	ADJ
iajs-2809	154	18	to	to	ADP
iajs-2809	154	19	:	:	PUNCT
iajs-2809	154	20	𝑏∗1	𝑏∗1	PROPN
iajs-2809	154	21	𝑢	𝑢	PRON
iajs-2809	154	22	𝑢1	𝑢1	PROPN
iajs-2809	154	23	+	+	CCONJ
iajs-2809	154	24	⋯	⋯	PROPN
iajs-2809	154	25	+	+	CCONJ
iajs-2809	154	26	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	154	27	𝑢	𝑢	X
iajs-2809	154	28	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	155	1	+	+	CCONJ
iajs-2809	155	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	155	3	𝑢	𝑢	X
iajs-2809	155	4	𝑡	𝑡	PROPN
iajs-2809	155	5	≤	≤	NUM
iajs-2809	155	6	1	1	NUM
iajs-2809	155	7	𝑏∗1	𝑏∗1	NOUN
iajs-2809	155	8	𝑙	𝑙	DET
iajs-2809	155	9	𝑢1	𝑢1	PROPN
iajs-2809	155	10	+	+	CCONJ
iajs-2809	155	11	⋯	⋯	PROPN
iajs-2809	155	12	+	+	CCONJ
iajs-2809	155	13	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	155	14	𝑙	𝑙	X
iajs-2809	155	15	𝑢𝑘	𝑢𝑘	NOUN
iajs-2809	156	1	+	+	CCONJ
iajs-2809	156	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	156	3	𝑙	𝑙	X
iajs-2809	156	4	𝑡	𝑡	NOUN
iajs-2809	156	5	≤	≤	ADV
iajs-2809	156	6	1	1	NUM
iajs-2809	156	7	(	(	PUNCT
iajs-2809	156	8	13	13	NUM
iajs-2809	156	9	)	)	PUNCT
iajs-2809	156	10	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	156	11	+	+	CCONJ
iajs-2809	157	1	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	158	1	+	+	SYM
iajs-2809	159	1	⋯	⋯	PROPN
iajs-2809	159	2	+	+	CCONJ
iajs-2809	159	3	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	159	4	−	−	NOUN
iajs-2809	159	5	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	159	6	≤	≤	NOUN
iajs-2809	159	7	0	0	NUM
iajs-2809	159	8	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	159	9	,	,	PUNCT
iajs-2809	159	10	𝑡	𝑡	X
iajs-2809	159	11	≥	≥	NOUN
iajs-2809	159	12	0	0	NUM
iajs-2809	159	13	,	,	PUNCT
iajs-2809	159	14	𝑖	𝑖	NOUN
iajs-2809	159	15	=	=	SYM
iajs-2809	159	16	1,2	1,2	NUM
iajs-2809	159	17	,	,	PUNCT
iajs-2809	159	18	…	…	PUNCT
iajs-2809	159	19	,	,	PUNCT
iajs-2809	159	20	𝑘.	𝑘.	NOUN
iajs-2809	159	21	and	and	CCONJ
iajs-2809	159	22	while	while	SCONJ
iajs-2809	159	23	,	,	PUNCT
iajs-2809	159	24	to	to	PART
iajs-2809	159	25	get	get	VERB
iajs-2809	159	26	the	the	DET
iajs-2809	159	27	surely	surely	ADV
iajs-2809	159	28	best	good	ADJ
iajs-2809	159	29	optimum	optimum	ADJ
iajs-2809	159	30	taken	take	VERB
iajs-2809	159	31	of	of	ADP
iajs-2809	159	32	the	the	DET
iajs-2809	159	33	surely	surely	ADV
iajs-2809	159	34	upper	upper	ADJ
iajs-2809	159	35	limit	limit	NOUN
iajs-2809	159	36	of	of	ADP
iajs-2809	159	37	the	the	DET
iajs-2809	159	38	interval	interval	NOUN
iajs-2809	159	39	coefficients	coefficient	NOUN
iajs-2809	159	40	in	in	ADP
iajs-2809	159	41	the	the	DET
iajs-2809	159	42	objective	objective	ADJ
iajs-2809	159	43	function	function	NOUN
iajs-2809	159	44	,	,	PUNCT
iajs-2809	159	45	thus	thus	ADV
iajs-2809	159	46	the	the	DET
iajs-2809	159	47	conversions	conversion	NOUN
iajs-2809	159	48	formula	formula	NOUN
iajs-2809	159	49	of	of	ADP
iajs-2809	159	50	lfp	lfp	NOUN
iajs-2809	159	51	problems	problem	NOUN
iajs-2809	159	52	with	with	ADP
iajs-2809	159	53	rough	rough	ADJ
iajs-2809	159	54	interval	interval	NOUN
iajs-2809	159	55	coefficients	coefficient	NOUN
iajs-2809	159	56	in	in	ADP
iajs-2809	159	57	the	the	DET
iajs-2809	159	58	objective	objective	ADJ
iajs-2809	159	59	function	function	NOUN
iajs-2809	159	60	is	be	AUX
iajs-2809	159	61	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	159	62	𝑍	𝑍	PROPN
iajs-2809	159	63	=	=	SYM
iajs-2809	159	64	𝑎∗1	𝑎∗1	NOUN
iajs-2809	159	65	𝑢	𝑢	PROPN
iajs-2809	159	66	𝑢1	𝑢1	PROPN
iajs-2809	159	67	+	+	CCONJ
iajs-2809	159	68	⋯	⋯	PROPN
iajs-2809	159	69	+	+	CCONJ
iajs-2809	159	70	𝑎∗𝑘	𝑎∗𝑘	PROPN
iajs-2809	159	71	𝑢	𝑢	X
iajs-2809	159	72	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	159	73	+	+	NOUN
iajs-2809	159	74	𝑎∗𝑘+1	𝑎∗𝑘+1	NOUN
iajs-2809	159	75	𝑢	𝑢	X
iajs-2809	159	76	𝑡	𝑡	NOUN
iajs-2809	159	77	subject	subject	ADJ
iajs-2809	159	78	to	to	ADP
iajs-2809	159	79	:	:	PUNCT
iajs-2809	159	80	𝑏∗1	𝑏∗1	PROPN
iajs-2809	159	81	𝑢	𝑢	PRON
iajs-2809	159	82	𝑢1	𝑢1	PROPN
iajs-2809	159	83	+	+	CCONJ
iajs-2809	159	84	⋯	⋯	PROPN
iajs-2809	159	85	+	+	CCONJ
iajs-2809	159	86	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	159	87	𝑢	𝑢	X
iajs-2809	159	88	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	160	1	+	+	CCONJ
iajs-2809	160	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	160	3	𝑢	𝑢	X
iajs-2809	160	4	𝑡	𝑡	PROPN
iajs-2809	160	5	≥	≥	VERB
iajs-2809	160	6	1	1	NUM
iajs-2809	160	7	𝑏∗1	𝑏∗1	ADJ
iajs-2809	160	8	𝑙	𝑙	DET
iajs-2809	160	9	𝑢1	𝑢1	PROPN
iajs-2809	160	10	+	+	CCONJ
iajs-2809	160	11	⋯	⋯	PROPN
iajs-2809	160	12	+	+	CCONJ
iajs-2809	160	13	𝑏∗𝑘	𝑏∗𝑘	PROPN
iajs-2809	160	14	𝑙	𝑙	X
iajs-2809	160	15	𝑢𝑘	𝑢𝑘	NOUN
iajs-2809	161	1	+	+	CCONJ
iajs-2809	161	2	𝑏∗𝑘+1	𝑏∗𝑘+1	NOUN
iajs-2809	161	3	𝑙	𝑙	X
iajs-2809	161	4	𝑡	𝑡	NOUN
iajs-2809	161	5	≤	≤	ADV
iajs-2809	161	6	1	1	NUM
iajs-2809	161	7	(	(	PUNCT
iajs-2809	161	8	14	14	NUM
iajs-2809	161	9	)	)	PUNCT
iajs-2809	161	10	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	161	11	+	+	CCONJ
iajs-2809	162	1	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	163	1	+	+	SYM
iajs-2809	164	1	⋯	⋯	PROPN
iajs-2809	164	2	+	+	CCONJ
iajs-2809	164	3	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	164	4	−	−	NOUN
iajs-2809	164	5	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	164	6	≤	≤	NOUN
iajs-2809	164	7	0	0	NUM
iajs-2809	164	8	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	164	9	,	,	PUNCT
iajs-2809	164	10	𝑡	𝑡	X
iajs-2809	164	11	≥	≥	NOUN
iajs-2809	164	12	0	0	NUM
iajs-2809	164	13	,	,	PUNCT
iajs-2809	164	14	𝑖	𝑖	NOUN
iajs-2809	164	15	=	=	SYM
iajs-2809	164	16	1,2	1,2	NUM
iajs-2809	164	17	,	,	PUNCT
iajs-2809	164	18	…	…	PUNCT
iajs-2809	164	19	,	,	PUNCT
iajs-2809	164	20	𝑘.	𝑘.	PROPN
iajs-2809	164	21	ibn	ibn	PROPN
iajs-2809	164	22	al	al	PROPN
iajs-2809	164	23	-	-	PUNCT
iajs-2809	164	24	haitham	haitham	PROPN
iajs-2809	164	25	jour	jour	X
iajs-2809	164	26	.	.	PROPN
iajs-2809	165	1	for	for	ADP
iajs-2809	165	2	pure	pure	ADJ
iajs-2809	165	3	&	&	CCONJ
iajs-2809	165	4	appl	appl	PROPN
iajs-2809	165	5	.	.	PUNCT
iajs-2809	166	1	sci	sci	PROPN
iajs-2809	166	2	.	.	PROPN
iajs-2809	167	1	53	53	NUM
iajs-2809	167	2	(	(	PUNCT
iajs-2809	167	3	2)2022	2)2022	VERB
iajs-2809	167	4	77	77	NUM
iajs-2809	167	5	the	the	DET
iajs-2809	167	6	optimal	optimal	ADJ
iajs-2809	167	7	solution	solution	NOUN
iajs-2809	167	8	(	(	PUNCT
iajs-2809	167	9	�	�	NOUN
iajs-2809	167	10	̅	̅	NOUN
iajs-2809	167	11	�	�	NOUN
iajs-2809	167	12	1	1	NUM
iajs-2809	167	13	,	,	PUNCT
iajs-2809	167	14	�	�	NOUN
iajs-2809	167	15	̅	̅	NOUN
iajs-2809	167	16	�	�	NOUN
iajs-2809	167	17	2	2	NUM
iajs-2809	167	18	,	,	PUNCT
iajs-2809	167	19	…	…	PUNCT
iajs-2809	167	20	,	,	PUNCT
iajs-2809	167	21	�	�	NOUN
iajs-2809	167	22	̅	̅	NOUN
iajs-2809	167	23	�	�	SYM
iajs-2809	167	24	𝑘	𝑘	NOUN
iajs-2809	167	25	,	,	PUNCT
iajs-2809	167	26	𝑡)̅	𝑡)̅	NOUN
iajs-2809	167	27	of	of	ADP
iajs-2809	167	28	problems	problem	NOUN
iajs-2809	167	29	(	(	PUNCT
iajs-2809	167	30	13	13	NUM
iajs-2809	167	31	)	)	PUNCT
iajs-2809	167	32	and	and	CCONJ
iajs-2809	167	33	(	(	PUNCT
iajs-2809	167	34	14	14	NUM
iajs-2809	167	35	)	)	PUNCT
iajs-2809	167	36	is	be	AUX
iajs-2809	167	37	the	the	DET
iajs-2809	167	38	same	same	ADJ
iajs-2809	167	39	as	as	ADP
iajs-2809	167	40	the	the	DET
iajs-2809	167	41	optimal	optimal	ADJ
iajs-2809	167	42	solution	solution	NOUN
iajs-2809	167	43	of	of	ADP
iajs-2809	167	44	the	the	DET
iajs-2809	167	45	original	original	ADJ
iajs-2809	167	46	problem	problem	NOUN
iajs-2809	167	47	(	(	PUNCT
iajs-2809	167	48	4	4	NUM
iajs-2809	167	49	)	)	PUNCT
iajs-2809	167	50	which	which	PRON
iajs-2809	167	51	can	can	AUX
iajs-2809	167	52	be	be	AUX
iajs-2809	167	53	easily	easily	ADV
iajs-2809	167	54	obtained	obtain	VERB
iajs-2809	167	55	by	by	ADP
iajs-2809	167	56	(	(	PUNCT
iajs-2809	167	57	𝑥1	𝑥1	NOUN
iajs-2809	167	58	∗	∗	NOUN
iajs-2809	167	59	,	,	PUNCT
iajs-2809	167	60	𝑥2	𝑥2	NOUN
iajs-2809	167	61	∗	∗	NOUN
iajs-2809	167	62	,	,	PUNCT
iajs-2809	167	63	…	…	PUNCT
iajs-2809	167	64	,	,	PUNCT
iajs-2809	167	65	𝑥𝑘	𝑥𝑘	NOUN
iajs-2809	167	66	∗	∗	NOUN
iajs-2809	167	67	)	)	PUNCT
iajs-2809	168	1	=	=	NOUN
iajs-2809	168	2	(	(	PUNCT
iajs-2809	168	3	𝑢1	𝑢1	PROPN
iajs-2809	168	4	�	�	PROPN
iajs-2809	168	5	̅	̅	PROPN
iajs-2809	168	6	�	�	PROPN
iajs-2809	168	7	,	,	PUNCT
iajs-2809	168	8	𝑢2	𝑢2	PROPN
iajs-2809	168	9	�	�	PROPN
iajs-2809	168	10	̅	̅	NOUN
iajs-2809	168	11	�	�	PROPN
iajs-2809	168	12	,	,	PUNCT
iajs-2809	168	13	…	…	PUNCT
iajs-2809	168	14	,	,	PUNCT
iajs-2809	168	15	𝑢𝑘	𝑢𝑘	ADP
iajs-2809	168	16	�	�	PROPN
iajs-2809	168	17	̅	̅	NOUN
iajs-2809	168	18	�	�	PROPN
iajs-2809	168	19	)	)	PUNCT
iajs-2809	168	20	.	.	PUNCT
iajs-2809	169	1	for	for	ADP
iajs-2809	169	2	solving	solve	VERB
iajs-2809	169	3	problem	problem	NOUN
iajs-2809	169	4	(	(	PUNCT
iajs-2809	169	5	5	5	NUM
iajs-2809	169	6	)	)	PUNCT
iajs-2809	169	7	,	,	PUNCT
iajs-2809	169	8	using	use	VERB
iajs-2809	169	9	the	the	DET
iajs-2809	169	10	same	same	ADJ
iajs-2809	169	11	derivation	derivation	NOUN
iajs-2809	169	12	and	and	CCONJ
iajs-2809	169	13	assumptions	assumption	NOUN
iajs-2809	169	14	of	of	ADP
iajs-2809	169	15	problem	problem	NOUN
iajs-2809	169	16	(	(	PUNCT
iajs-2809	169	17	4	4	NUM
iajs-2809	169	18	)	)	PUNCT
iajs-2809	169	19	,	,	PUNCT
iajs-2809	169	20	we	we	PRON
iajs-2809	169	21	can	can	AUX
iajs-2809	169	22	obtain	obtain	VERB
iajs-2809	169	23	problem	problem	NOUN
iajs-2809	169	24	(	(	PUNCT
iajs-2809	169	25	15	15	NUM
iajs-2809	169	26	)	)	PUNCT
iajs-2809	169	27	&	&	CCONJ
iajs-2809	169	28	(	(	PUNCT
iajs-2809	169	29	16	16	NUM
iajs-2809	169	30	)	)	PUNCT
iajs-2809	169	31	,	,	PUNCT
iajs-2809	169	32	with	with	ADP
iajs-2809	169	33	possibly	possibly	ADV
iajs-2809	169	34	worst	bad	ADJ
iajs-2809	169	35	optimum	optimum	ADJ
iajs-2809	169	36	,	,	PUNCT
iajs-2809	169	37	and	and	CCONJ
iajs-2809	169	38	best	good	ADJ
iajs-2809	169	39	optimum	optimum	ADJ
iajs-2809	169	40	.	.	PUNCT
iajs-2809	170	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	170	2	𝑍	𝑍	PROPN
iajs-2809	170	3	=	=	SYM
iajs-2809	170	4	𝑎1	𝑎1	ADP
iajs-2809	170	5	∗𝑙𝑢1	∗𝑙𝑢1	NOUN
iajs-2809	170	6	+	+	CCONJ
iajs-2809	170	7	⋯	⋯	PROPN
iajs-2809	170	8	+	+	CCONJ
iajs-2809	170	9	𝑎𝑘	𝑎𝑘	NOUN
iajs-2809	171	1	∗𝑙𝑢𝑘	∗𝑙𝑢𝑘	INTJ
iajs-2809	171	2	+	+	NUM
iajs-2809	171	3	𝑎𝑘+1	𝑎𝑘+1	PROPN
iajs-2809	171	4	∗𝑙	∗𝑙	PROPN
iajs-2809	171	5	𝑡	𝑡	PROPN
iajs-2809	171	6	subject	subject	ADJ
iajs-2809	171	7	to	to	ADP
iajs-2809	171	8	:	:	PUNCT
iajs-2809	171	9	𝑏1	𝑏1	VERB
iajs-2809	171	10	∗𝑢𝑢1	∗𝑢𝑢1	NOUN
iajs-2809	172	1	+	+	CCONJ
iajs-2809	172	2	⋯	⋯	VERB
iajs-2809	172	3	+	+	CCONJ
iajs-2809	172	4	𝑏𝑘	𝑏𝑘	NUM
iajs-2809	172	5	∗𝑢𝑢𝑘	∗𝑢𝑢𝑘	VERB
iajs-2809	172	6	+	+	CCONJ
iajs-2809	172	7	𝑏𝑘+1	𝑏𝑘+1	PROPN
iajs-2809	172	8	∗𝑢	∗𝑢	PROPN
iajs-2809	172	9	𝑡	𝑡	VERB
iajs-2809	172	10	≤	≤	ADV
iajs-2809	172	11	1	1	NUM
iajs-2809	172	12	𝑏1	𝑏1	ADJ
iajs-2809	172	13	∗𝑙𝑢1	∗𝑙𝑢1	NOUN
iajs-2809	172	14	+	+	CCONJ
iajs-2809	172	15	⋯	⋯	VERB
iajs-2809	172	16	+	+	NUM
iajs-2809	172	17	𝑏𝑘	𝑏𝑘	NOUN
iajs-2809	173	1	∗𝑙𝑢𝑘	∗𝑙𝑢𝑘	INTJ
iajs-2809	173	2	+	+	NUM
iajs-2809	173	3	𝑏𝑘+1	𝑏𝑘+1	NOUN
iajs-2809	173	4	∗𝑙	∗𝑙	PROPN
iajs-2809	173	5	𝑡	𝑡	VERB
iajs-2809	173	6	≤	≤	ADV
iajs-2809	173	7	1	1	NUM
iajs-2809	173	8	(	(	PUNCT
iajs-2809	173	9	15	15	NUM
iajs-2809	173	10	)	)	PUNCT
iajs-2809	173	11	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	173	12	+	+	CCONJ
iajs-2809	173	13	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	173	14	+	+	SYM
iajs-2809	173	15	⋯	⋯	PROPN
iajs-2809	173	16	+	+	CCONJ
iajs-2809	173	17	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	173	18	−	−	NOUN
iajs-2809	173	19	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	173	20	≤	≤	NOUN
iajs-2809	173	21	0	0	NUM
iajs-2809	173	22	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	173	23	,	,	PUNCT
iajs-2809	173	24	𝑡	𝑡	X
iajs-2809	173	25	≥	≥	NOUN
iajs-2809	173	26	0	0	NUM
iajs-2809	173	27	,	,	PUNCT
iajs-2809	173	28	𝑖	𝑖	NOUN
iajs-2809	173	29	=	=	SYM
iajs-2809	173	30	1,2	1,2	NUM
iajs-2809	173	31	,	,	PUNCT
iajs-2809	173	32	…	…	PUNCT
iajs-2809	173	33	,	,	PUNCT
iajs-2809	173	34	𝑘.	𝑘.	NOUN
iajs-2809	173	35	and	and	CCONJ
iajs-2809	173	36	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	173	37	𝑍	𝑍	PROPN
iajs-2809	173	38	=	=	NOUN
iajs-2809	173	39	𝑎1	𝑎1	NOUN
iajs-2809	173	40	∗𝑢𝑢1	∗𝑢𝑢1	NOUN
iajs-2809	174	1	+	+	CCONJ
iajs-2809	174	2	⋯	⋯	VERB
iajs-2809	175	1	+	+	CCONJ
iajs-2809	175	2	𝑎𝑘	𝑎𝑘	ADP
iajs-2809	175	3	∗𝑢𝑢𝑘	∗𝑢𝑢𝑘	AUX
iajs-2809	175	4	+	+	CCONJ
iajs-2809	175	5	𝑎𝑘+1	𝑎𝑘+1	PROPN
iajs-2809	175	6	∗𝑢	∗𝑢	PROPN
iajs-2809	175	7	𝑡	𝑡	PROPN
iajs-2809	175	8	subject	subject	ADJ
iajs-2809	175	9	to	to	ADP
iajs-2809	175	10	:	:	PUNCT
iajs-2809	175	11	𝑏1	𝑏1	VERB
iajs-2809	175	12	∗𝑢𝑢1	∗𝑢𝑢1	NOUN
iajs-2809	176	1	+	+	CCONJ
iajs-2809	176	2	⋯	⋯	VERB
iajs-2809	176	3	+	+	CCONJ
iajs-2809	176	4	𝑏𝑘	𝑏𝑘	NUM
iajs-2809	176	5	∗𝑢𝑢𝑘	∗𝑢𝑢𝑘	VERB
iajs-2809	176	6	+	+	CCONJ
iajs-2809	176	7	𝑏𝑘+1	𝑏𝑘+1	PROPN
iajs-2809	176	8	∗𝑢	∗𝑢	PROPN
iajs-2809	176	9	𝑡	𝑡	PROPN
iajs-2809	176	10	≥	≥	NOUN
iajs-2809	176	11	1	1	NUM
iajs-2809	176	12	𝑏1	𝑏1	ADJ
iajs-2809	176	13	∗𝑙𝑢1	∗𝑙𝑢1	NOUN
iajs-2809	176	14	+	+	CCONJ
iajs-2809	176	15	⋯	⋯	VERB
iajs-2809	176	16	+	+	NUM
iajs-2809	176	17	𝑏𝑘	𝑏𝑘	NOUN
iajs-2809	177	1	∗𝑙𝑢𝑘	∗𝑙𝑢𝑘	INTJ
iajs-2809	177	2	+	+	NUM
iajs-2809	177	3	𝑏𝑘+1	𝑏𝑘+1	NOUN
iajs-2809	177	4	∗𝑙	∗𝑙	PROPN
iajs-2809	177	5	𝑡	𝑡	VERB
iajs-2809	177	6	≤	≤	ADV
iajs-2809	177	7	1	1	NUM
iajs-2809	177	8	(	(	PUNCT
iajs-2809	177	9	16	16	NUM
iajs-2809	177	10	)	)	PUNCT
iajs-2809	177	11	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	177	12	+	+	CCONJ
iajs-2809	178	1	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	178	2	+	+	SYM
iajs-2809	178	3	⋯	⋯	PROPN
iajs-2809	178	4	+	+	CCONJ
iajs-2809	178	5	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	178	6	−	−	NOUN
iajs-2809	178	7	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	178	8	≤	≤	NOUN
iajs-2809	178	9	0	0	NUM
iajs-2809	178	10	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	178	11	,	,	PUNCT
iajs-2809	178	12	𝑡	𝑡	X
iajs-2809	178	13	≥	≥	NOUN
iajs-2809	178	14	0	0	NUM
iajs-2809	178	15	,	,	PUNCT
iajs-2809	178	16	𝑖	𝑖	NOUN
iajs-2809	178	17	=	=	SYM
iajs-2809	178	18	1,2	1,2	NUM
iajs-2809	178	19	,	,	PUNCT
iajs-2809	178	20	…	…	PUNCT
iajs-2809	178	21	,	,	PUNCT
iajs-2809	178	22	𝑘.	𝑘.	NOUN
iajs-2809	178	23	then	then	ADV
iajs-2809	178	24	to	to	PART
iajs-2809	178	25	solve	solve	VERB
iajs-2809	178	26	problem	problem	NOUN
iajs-2809	178	27	(	(	PUNCT
iajs-2809	178	28	13	13	NUM
iajs-2809	178	29	,	,	PUNCT
iajs-2809	178	30	14	14	NUM
iajs-2809	178	31	,	,	PUNCT
iajs-2809	178	32	15	15	NUM
iajs-2809	178	33	,	,	PUNCT
iajs-2809	178	34	16	16	NUM
iajs-2809	178	35	)	)	PUNCT
iajs-2809	178	36	using	use	VERB
iajs-2809	178	37	the	the	DET
iajs-2809	178	38	simplex	simplex	NOUN
iajs-2809	178	39	routine	routine	NOUN
iajs-2809	178	40	,	,	PUNCT
iajs-2809	178	41	we	we	PRON
iajs-2809	178	42	obtain	obtain	VERB
iajs-2809	178	43	the	the	DET
iajs-2809	178	44	optimal	optimal	ADJ
iajs-2809	178	45	value	value	NOUN
iajs-2809	178	46	and	and	CCONJ
iajs-2809	178	47	solution	solution	NOUN
iajs-2809	178	48	of	of	ADP
iajs-2809	178	49	the	the	DET
iajs-2809	178	50	original	original	ADJ
iajs-2809	178	51	problem	problem	NOUN
iajs-2809	178	52	(	(	PUNCT
iajs-2809	178	53	1	1	NUM
iajs-2809	178	54	)	)	PUNCT
iajs-2809	178	55	.	.	PUNCT
iajs-2809	179	1	next	next	ADV
iajs-2809	179	2	,	,	PUNCT
iajs-2809	179	3	will	will	AUX
iajs-2809	179	4	be	be	AUX
iajs-2809	179	5	confirmed	confirm	VERB
iajs-2809	179	6	that	that	SCONJ
iajs-2809	179	7	lfp	lfp	PROPN
iajs-2809	179	8	problems	problem	NOUN
iajs-2809	179	9	with	with	ADP
iajs-2809	179	10	rics	ric	NOUN
iajs-2809	179	11	in	in	ADP
iajs-2809	179	12	the	the	DET
iajs-2809	179	13	objective	objective	ADJ
iajs-2809	179	14	function	function	NOUN
iajs-2809	179	15	according	accord	VERB
iajs-2809	179	16	to	to	ADP
iajs-2809	179	17	the	the	DET
iajs-2809	179	18	fact	fact	NOUN
iajs-2809	179	19	that	that	SCONJ
iajs-2809	179	20	each	each	DET
iajs-2809	179	21	fixed	fix	VERB
iajs-2809	179	22	number	number	NOUN
iajs-2809	179	23	𝑛	𝑛	PROPN
iajs-2809	179	24	can	can	AUX
iajs-2809	179	25	be	be	AUX
iajs-2809	179	26	equivalently	equivalently	ADV
iajs-2809	179	27	written	write	VERB
iajs-2809	179	28	as	as	ADP
iajs-2809	179	29	the	the	DET
iajs-2809	179	30	interval	interval	NOUN
iajs-2809	179	31	[	[	X
iajs-2809	179	32	𝑛	𝑛	PROPN
iajs-2809	179	33	,	,	PUNCT
iajs-2809	179	34	𝑛	𝑛	NOUN
iajs-2809	179	35	]	]	PUNCT
iajs-2809	179	36	,	,	PUNCT
iajs-2809	179	37	and	and	CCONJ
iajs-2809	179	38	also	also	ADV
iajs-2809	179	39	equivalently	equivalently	ADV
iajs-2809	179	40	as	as	ADP
iajs-2809	179	41	the	the	DET
iajs-2809	179	42	rough	rough	ADJ
iajs-2809	179	43	interval	interval	NOUN
iajs-2809	179	44	(	(	PUNCT
iajs-2809	179	45	[	[	X
iajs-2809	179	46	𝑛	𝑛	NOUN
iajs-2809	179	47	,	,	PUNCT
iajs-2809	179	48	𝑛	𝑛	PROPN
iajs-2809	179	49	]	]	PUNCT
iajs-2809	179	50	,	,	PUNCT
iajs-2809	179	51	[	[	X
iajs-2809	179	52	𝑛	𝑛	X
iajs-2809	179	53	,	,	PUNCT
iajs-2809	179	54	𝑛	𝑛	PROPN
iajs-2809	179	55	]	]	X
iajs-2809	179	56	)	)	PUNCT
iajs-2809	179	57	,	,	PUNCT
iajs-2809	179	58	we	we	PRON
iajs-2809	179	59	claim	claim	VERB
iajs-2809	179	60	that	that	SCONJ
iajs-2809	179	61	lfp	lfp	PROPN
iajs-2809	179	62	is	be	AUX
iajs-2809	179	63	a	a	DET
iajs-2809	179	64	special	special	ADJ
iajs-2809	179	65	case	case	NOUN
iajs-2809	179	66	of	of	ADP
iajs-2809	179	67	lfp	lfp	PROPN
iajs-2809	179	68	with	with	ADP
iajs-2809	179	69	ics	ic	NOUN
iajs-2809	179	70	in	in	ADP
iajs-2809	179	71	the	the	DET
iajs-2809	179	72	objective	objective	ADJ
iajs-2809	179	73	function	function	NOUN
iajs-2809	179	74	and	and	CCONJ
iajs-2809	179	75	the	the	DET
iajs-2809	179	76	lfp	lfp	NOUN
iajs-2809	179	77	with	with	ADP
iajs-2809	179	78	ics	ic	NOUN
iajs-2809	179	79	in	in	ADP
iajs-2809	179	80	the	the	DET
iajs-2809	179	81	objective	objective	ADJ
iajs-2809	179	82	function	function	NOUN
iajs-2809	179	83	is	be	AUX
iajs-2809	179	84	a	a	DET
iajs-2809	179	85	special	special	ADJ
iajs-2809	179	86	case	case	NOUN
iajs-2809	179	87	of	of	ADP
iajs-2809	179	88	lfp	lfp	PROPN
iajs-2809	179	89	with	with	ADP
iajs-2809	179	90	rics	ric	NOUN
iajs-2809	179	91	in	in	ADP
iajs-2809	179	92	the	the	DET
iajs-2809	179	93	objective	objective	ADJ
iajs-2809	179	94	function	function	NOUN
iajs-2809	179	95	.	.	PUNCT
iajs-2809	180	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	180	2	𝑍	𝑍	PROPN
iajs-2809	180	3	=	=	SYM
iajs-2809	180	4	(	(	PUNCT
iajs-2809	180	5	[	[	X
iajs-2809	180	6	𝑛1	𝑛1	NOUN
iajs-2809	180	7	,	,	PUNCT
iajs-2809	180	8	𝑛1	𝑛1	NOUN
iajs-2809	180	9	]	]	PUNCT
iajs-2809	180	10	,	,	PUNCT
iajs-2809	181	1	[	[	X
iajs-2809	181	2	𝑛1	𝑛1	NOUN
iajs-2809	181	3	,	,	PUNCT
iajs-2809	181	4	𝑛1])𝑥1	𝑛1])𝑥1	ADV
iajs-2809	181	5	+	+	CCONJ
iajs-2809	181	6	(	(	PUNCT
iajs-2809	181	7	[	[	X
iajs-2809	181	8	𝑛2	𝑛2	NOUN
iajs-2809	181	9	,	,	PUNCT
iajs-2809	181	10	𝑛2	𝑛2	NOUN
iajs-2809	181	11	]	]	PUNCT
iajs-2809	181	12	,	,	PUNCT
iajs-2809	181	13	[	[	X
iajs-2809	181	14	𝑛2	𝑛2	NOUN
iajs-2809	181	15	,	,	PUNCT
iajs-2809	181	16	𝑛2])𝑥2	𝑛2])𝑥2	PROPN
iajs-2809	181	17	+	+	CCONJ
iajs-2809	181	18	⋯	⋯	VERB
iajs-2809	181	19	+	+	CCONJ
iajs-2809	181	20	(	(	PUNCT
iajs-2809	181	21	[	[	X
iajs-2809	181	22	𝑛𝑘	𝑛𝑘	X
iajs-2809	181	23	,	,	PUNCT
iajs-2809	181	24	𝑛𝑘	𝑛𝑘	NOUN
iajs-2809	181	25	]	]	PUNCT
iajs-2809	181	26	,	,	PUNCT
iajs-2809	181	27	[	[	X
iajs-2809	181	28	𝑛𝑘	𝑛𝑘	X
iajs-2809	181	29	,	,	PUNCT
iajs-2809	181	30	𝑛𝑘])𝑥𝑘	𝑛𝑘])𝑥𝑘	AUX
iajs-2809	181	31	+	+	PUNCT
iajs-2809	181	32	(	(	PUNCT
iajs-2809	181	33	[	[	X
iajs-2809	181	34	𝑛𝑘+1	𝑛𝑘+1	NOUN
iajs-2809	181	35	,	,	PUNCT
iajs-2809	181	36	𝑛𝑘+1	𝑛𝑘+1	PRON
iajs-2809	181	37	]	]	PUNCT
iajs-2809	181	38	,	,	PUNCT
iajs-2809	181	39	[	[	X
iajs-2809	181	40	𝑛𝑘+1	𝑛𝑘+1	NOUN
iajs-2809	181	41	,	,	PUNCT
iajs-2809	181	42	𝑛𝑘+1	𝑛𝑘+1	NOUN
iajs-2809	181	43	]	]	PUNCT
iajs-2809	181	44	)	)	PUNCT
iajs-2809	181	45	(	(	PUNCT
iajs-2809	182	1	[	[	X
iajs-2809	182	2	ℎ1	ℎ1	PROPN
iajs-2809	182	3	,	,	PUNCT
iajs-2809	182	4	ℎ1	ℎ1	PROPN
iajs-2809	182	5	]	]	X
iajs-2809	182	6	,	,	PUNCT
iajs-2809	183	1	[	[	X
iajs-2809	183	2	ℎ1	ℎ1	PROPN
iajs-2809	183	3	,	,	PUNCT
iajs-2809	183	4	ℎ1])𝑥1	ℎ1])𝑥1	ADV
iajs-2809	183	5	+	+	CCONJ
iajs-2809	183	6	(	(	PUNCT
iajs-2809	183	7	[	[	X
iajs-2809	183	8	ℎ2	ℎ2	NOUN
iajs-2809	183	9	,	,	PUNCT
iajs-2809	183	10	ℎ2	ℎ2	NOUN
iajs-2809	183	11	]	]	PUNCT
iajs-2809	183	12	,	,	PUNCT
iajs-2809	183	13	[	[	X
iajs-2809	183	14	ℎ2	ℎ2	NOUN
iajs-2809	183	15	,	,	PUNCT
iajs-2809	183	16	ℎ2])𝑥2	ℎ2])𝑥2	NOUN
iajs-2809	183	17	+	+	CCONJ
iajs-2809	183	18	⋯	⋯	PROPN
iajs-2809	183	19	+	+	CCONJ
iajs-2809	183	20	(	(	PUNCT
iajs-2809	183	21	[	[	X
iajs-2809	183	22	ℎ𝑘	ℎ𝑘	PROPN
iajs-2809	183	23	,	,	PUNCT
iajs-2809	183	24	ℎ𝑘	ℎ𝑘	PROPN
iajs-2809	183	25	]	]	PUNCT
iajs-2809	183	26	,	,	PUNCT
iajs-2809	184	1	[	[	X
iajs-2809	184	2	ℎ𝑘	ℎ𝑘	NOUN
iajs-2809	184	3	,	,	PUNCT
iajs-2809	184	4	ℎ𝑘])𝑥𝑘	ℎ𝑘])𝑥𝑘	PROPN
iajs-2809	184	5	+	+	X
iajs-2809	184	6	(	(	PUNCT
iajs-2809	184	7	[	[	X
iajs-2809	184	8	ℎ𝑘+1	ℎ𝑘+1	INTJ
iajs-2809	184	9	,	,	PUNCT
iajs-2809	184	10	ℎ𝑘+1	ℎ𝑘+1	NOUN
iajs-2809	184	11	]	]	PUNCT
iajs-2809	184	12	,	,	PUNCT
iajs-2809	184	13	[	[	X
iajs-2809	184	14	ℎ𝑘+1	ℎ𝑘+1	INTJ
iajs-2809	184	15	,	,	PUNCT
iajs-2809	184	16	ℎ𝑘+1	ℎ𝑘+1	NOUN
iajs-2809	184	17	]	]	PUNCT
iajs-2809	184	18	)	)	PUNCT
iajs-2809	184	19	subject	subject	ADJ
iajs-2809	184	20	to	to	ADP
iajs-2809	184	21	:	:	PUNCT
iajs-2809	184	22	(	(	PUNCT
iajs-2809	184	23	17	17	NUM
iajs-2809	184	24	)	)	PUNCT
iajs-2809	184	25	∑	∑	PUNCT
iajs-2809	184	26	𝐴𝑖𝑥𝑖	𝐴𝑖𝑥𝑖	PROPN
iajs-2809	184	27	≤	≤	NOUN
iajs-2809	184	28	𝑏	𝑏	DET
iajs-2809	184	29	𝑘	𝑘	X
iajs-2809	184	30	𝑖=1	𝑖=1	PROPN
iajs-2809	184	31	ibn	ibn	PROPN
iajs-2809	184	32	al	al	PROPN
iajs-2809	184	33	-	-	PUNCT
iajs-2809	184	34	haitham	haitham	PROPN
iajs-2809	184	35	jour	jour	X
iajs-2809	184	36	.	.	PROPN
iajs-2809	184	37	for	for	ADP
iajs-2809	184	38	pure	pure	ADJ
iajs-2809	184	39	&	&	CCONJ
iajs-2809	184	40	appl	appl	PROPN
iajs-2809	184	41	.	.	PUNCT
iajs-2809	185	1	sci	sci	PROPN
iajs-2809	185	2	.	.	PROPN
iajs-2809	186	1	53	53	NUM
iajs-2809	186	2	(	(	PUNCT
iajs-2809	186	3	2)2022	2)2022	VERB
iajs-2809	186	4	78	78	NUM
iajs-2809	186	5	proof	proof	NOUN
iajs-2809	186	6	:	:	PUNCT
iajs-2809	186	7	because	because	SCONJ
iajs-2809	186	8	the	the	DET
iajs-2809	186	9	rics	ric	NOUN
iajs-2809	186	10	in	in	ADP
iajs-2809	186	11	the	the	DET
iajs-2809	186	12	numerator	numerator	NOUN
iajs-2809	186	13	of	of	ADP
iajs-2809	186	14	the	the	DET
iajs-2809	186	15	objective	objective	ADJ
iajs-2809	186	16	function	function	NOUN
iajs-2809	186	17	has	have	VERB
iajs-2809	186	18	a	a	DET
iajs-2809	186	19	similar	similar	ADJ
iajs-2809	186	20	cost	cost	NOUN
iajs-2809	186	21	.	.	PUNCT
iajs-2809	187	1	the	the	DET
iajs-2809	187	2	coefficients	coefficient	NOUN
iajs-2809	187	3	of	of	ADP
iajs-2809	187	4	the	the	DET
iajs-2809	187	5	objective	objective	ADJ
iajs-2809	187	6	function	function	NOUN
iajs-2809	187	7	at	at	ADP
iajs-2809	187	8	its	its	PRON
iajs-2809	187	9	optimum	optimum	NOUN
iajs-2809	187	10	surely	surely	ADV
iajs-2809	187	11	the	the	DET
iajs-2809	187	12	best	good	ADJ
iajs-2809	187	13	,	,	PUNCT
iajs-2809	187	14	surely	surely	ADV
iajs-2809	187	15	the	the	DET
iajs-2809	187	16	worst	bad	ADJ
iajs-2809	187	17	,	,	PUNCT
iajs-2809	187	18	possibly	possibly	ADV
iajs-2809	187	19	the	the	DET
iajs-2809	187	20	best	good	ADJ
iajs-2809	187	21	,	,	PUNCT
iajs-2809	187	22	and	and	CCONJ
iajs-2809	187	23	possibly	possibly	ADV
iajs-2809	187	24	the	the	DET
iajs-2809	187	25	worst	bad	ADJ
iajs-2809	187	26	are	be	AUX
iajs-2809	187	27	the	the	DET
iajs-2809	187	28	same	same	ADJ
iajs-2809	187	29	optimum	optimum	NOUN
iajs-2809	187	30	.	.	PUNCT
iajs-2809	188	1	by	by	ADP
iajs-2809	188	2	charnes	charne	NOUN
iajs-2809	188	3	-	-	PUNCT
iajs-2809	188	4	cooper	cooper	NOUN
iajs-2809	188	5	’s	’s	PART
iajs-2809	188	6	method	method	NOUN
iajs-2809	188	7	,	,	PUNCT
iajs-2809	188	8	problem	problem	NOUN
iajs-2809	188	9	(	(	PUNCT
iajs-2809	188	10	17	17	NUM
iajs-2809	188	11	)	)	PUNCT
iajs-2809	188	12	is	be	AUX
iajs-2809	188	13	transformed	transform	VERB
iajs-2809	188	14	into	into	ADP
iajs-2809	188	15	the	the	DET
iajs-2809	188	16	following	follow	VERB
iajs-2809	188	17	problem	problem	NOUN
iajs-2809	188	18	.	.	PUNCT
iajs-2809	189	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	189	2	𝑍	𝑍	PROPN
iajs-2809	189	3	=	=	SYM
iajs-2809	189	4	𝑛1𝑢1	𝑛1𝑢1	PROPN
iajs-2809	190	1	+	+	PUNCT
iajs-2809	190	2	𝑛2𝑢2	𝑛2𝑢2	PUNCT
iajs-2809	191	1	+	+	NUM
iajs-2809	191	2	⋯	⋯	VERB
iajs-2809	191	3	+	+	CCONJ
iajs-2809	191	4	𝑛𝑘𝑢𝑘	𝑛𝑘𝑢𝑘	NOUN
iajs-2809	191	5	+	+	CCONJ
iajs-2809	191	6	𝑛𝑘+1𝑡	𝑛𝑘+1𝑡	ADJ
iajs-2809	191	7	subject	subject	NOUN
iajs-2809	191	8	to	to	ADP
iajs-2809	191	9	:	:	PUNCT
iajs-2809	191	10	ℎ1𝑢1	ℎ1𝑢1	PROPN
iajs-2809	192	1	+	+	NUM
iajs-2809	192	2	⋯	⋯	PROPN
iajs-2809	192	3	+	+	PROPN
iajs-2809	192	4	ℎ𝑘𝑢𝑘	ℎ𝑘𝑢𝑘	NOUN
iajs-2809	192	5	+	+	NOUN
iajs-2809	192	6	ℎ𝑘+1𝑡	ℎ𝑘+1𝑡	NOUN
iajs-2809	192	7	≥	≥	NOUN
iajs-2809	192	8	1	1	NUM
iajs-2809	192	9	ℎ1𝑢1	ℎ1𝑢1	PUNCT
iajs-2809	192	10	+	+	PROPN
iajs-2809	192	11	⋯	⋯	PROPN
iajs-2809	192	12	+	+	PROPN
iajs-2809	192	13	ℎ𝑘𝑢𝑘	ℎ𝑘𝑢𝑘	NOUN
iajs-2809	192	14	+	+	CCONJ
iajs-2809	192	15	ℎ𝑘+1𝑡	ℎ𝑘+1𝑡	ADJ
iajs-2809	192	16	≤	≤	NUM
iajs-2809	192	17	1	1	NUM
iajs-2809	192	18	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	192	19	+	+	CCONJ
iajs-2809	192	20	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	192	21	+	+	SYM
iajs-2809	192	22	⋯	⋯	PROPN
iajs-2809	192	23	+	+	CCONJ
iajs-2809	192	24	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	192	25	−	−	NOUN
iajs-2809	192	26	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	192	27	≤	≤	NOUN
iajs-2809	192	28	0	0	NUM
iajs-2809	192	29	(	(	PUNCT
iajs-2809	192	30	18	18	NUM
iajs-2809	192	31	)	)	PUNCT
iajs-2809	192	32	𝑢𝑖	𝑢𝑖	NOUN
iajs-2809	192	33	,	,	PUNCT
iajs-2809	192	34	𝑡	𝑡	X
iajs-2809	192	35	≥	≥	NOUN
iajs-2809	192	36	0	0	NUM
iajs-2809	192	37	,	,	PUNCT
iajs-2809	192	38	𝑖	𝑖	NOUN
iajs-2809	192	39	=	=	SYM
iajs-2809	192	40	1,2	1,2	NUM
iajs-2809	192	41	,	,	PUNCT
iajs-2809	192	42	…	…	PUNCT
iajs-2809	192	43	,	,	PUNCT
iajs-2809	192	44	𝑘.	𝑘.	VERB
iajs-2809	192	45	the	the	DET
iajs-2809	192	46	combination	combination	NOUN
iajs-2809	192	47	of	of	ADP
iajs-2809	192	48	the	the	DET
iajs-2809	192	49	first	first	ADJ
iajs-2809	192	50	two	two	NUM
iajs-2809	192	51	constraints	constraint	NOUN
iajs-2809	192	52	causes	cause	VERB
iajs-2809	192	53	the	the	DET
iajs-2809	192	54	following	follow	VERB
iajs-2809	192	55	problem	problem	NOUN
iajs-2809	192	56	:	:	PUNCT
iajs-2809	192	57	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	192	58	𝑍	𝑍	PROPN
iajs-2809	192	59	=	=	SYM
iajs-2809	192	60	𝑛1𝑢1	𝑛1𝑢1	PROPN
iajs-2809	193	1	+	+	PUNCT
iajs-2809	193	2	𝑛2𝑢2	𝑛2𝑢2	PUNCT
iajs-2809	194	1	+	+	NUM
iajs-2809	194	2	⋯	⋯	VERB
iajs-2809	194	3	+	+	CCONJ
iajs-2809	194	4	𝑛𝑘𝑢𝑘	𝑛𝑘𝑢𝑘	NOUN
iajs-2809	194	5	+	+	CCONJ
iajs-2809	194	6	𝑛𝑘+1𝑡	𝑛𝑘+1𝑡	ADJ
iajs-2809	194	7	subject	subject	NOUN
iajs-2809	194	8	to	to	ADP
iajs-2809	194	9	:	:	PUNCT
iajs-2809	194	10	ℎ1𝑢1	ℎ1𝑢1	PROPN
iajs-2809	195	1	+	+	NUM
iajs-2809	195	2	⋯	⋯	PROPN
iajs-2809	195	3	+	+	PROPN
iajs-2809	195	4	ℎ𝑘𝑢𝑘	ℎ𝑘𝑢𝑘	ADJ
iajs-2809	195	5	+	+	CCONJ
iajs-2809	195	6	ℎ𝑘+1𝑡	ℎ𝑘+1𝑡	NOUN
iajs-2809	195	7	=	=	SYM
iajs-2809	195	8	1	1	NUM
iajs-2809	195	9	𝐴1𝑢1	𝐴1𝑢1	NOUN
iajs-2809	195	10	+	+	CCONJ
iajs-2809	195	11	𝐴2𝑢2	𝐴2𝑢2	PROPN
iajs-2809	195	12	+	+	SYM
iajs-2809	195	13	⋯	⋯	PROPN
iajs-2809	195	14	+	+	CCONJ
iajs-2809	195	15	𝐴𝑘𝑢𝑘	𝐴𝑘𝑢𝑘	PROPN
iajs-2809	195	16	−	−	NOUN
iajs-2809	195	17	𝑏𝑡	𝑏𝑡	NOUN
iajs-2809	195	18	≤	≤	NOUN
iajs-2809	195	19	0	0	NUM
iajs-2809	195	20	,	,	PUNCT
iajs-2809	195	21	where	where	SCONJ
iajs-2809	195	22	𝑢𝑖	𝑢𝑖	INTJ
iajs-2809	195	23	,	,	PUNCT
iajs-2809	195	24	𝑡	𝑡	X
iajs-2809	195	25	≥	≥	NOUN
iajs-2809	195	26	0	0	NUM
iajs-2809	195	27	,	,	PUNCT
iajs-2809	195	28	𝑖	𝑖	NOUN
iajs-2809	195	29	=	=	SYM
iajs-2809	195	30	1,2	1,2	NUM
iajs-2809	195	31	,	,	PUNCT
iajs-2809	195	32	…	…	PUNCT
iajs-2809	195	33	,	,	PUNCT
iajs-2809	195	34	𝑘.	𝑘.	NOUN
iajs-2809	195	35	(	(	PUNCT
iajs-2809	195	36	19	19	NUM
iajs-2809	195	37	)	)	PUNCT
iajs-2809	195	38	4	4	NUM
iajs-2809	195	39	.	.	PUNCT
iajs-2809	195	40	numerical	numerical	ADJ
iajs-2809	195	41	examples	example	NOUN
iajs-2809	195	42	example1	example1	VERB
iajs-2809	195	43	:	:	PUNCT
iajs-2809	195	44	consider	consider	VERB
iajs-2809	195	45	the	the	DET
iajs-2809	195	46	following	follow	VERB
iajs-2809	195	47	lfpp	lfpp	NOUN
iajs-2809	195	48	with	with	ADP
iajs-2809	195	49	rics	ric	NOUN
iajs-2809	195	50	in	in	ADP
iajs-2809	195	51	the	the	DET
iajs-2809	195	52	objective	objective	ADJ
iajs-2809	195	53	function	function	NOUN
iajs-2809	195	54	:	:	PUNCT
iajs-2809	195	55	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	195	56	𝑍	𝑍	PROPN
iajs-2809	195	57	=	=	SYM
iajs-2809	195	58	(	(	PUNCT
iajs-2809	195	59	[	[	X
iajs-2809	195	60	1,3	1,3	NUM
iajs-2809	195	61	]	]	PUNCT
iajs-2809	195	62	,	,	PUNCT
iajs-2809	196	1	[	[	X
iajs-2809	196	2	0.5,4])𝑥1	0.5,4])𝑥1	NOUN
iajs-2809	196	3	+	+	CCONJ
iajs-2809	196	4	(	(	PUNCT
iajs-2809	196	5	[	[	X
iajs-2809	196	6	2,4	2,4	NUM
iajs-2809	196	7	]	]	X
iajs-2809	196	8	,	,	PUNCT
iajs-2809	196	9	[	[	X
iajs-2809	196	10	1,4.5])𝑥2	1,4.5])𝑥2	NUM
iajs-2809	196	11	(	(	PUNCT
iajs-2809	196	12	[	[	X
iajs-2809	196	13	0.5,1.5	0.5,1.5	PROPN
iajs-2809	196	14	]	]	X
iajs-2809	196	15	,	,	PUNCT
iajs-2809	197	1	[	[	X
iajs-2809	197	2	0.25,2])𝑥1	0.25,2])𝑥1	X
iajs-2809	197	3	+	+	SYM
iajs-2809	197	4	(	(	PUNCT
iajs-2809	197	5	[	[	X
iajs-2809	197	6	0.5,1.5	0.5,1.5	X
iajs-2809	197	7	]	]	X
iajs-2809	197	8	,	,	PUNCT
iajs-2809	197	9	[	[	X
iajs-2809	197	10	0.25,2])𝑥2	0.25,2])𝑥2	X
iajs-2809	197	11	+	+	CCONJ
iajs-2809	197	12	(	(	PUNCT
iajs-2809	197	13	[	[	X
iajs-2809	197	14	1,3	1,3	NUM
iajs-2809	197	15	]	]	X
iajs-2809	197	16	,	,	PUNCT
iajs-2809	197	17	[	[	PUNCT
iajs-2809	197	18	1	1	NUM
iajs-2809	197	19	3	3	NUM
iajs-2809	197	20	,	,	PUNCT
iajs-2809	197	21	3.5	3.5	NUM
iajs-2809	197	22	]	]	PUNCT
iajs-2809	197	23	)	)	PUNCT
iajs-2809	197	24	subject	subject	ADJ
iajs-2809	197	25	to	to	ADP
iajs-2809	197	26	:	:	PUNCT
iajs-2809	197	27	𝑥1	𝑥1	NOUN
iajs-2809	197	28	−	−	PROPN
iajs-2809	197	29	𝑥2	𝑥2	PROPN
iajs-2809	197	30	≥	≥	NOUN
iajs-2809	197	31	1	1	NUM
iajs-2809	197	32	,	,	PUNCT
iajs-2809	197	33	2𝑥1	2𝑥1	NUM
iajs-2809	197	34	+	+	CCONJ
iajs-2809	197	35	3𝑥2	3𝑥2	NUM
iajs-2809	197	36	≤	≤	NUM
iajs-2809	197	37	15	15	NUM
iajs-2809	197	38	,	,	PUNCT
iajs-2809	197	39	𝑥1	𝑥1	NOUN
iajs-2809	197	40	≥	≥	NOUN
iajs-2809	197	41	3	3	NUM
iajs-2809	197	42	,	,	PUNCT
iajs-2809	197	43	𝑥1	𝑥1	PROPN
iajs-2809	197	44	,	,	PUNCT
iajs-2809	197	45	𝑥1	𝑥1	NOUN
iajs-2809	197	46	≥	≥	NOUN
iajs-2809	197	47	0	0	NUM
iajs-2809	197	48	.	.	PUNCT
iajs-2809	197	49	solution	solution	NOUN
iajs-2809	197	50	problem	problem	NOUN
iajs-2809	197	51	(	(	PUNCT
iajs-2809	197	52	20	20	NUM
iajs-2809	197	53	)	)	PUNCT
iajs-2809	197	54	is	be	AUX
iajs-2809	197	55	transformed	transform	VERB
iajs-2809	197	56	into	into	ADP
iajs-2809	197	57	rough	rough	ADJ
iajs-2809	197	58	interval	interval	NOUN
iajs-2809	197	59	linear	linear	NOUN
iajs-2809	197	60	programming	programming	NOUN
iajs-2809	197	61	problem	problem	NOUN
iajs-2809	197	62	by	by	ADP
iajs-2809	197	63	charnes	charne	NOUN
iajs-2809	197	64	&	&	CCONJ
iajs-2809	197	65	cooper	cooper	PROPN
iajs-2809	197	66	method	method	VERB
iajs-2809	197	67	.	.	PUNCT
iajs-2809	198	1	we	we	PRON
iajs-2809	198	2	introduce	introduce	VERB
iajs-2809	198	3	𝑡	𝑡	NOUN
iajs-2809	198	4	=	=	NOUN
iajs-2809	198	5	1	1	NUM
iajs-2809	198	6	(	(	PUNCT
iajs-2809	198	7	[	[	X
iajs-2809	198	8	0.5,1.5],[0.25,2])𝑥1+([0.5,1.5],[0.25,2])𝑥2+([1,3	0.5,1.5],[0.25,2])𝑥1+([0.5,1.5],[0.25,2])𝑥2+([1,3	X
iajs-2809	198	9	]	]	X
iajs-2809	198	10	,	,	PUNCT
iajs-2809	198	11	[	[	PUNCT
iajs-2809	198	12	1	1	NUM
iajs-2809	198	13	3	3	NUM
iajs-2809	198	14	,	,	PUNCT
iajs-2809	198	15	3.5	3.5	NUM
iajs-2809	198	16	]	]	PUNCT
iajs-2809	198	17	)	)	PUNCT
iajs-2809	199	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	199	2	𝑍	𝑍	PROPN
iajs-2809	199	3	=	=	SYM
iajs-2809	199	4	(	(	PUNCT
iajs-2809	199	5	[	[	X
iajs-2809	199	6	1,3	1,3	NUM
iajs-2809	199	7	]	]	PUNCT
iajs-2809	199	8	,	,	PUNCT
iajs-2809	200	1	[	[	X
iajs-2809	200	2	0.5,4])𝑥1𝑡	0.5,4])𝑥1𝑡	NOUN
iajs-2809	200	3	+	+	PUNCT
iajs-2809	200	4	(	(	PUNCT
iajs-2809	200	5	[	[	X
iajs-2809	200	6	2,4	2,4	NUM
iajs-2809	200	7	]	]	X
iajs-2809	200	8	,	,	PUNCT
iajs-2809	201	1	[	[	X
iajs-2809	201	2	1,4.5])𝑥2𝑡	1,4.5])𝑥2𝑡	NUM
iajs-2809	201	3	+	+	NUM
iajs-2809	201	4	0𝑡	0𝑡	NOUN
iajs-2809	201	5	subject	subject	ADJ
iajs-2809	201	6	to	to	ADP
iajs-2809	201	7	:	:	PUNCT
iajs-2809	201	8	(	(	PUNCT
iajs-2809	201	9	20	20	X
iajs-2809	201	10	)	)	PUNCT
iajs-2809	201	11	ibn	ibn	PROPN
iajs-2809	201	12	al	al	PROPN
iajs-2809	201	13	-	-	PUNCT
iajs-2809	201	14	haitham	haitham	PROPN
iajs-2809	201	15	jour	jour	X
iajs-2809	201	16	.	.	PROPN
iajs-2809	201	17	for	for	ADP
iajs-2809	201	18	pure	pure	ADJ
iajs-2809	201	19	&	&	CCONJ
iajs-2809	201	20	appl	appl	PROPN
iajs-2809	201	21	.	.	PUNCT
iajs-2809	202	1	sci	sci	PROPN
iajs-2809	202	2	.	.	PROPN
iajs-2809	203	1	53	53	NUM
iajs-2809	203	2	(	(	PUNCT
iajs-2809	203	3	2)2022	2)2022	NUM
iajs-2809	203	4	79	79	NUM
iajs-2809	203	5	(	(	PUNCT
iajs-2809	203	6	[	[	X
iajs-2809	203	7	0.5,1.5	0.5,1.5	PROPN
iajs-2809	203	8	]	]	X
iajs-2809	203	9	,	,	PUNCT
iajs-2809	203	10	[	[	X
iajs-2809	203	11	0.25,2])𝑥1𝑡	0.25,2])𝑥1𝑡	X
iajs-2809	203	12	+	+	CCONJ
iajs-2809	203	13	(	(	PUNCT
iajs-2809	203	14	[	[	X
iajs-2809	203	15	0.5,1.5	0.5,1.5	X
iajs-2809	203	16	]	]	X
iajs-2809	203	17	,	,	PUNCT
iajs-2809	203	18	[	[	X
iajs-2809	203	19	0.25,2])𝑥2𝑡	0.25,2])𝑥2𝑡	X
iajs-2809	203	20	+	+	CCONJ
iajs-2809	203	21	(	(	PUNCT
iajs-2809	203	22	[	[	X
iajs-2809	203	23	1,3	1,3	NUM
iajs-2809	203	24	]	]	X
iajs-2809	203	25	,	,	PUNCT
iajs-2809	203	26	[	[	PUNCT
iajs-2809	203	27	1	1	NUM
iajs-2809	203	28	3	3	NUM
iajs-2809	203	29	,	,	PUNCT
iajs-2809	203	30	3.5	3.5	NUM
iajs-2809	203	31	]	]	PUNCT
iajs-2809	203	32	)	)	PUNCT
iajs-2809	203	33	𝑡	𝑡	PROPN
iajs-2809	203	34	=	=	SYM
iajs-2809	203	35	1	1	NUM
iajs-2809	203	36	−𝑥1𝑡	−𝑥1𝑡	PROPN
iajs-2809	203	37	+	+	CCONJ
iajs-2809	203	38	𝑥2𝑡	𝑥2𝑡	PROPN
iajs-2809	203	39	+	+	CCONJ
iajs-2809	203	40	𝑡	𝑡	VERB
iajs-2809	203	41	≤	≤	NUM
iajs-2809	203	42	0	0	NUM
iajs-2809	203	43	,	,	PUNCT
iajs-2809	203	44	2𝑥1𝑡	2𝑥1𝑡	NUM
iajs-2809	203	45	+	+	NUM
iajs-2809	203	46	3𝑥2𝑡	3𝑥2𝑡	NOUN
iajs-2809	203	47	−	−	PROPN
iajs-2809	203	48	15𝑡	15𝑡	NOUN
iajs-2809	203	49	≤	≤	ADV
iajs-2809	203	50	0	0	NUM
iajs-2809	203	51	,	,	PUNCT
iajs-2809	203	52	−𝑥1𝑡	−𝑥1𝑡	PUNCT
iajs-2809	203	53	+	+	CCONJ
iajs-2809	203	54	3𝑡	3𝑡	NUM
iajs-2809	203	55	≤	≤	NUM
iajs-2809	203	56	0	0	NUM
iajs-2809	203	57	,	,	PUNCT
iajs-2809	203	58	𝑥1	𝑥1	PROPN
iajs-2809	203	59	,	,	PUNCT
iajs-2809	203	60	𝑥1	𝑥1	PROPN
iajs-2809	203	61	,	,	PUNCT
iajs-2809	203	62	𝑡	𝑡	X
iajs-2809	203	63	≥	≥	NOUN
iajs-2809	203	64	0	0	NUM
iajs-2809	203	65	.	.	PUNCT
iajs-2809	204	1	let𝑢𝑖	let𝑢𝑖	PROPN
iajs-2809	204	2	=	=	SYM
iajs-2809	204	3	𝑥𝑖𝑡	𝑥𝑖𝑡	PROPN
iajs-2809	204	4	,	,	PUNCT
iajs-2809	204	5	𝑖	𝑖	X
iajs-2809	204	6	=	=	SYM
iajs-2809	204	7	1,2	1,2	NUM
iajs-2809	204	8	.	.	PUNCT
iajs-2809	205	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	205	2	𝑍	𝑍	PROPN
iajs-2809	205	3	=	=	SYM
iajs-2809	205	4	(	(	PUNCT
iajs-2809	205	5	[	[	X
iajs-2809	205	6	1,3	1,3	NUM
iajs-2809	205	7	]	]	PUNCT
iajs-2809	205	8	,	,	PUNCT
iajs-2809	206	1	[	[	X
iajs-2809	206	2	0.5,4])𝑢1	0.5,4])𝑢1	NOUN
iajs-2809	206	3	+	+	X
iajs-2809	206	4	(	(	PUNCT
iajs-2809	206	5	[	[	X
iajs-2809	206	6	2,4	2,4	NUM
iajs-2809	206	7	]	]	X
iajs-2809	206	8	,	,	PUNCT
iajs-2809	206	9	[	[	X
iajs-2809	206	10	1,4.5])𝑢2	1,4.5])𝑢2	NUM
iajs-2809	206	11	+	+	NUM
iajs-2809	206	12	0𝑡	0𝑡	NOUN
iajs-2809	206	13	subject	subject	ADJ
iajs-2809	206	14	to	to	ADP
iajs-2809	206	15	:	:	PUNCT
iajs-2809	206	16	(	(	PUNCT
iajs-2809	206	17	21	21	NUM
iajs-2809	206	18	)	)	PUNCT
iajs-2809	206	19	(	(	PUNCT
iajs-2809	207	1	[	[	X
iajs-2809	207	2	0.5,1.5	0.5,1.5	X
iajs-2809	207	3	]	]	X
iajs-2809	207	4	,	,	PUNCT
iajs-2809	207	5	[	[	X
iajs-2809	207	6	0.25,2])𝑢1	0.25,2])𝑢1	X
iajs-2809	208	1	+	+	CCONJ
iajs-2809	208	2	(	(	PUNCT
iajs-2809	208	3	[	[	X
iajs-2809	208	4	0.5,1.5	0.5,1.5	X
iajs-2809	208	5	]	]	X
iajs-2809	208	6	,	,	PUNCT
iajs-2809	209	1	[	[	X
iajs-2809	209	2	0.25,2])𝑢2	0.25,2])𝑢2	X
iajs-2809	209	3	+	+	CCONJ
iajs-2809	209	4	(	(	PUNCT
iajs-2809	209	5	[	[	X
iajs-2809	209	6	1,3	1,3	NUM
iajs-2809	209	7	]	]	X
iajs-2809	209	8	,	,	PUNCT
iajs-2809	209	9	[	[	PUNCT
iajs-2809	209	10	1	1	NUM
iajs-2809	209	11	3	3	NUM
iajs-2809	209	12	,	,	PUNCT
iajs-2809	209	13	3.5	3.5	NUM
iajs-2809	209	14	]	]	PUNCT
iajs-2809	209	15	)	)	PUNCT
iajs-2809	209	16	𝑡	𝑡	PROPN
iajs-2809	209	17	=	=	SYM
iajs-2809	209	18	1	1	NUM
iajs-2809	209	19	−𝑢1	−𝑢1	NOUN
iajs-2809	209	20	+	+	CCONJ
iajs-2809	209	21	𝑢2	𝑢2	PROPN
iajs-2809	209	22	+	+	CCONJ
iajs-2809	209	23	𝑡	𝑡	PROPN
iajs-2809	209	24	≤	≤	ADV
iajs-2809	209	25	0	0	NUM
iajs-2809	209	26	,	,	PUNCT
iajs-2809	209	27	2𝑢1	2𝑢1	NUM
iajs-2809	210	1	+	+	SYM
iajs-2809	210	2	3𝑢2	3𝑢2	NUM
iajs-2809	210	3	−	−	PROPN
iajs-2809	210	4	15𝑡	15𝑡	NOUN
iajs-2809	210	5	≤	≤	ADV
iajs-2809	210	6	0	0	NUM
iajs-2809	210	7	,	,	PUNCT
iajs-2809	210	8	−𝑢1	−𝑢1	PROPN
iajs-2809	210	9	+	+	CCONJ
iajs-2809	210	10	3𝑡	3𝑡	NUM
iajs-2809	210	11	≤	≤	NUM
iajs-2809	210	12	0	0	NUM
iajs-2809	210	13	,	,	PUNCT
iajs-2809	210	14	𝑢1	𝑢1	NOUN
iajs-2809	210	15	,	,	PUNCT
iajs-2809	210	16	𝑢1	𝑢1	PROPN
iajs-2809	210	17	,	,	PUNCT
iajs-2809	210	18	𝑡	𝑡	X
iajs-2809	210	19	≥	≥	NOUN
iajs-2809	210	20	0	0	NUM
iajs-2809	210	21	.	.	PUNCT
iajs-2809	211	1	now	now	ADV
iajs-2809	211	2	,	,	PUNCT
iajs-2809	211	3	by	by	ADP
iajs-2809	211	4	hamzehee	hamzehee	PROPN
iajs-2809	211	5	et	et	PROPN
iajs-2809	211	6	al	al	PROPN
iajs-2809	211	7	.	.	PUNCT
iajs-2809	212	1	[	[	X
iajs-2809	212	2	7	7	NUM
iajs-2809	212	3	]	]	X
iajs-2809	212	4	problem	problem	NOUN
iajs-2809	212	5	(	(	PUNCT
iajs-2809	212	6	21	21	NUM
iajs-2809	212	7	)	)	PUNCT
iajs-2809	212	8	can	can	AUX
iajs-2809	212	9	be	be	AUX
iajs-2809	212	10	changed	change	VERB
iajs-2809	212	11	into	into	ADP
iajs-2809	212	12	two	two	NUM
iajs-2809	212	13	linear	linear	ADJ
iajs-2809	212	14	programming	programming	NOUN
iajs-2809	212	15	with	with	ADP
iajs-2809	212	16	interval	interval	NOUN
iajs-2809	212	17	coefficients	coefficient	NOUN
iajs-2809	212	18	(	(	PUNCT
iajs-2809	212	19	22	22	NUM
iajs-2809	212	20	)	)	PUNCT
iajs-2809	212	21	&	&	CCONJ
iajs-2809	212	22	(	(	PUNCT
iajs-2809	212	23	23	23	NUM
iajs-2809	212	24	)	)	PUNCT
iajs-2809	212	25	.	.	PUNCT
iajs-2809	213	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	213	2	𝑍	𝑍	PROPN
iajs-2809	213	3	=	=	SYM
iajs-2809	213	4	(	(	PUNCT
iajs-2809	213	5	[	[	X
iajs-2809	213	6	1,3])𝑢1	1,3])𝑢1	NUM
iajs-2809	213	7	+	+	CCONJ
iajs-2809	213	8	(	(	PUNCT
iajs-2809	213	9	[	[	X
iajs-2809	213	10	2,4])𝑢2	2,4])𝑢2	NUM
iajs-2809	213	11	+	+	CCONJ
iajs-2809	213	12	0𝑡	0𝑡	NOUN
iajs-2809	213	13	subject	subject	ADJ
iajs-2809	213	14	to	to	ADP
iajs-2809	213	15	:	:	PUNCT
iajs-2809	213	16	(	(	PUNCT
iajs-2809	213	17	22	22	NUM
iajs-2809	213	18	)	)	PUNCT
iajs-2809	213	19	(	(	PUNCT
iajs-2809	214	1	[	[	X
iajs-2809	214	2	0.5,1.5])𝑢1	0.5,1.5])𝑢1	X
iajs-2809	215	1	+	+	CCONJ
iajs-2809	215	2	(	(	PUNCT
iajs-2809	215	3	[	[	X
iajs-2809	215	4	0.5,1.5])𝑢2	0.5,1.5])𝑢2	X
iajs-2809	215	5	+	+	CCONJ
iajs-2809	215	6	(	(	PUNCT
iajs-2809	215	7	[	[	X
iajs-2809	215	8	1,3])𝑡	1,3])𝑡	NUM
iajs-2809	215	9	=	=	SYM
iajs-2809	215	10	1	1	NUM
iajs-2809	215	11	−𝑢1	−𝑢1	NOUN
iajs-2809	215	12	+	+	CCONJ
iajs-2809	215	13	𝑢2	𝑢2	PROPN
iajs-2809	215	14	+	+	CCONJ
iajs-2809	215	15	𝑡	𝑡	PROPN
iajs-2809	215	16	≤	≤	ADV
iajs-2809	215	17	0	0	NUM
iajs-2809	215	18	,	,	PUNCT
iajs-2809	215	19	2𝑢1	2𝑢1	NUM
iajs-2809	215	20	+	+	SYM
iajs-2809	215	21	3𝑢2	3𝑢2	NUM
iajs-2809	215	22	−	−	PROPN
iajs-2809	215	23	15𝑡	15𝑡	NOUN
iajs-2809	215	24	≤	≤	ADV
iajs-2809	215	25	0	0	NUM
iajs-2809	215	26	,	,	PUNCT
iajs-2809	215	27	−𝑢1	−𝑢1	PROPN
iajs-2809	215	28	+	+	CCONJ
iajs-2809	215	29	3𝑡	3𝑡	NUM
iajs-2809	215	30	≤	≤	NUM
iajs-2809	215	31	0	0	NUM
iajs-2809	215	32	,	,	PUNCT
iajs-2809	215	33	𝑢1	𝑢1	NOUN
iajs-2809	215	34	,	,	PUNCT
iajs-2809	215	35	𝑢1	𝑢1	PROPN
iajs-2809	215	36	,	,	PUNCT
iajs-2809	215	37	𝑡	𝑡	X
iajs-2809	215	38	≥	≥	NOUN
iajs-2809	215	39	0	0	NUM
iajs-2809	215	40	.	.	PUNCT
iajs-2809	216	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	216	2	𝑍	𝑍	PROPN
iajs-2809	216	3	=	=	SYM
iajs-2809	216	4	(	(	PUNCT
iajs-2809	216	5	[	[	X
iajs-2809	216	6	0.5,4])𝑢1	0.5,4])𝑢1	NOUN
iajs-2809	216	7	+	+	CCONJ
iajs-2809	216	8	(	(	PUNCT
iajs-2809	216	9	[	[	X
iajs-2809	216	10	1,4.5])𝑢2	1,4.5])𝑢2	NUM
iajs-2809	216	11	+	+	NUM
iajs-2809	216	12	0𝑡	0𝑡	NOUN
iajs-2809	216	13	subject	subject	ADJ
iajs-2809	216	14	to	to	ADP
iajs-2809	216	15	:	:	PUNCT
iajs-2809	216	16	(	(	PUNCT
iajs-2809	216	17	23	23	NUM
iajs-2809	216	18	)	)	PUNCT
iajs-2809	216	19	(	(	PUNCT
iajs-2809	216	20	[	[	X
iajs-2809	216	21	0.25,2])𝑢1	0.25,2])𝑢1	X
iajs-2809	216	22	+	+	CCONJ
iajs-2809	216	23	(	(	PUNCT
iajs-2809	216	24	[	[	X
iajs-2809	216	25	0.25,2])𝑢2	0.25,2])𝑢2	X
iajs-2809	216	26	+	+	CCONJ
iajs-2809	216	27	(	(	PUNCT
iajs-2809	216	28	[	[	PUNCT
iajs-2809	216	29	1	1	NUM
iajs-2809	216	30	3	3	NUM
iajs-2809	216	31	,	,	PUNCT
iajs-2809	216	32	3.5	3.5	NUM
iajs-2809	216	33	]	]	PUNCT
iajs-2809	216	34	)	)	PUNCT
iajs-2809	216	35	𝑡	𝑡	PROPN
iajs-2809	216	36	=	=	SYM
iajs-2809	216	37	1	1	NUM
iajs-2809	216	38	−𝑢1	−𝑢1	NOUN
iajs-2809	216	39	+	+	CCONJ
iajs-2809	216	40	𝑢2	𝑢2	PROPN
iajs-2809	216	41	+	+	CCONJ
iajs-2809	216	42	𝑡	𝑡	PROPN
iajs-2809	216	43	≤	≤	ADV
iajs-2809	216	44	0	0	NUM
iajs-2809	216	45	,	,	PUNCT
iajs-2809	216	46	2𝑢1	2𝑢1	NUM
iajs-2809	216	47	+	+	SYM
iajs-2809	216	48	3𝑢2	3𝑢2	NUM
iajs-2809	216	49	−	−	PROPN
iajs-2809	216	50	15𝑡	15𝑡	NOUN
iajs-2809	216	51	≤	≤	ADV
iajs-2809	216	52	0	0	NUM
iajs-2809	216	53	,	,	PUNCT
iajs-2809	216	54	−𝑢1	−𝑢1	PROPN
iajs-2809	216	55	+	+	CCONJ
iajs-2809	216	56	3𝑡	3𝑡	NUM
iajs-2809	216	57	≤	≤	NUM
iajs-2809	216	58	0	0	NUM
iajs-2809	216	59	,	,	PUNCT
iajs-2809	216	60	𝑢1	𝑢1	NOUN
iajs-2809	216	61	,	,	PUNCT
iajs-2809	216	62	𝑢1	𝑢1	PROPN
iajs-2809	216	63	,	,	PUNCT
iajs-2809	216	64	𝑡	𝑡	X
iajs-2809	216	65	≥	≥	NOUN
iajs-2809	216	66	0	0	NUM
iajs-2809	216	67	.	.	PUNCT
iajs-2809	216	68	problem	problem	NOUN
iajs-2809	216	69	(	(	PUNCT
iajs-2809	216	70	22	22	NUM
iajs-2809	216	71	)	)	PUNCT
iajs-2809	216	72	is	be	AUX
iajs-2809	216	73	converted	convert	VERB
iajs-2809	216	74	to	to	ADP
iajs-2809	216	75	linear	linear	ADJ
iajs-2809	216	76	programming	programming	NOUN
iajs-2809	216	77	problems	problem	NOUN
iajs-2809	216	78	(	(	PUNCT
iajs-2809	216	79	24	24	NUM
iajs-2809	216	80	)	)	PUNCT
iajs-2809	216	81	&	&	CCONJ
iajs-2809	216	82	(	(	PUNCT
iajs-2809	216	83	25	25	NUM
iajs-2809	216	84	)	)	PUNCT
iajs-2809	216	85	.	.	PUNCT
iajs-2809	217	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	217	2	𝑍	𝑍	PROPN
iajs-2809	217	3	=	=	SYM
iajs-2809	217	4	𝑢1	𝑢1	NOUN
iajs-2809	217	5	+	+	CCONJ
iajs-2809	217	6	2𝑢2	2𝑢2	NUM
iajs-2809	217	7	+	+	CCONJ
iajs-2809	217	8	0𝑡	0𝑡	NOUN
iajs-2809	217	9	subject	subject	ADJ
iajs-2809	217	10	to	to	ADP
iajs-2809	217	11	:	:	PUNCT
iajs-2809	217	12	(	(	PUNCT
iajs-2809	217	13	24	24	NUM
iajs-2809	217	14	)	)	PUNCT
iajs-2809	217	15	0.5𝑢1	0.5𝑢1	NUM
iajs-2809	218	1	+	+	CCONJ
iajs-2809	218	2	0.5𝑢2	0.5𝑢2	NUM
iajs-2809	218	3	+	+	CCONJ
iajs-2809	218	4	𝑡	𝑡	X
iajs-2809	218	5	≤	≤	NUM
iajs-2809	218	6	1	1	NUM
iajs-2809	218	7	1.5𝑢1	1.5𝑢1	NUM
iajs-2809	218	8	+	+	NUM
iajs-2809	218	9	1.5𝑢2	1.5𝑢2	NUM
iajs-2809	218	10	+	+	CCONJ
iajs-2809	218	11	3𝑡	3𝑡	NUM
iajs-2809	218	12	≤	≤	NUM
iajs-2809	218	13	1	1	NUM
iajs-2809	218	14	−𝑢1	−𝑢1	NOUN
iajs-2809	218	15	+	+	CCONJ
iajs-2809	218	16	𝑢2	𝑢2	PROPN
iajs-2809	218	17	+	+	CCONJ
iajs-2809	218	18	𝑡	𝑡	PROPN
iajs-2809	218	19	≤	≤	ADV
iajs-2809	218	20	0	0	NUM
iajs-2809	218	21	,	,	PUNCT
iajs-2809	218	22	2𝑢1	2𝑢1	NUM
iajs-2809	218	23	+	+	SYM
iajs-2809	218	24	3𝑢2	3𝑢2	NUM
iajs-2809	218	25	−	−	PROPN
iajs-2809	218	26	15𝑡	15𝑡	NOUN
iajs-2809	218	27	≤	≤	ADV
iajs-2809	218	28	0	0	NUM
iajs-2809	218	29	,	,	PUNCT
iajs-2809	218	30	−𝑢1	−𝑢1	PROPN
iajs-2809	218	31	+	+	CCONJ
iajs-2809	218	32	3𝑡	3𝑡	NUM
iajs-2809	218	33	≤	≤	NUM
iajs-2809	218	34	0	0	NUM
iajs-2809	218	35	,	,	PUNCT
iajs-2809	218	36	𝑢1	𝑢1	NOUN
iajs-2809	218	37	,	,	PUNCT
iajs-2809	218	38	𝑢1	𝑢1	PROPN
iajs-2809	218	39	,	,	PUNCT
iajs-2809	218	40	𝑡	𝑡	X
iajs-2809	218	41	≥	≥	NOUN
iajs-2809	218	42	0	0	NUM
iajs-2809	218	43	.	.	PUNCT
iajs-2809	219	1	ibn	ibn	PROPN
iajs-2809	219	2	al	al	PROPN
iajs-2809	219	3	-	-	PUNCT
iajs-2809	219	4	haitham	haitham	PROPN
iajs-2809	219	5	jour	jour	X
iajs-2809	219	6	.	.	PROPN
iajs-2809	219	7	for	for	ADP
iajs-2809	219	8	pure	pure	ADJ
iajs-2809	219	9	&	&	CCONJ
iajs-2809	219	10	appl	appl	PROPN
iajs-2809	219	11	.	.	PUNCT
iajs-2809	220	1	sci	sci	PROPN
iajs-2809	220	2	.	.	PROPN
iajs-2809	221	1	53	53	NUM
iajs-2809	221	2	(	(	PUNCT
iajs-2809	221	3	2)2022	2)2022	VERB
iajs-2809	221	4	80	80	NUM
iajs-2809	221	5	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	221	6	𝑍	𝑍	PROPN
iajs-2809	221	7	=	=	SYM
iajs-2809	221	8	3𝑢1	3𝑢1	NUM
iajs-2809	221	9	+	+	NUM
iajs-2809	221	10	4𝑢2	4𝑢2	NUM
iajs-2809	222	1	+	+	CCONJ
iajs-2809	222	2	0𝑡	0𝑡	NOUN
iajs-2809	222	3	subject	subject	ADJ
iajs-2809	222	4	to	to	ADP
iajs-2809	222	5	:	:	PUNCT
iajs-2809	222	6	(	(	PUNCT
iajs-2809	222	7	25	25	NUM
iajs-2809	222	8	)	)	PUNCT
iajs-2809	222	9	0.5𝑢1	0.5𝑢1	NUM
iajs-2809	223	1	+	+	CCONJ
iajs-2809	223	2	0.5𝑢2	0.5𝑢2	NUM
iajs-2809	223	3	+	+	CCONJ
iajs-2809	223	4	𝑡	𝑡	X
iajs-2809	223	5	≤	≤	NUM
iajs-2809	223	6	1	1	NUM
iajs-2809	223	7	1.5𝑢1	1.5𝑢1	NUM
iajs-2809	223	8	+	+	NUM
iajs-2809	223	9	1.5𝑢2	1.5𝑢2	NUM
iajs-2809	223	10	+	+	CCONJ
iajs-2809	223	11	3𝑡	3𝑡	NUM
iajs-2809	223	12	≥	≥	NUM
iajs-2809	223	13	1	1	NUM
iajs-2809	223	14	−𝑢1	−𝑢1	NOUN
iajs-2809	223	15	+	+	CCONJ
iajs-2809	223	16	𝑢2	𝑢2	PROPN
iajs-2809	223	17	+	+	CCONJ
iajs-2809	223	18	𝑡	𝑡	PROPN
iajs-2809	223	19	≤	≤	ADV
iajs-2809	223	20	0	0	NUM
iajs-2809	223	21	,	,	PUNCT
iajs-2809	223	22	2𝑢1	2𝑢1	NUM
iajs-2809	223	23	+	+	SYM
iajs-2809	223	24	3𝑢2	3𝑢2	NUM
iajs-2809	223	25	−	−	PROPN
iajs-2809	223	26	15𝑡	15𝑡	NOUN
iajs-2809	223	27	≤	≤	ADV
iajs-2809	223	28	0	0	NUM
iajs-2809	223	29	,	,	PUNCT
iajs-2809	223	30	−𝑢1	−𝑢1	PROPN
iajs-2809	223	31	+	+	CCONJ
iajs-2809	223	32	3𝑡	3𝑡	NUM
iajs-2809	223	33	≤	≤	NUM
iajs-2809	223	34	0	0	NUM
iajs-2809	223	35	,	,	PUNCT
iajs-2809	223	36	𝑢1	𝑢1	NOUN
iajs-2809	223	37	,	,	PUNCT
iajs-2809	223	38	𝑢1	𝑢1	PROPN
iajs-2809	223	39	,	,	PUNCT
iajs-2809	223	40	𝑡	𝑡	X
iajs-2809	223	41	≥	≥	NOUN
iajs-2809	223	42	0	0	NUM
iajs-2809	223	43	.	.	PUNCT
iajs-2809	224	1	and	and	CCONJ
iajs-2809	224	2	problem	problem	NOUN
iajs-2809	224	3	(	(	PUNCT
iajs-2809	224	4	23	23	NUM
iajs-2809	224	5	)	)	PUNCT
iajs-2809	224	6	is	be	AUX
iajs-2809	224	7	converted	convert	VERB
iajs-2809	224	8	to	to	ADP
iajs-2809	224	9	linear	linear	ADJ
iajs-2809	224	10	programming	programming	NOUN
iajs-2809	224	11	problems	problem	NOUN
iajs-2809	224	12	(	(	PUNCT
iajs-2809	224	13	26	26	NUM
iajs-2809	224	14	)	)	PUNCT
iajs-2809	224	15	&	&	CCONJ
iajs-2809	224	16	(	(	PUNCT
iajs-2809	224	17	27	27	NUM
iajs-2809	224	18	)	)	PUNCT
iajs-2809	224	19	.	.	PUNCT
iajs-2809	225	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	225	2	𝑍	𝑍	PROPN
iajs-2809	225	3	=	=	SYM
iajs-2809	225	4	0.5𝑢1	0.5𝑢1	NUM
iajs-2809	225	5	+	+	CCONJ
iajs-2809	225	6	𝑢2	𝑢2	PROPN
iajs-2809	225	7	+	+	CCONJ
iajs-2809	225	8	0𝑡	0𝑡	NOUN
iajs-2809	225	9	subject	subject	ADJ
iajs-2809	225	10	to	to	ADP
iajs-2809	225	11	:	:	PUNCT
iajs-2809	225	12	(	(	PUNCT
iajs-2809	225	13	26	26	NUM
iajs-2809	225	14	)	)	PUNCT
iajs-2809	225	15	0.25𝑢1	0.25𝑢1	NOUN
iajs-2809	226	1	+	+	CCONJ
iajs-2809	226	2	0.25𝑢2	0.25𝑢2	NUM
iajs-2809	227	1	+	+	CCONJ
iajs-2809	227	2	1	1	NUM
iajs-2809	227	3	3	3	NUM
iajs-2809	227	4	𝑡	𝑡	NOUN
iajs-2809	227	5	≤	≤	NUM
iajs-2809	227	6	1	1	NUM
iajs-2809	227	7	2𝑢1	2𝑢1	NUM
iajs-2809	227	8	+	+	CCONJ
iajs-2809	227	9	2𝑢2	2𝑢2	NUM
iajs-2809	227	10	+	+	CCONJ
iajs-2809	227	11	3.5𝑡	3.5𝑡	NUM
iajs-2809	227	12	≤	≤	NUM
iajs-2809	227	13	1	1	NUM
iajs-2809	227	14	−𝑢1	−𝑢1	NOUN
iajs-2809	227	15	+	+	CCONJ
iajs-2809	227	16	𝑢2	𝑢2	PROPN
iajs-2809	227	17	+	+	CCONJ
iajs-2809	227	18	𝑡	𝑡	PROPN
iajs-2809	227	19	≤	≤	ADV
iajs-2809	227	20	0	0	NUM
iajs-2809	227	21	,	,	PUNCT
iajs-2809	227	22	2𝑢1	2𝑢1	NUM
iajs-2809	227	23	+	+	SYM
iajs-2809	227	24	3𝑢2	3𝑢2	NUM
iajs-2809	227	25	−	−	PROPN
iajs-2809	227	26	15𝑡	15𝑡	NOUN
iajs-2809	227	27	≤	≤	ADV
iajs-2809	227	28	0	0	NUM
iajs-2809	227	29	,	,	PUNCT
iajs-2809	227	30	−𝑢1	−𝑢1	PROPN
iajs-2809	227	31	+	+	CCONJ
iajs-2809	227	32	3𝑡	3𝑡	NUM
iajs-2809	227	33	≤	≤	NUM
iajs-2809	227	34	0	0	NUM
iajs-2809	227	35	,	,	PUNCT
iajs-2809	227	36	𝑢1	𝑢1	NOUN
iajs-2809	227	37	,	,	PUNCT
iajs-2809	227	38	𝑢1	𝑢1	PROPN
iajs-2809	227	39	,	,	PUNCT
iajs-2809	227	40	𝑡	𝑡	X
iajs-2809	227	41	≥	≥	NOUN
iajs-2809	227	42	0	0	NUM
iajs-2809	227	43	.	.	PUNCT
iajs-2809	228	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	228	2	𝑍	𝑍	PROPN
iajs-2809	228	3	=	=	SYM
iajs-2809	228	4	4𝑢1	4𝑢1	NUM
iajs-2809	229	1	+	+	CCONJ
iajs-2809	229	2	4.5𝑢2	4.5𝑢2	NUM
iajs-2809	229	3	+	+	CCONJ
iajs-2809	229	4	0𝑡	0𝑡	NOUN
iajs-2809	229	5	subject	subject	ADJ
iajs-2809	229	6	to	to	ADP
iajs-2809	229	7	:	:	PUNCT
iajs-2809	229	8	(	(	PUNCT
iajs-2809	229	9	27	27	NUM
iajs-2809	229	10	)	)	PUNCT
iajs-2809	229	11	0.25𝑢1	0.25𝑢1	NOUN
iajs-2809	230	1	+	+	CCONJ
iajs-2809	230	2	0.25𝑢2	0.25𝑢2	NUM
iajs-2809	231	1	+	+	CCONJ
iajs-2809	231	2	1	1	NUM
iajs-2809	231	3	3	3	NUM
iajs-2809	231	4	𝑡	𝑡	NOUN
iajs-2809	231	5	≤	≤	NUM
iajs-2809	231	6	1	1	NUM
iajs-2809	231	7	2𝑢1	2𝑢1	NUM
iajs-2809	231	8	+	+	CCONJ
iajs-2809	231	9	2𝑢2	2𝑢2	NUM
iajs-2809	231	10	+	+	CCONJ
iajs-2809	231	11	3.5𝑡	3.5𝑡	NUM
iajs-2809	231	12	≥	≥	NUM
iajs-2809	231	13	1	1	NUM
iajs-2809	231	14	−𝑢1	−𝑢1	NOUN
iajs-2809	231	15	+	+	CCONJ
iajs-2809	231	16	𝑢2	𝑢2	PROPN
iajs-2809	231	17	+	+	CCONJ
iajs-2809	231	18	𝑡	𝑡	PROPN
iajs-2809	231	19	≤	≤	ADV
iajs-2809	231	20	0	0	NUM
iajs-2809	231	21	,	,	PUNCT
iajs-2809	231	22	2𝑢1	2𝑢1	NUM
iajs-2809	231	23	+	+	SYM
iajs-2809	231	24	3𝑢2	3𝑢2	NUM
iajs-2809	231	25	−	−	PROPN
iajs-2809	231	26	15𝑡	15𝑡	NOUN
iajs-2809	231	27	≤	≤	ADV
iajs-2809	231	28	0	0	NUM
iajs-2809	231	29	,	,	PUNCT
iajs-2809	231	30	−𝑢1	−𝑢1	PROPN
iajs-2809	231	31	+	+	CCONJ
iajs-2809	231	32	3𝑡	3𝑡	NUM
iajs-2809	231	33	≤	≤	NUM
iajs-2809	231	34	0	0	NUM
iajs-2809	231	35	,	,	PUNCT
iajs-2809	231	36	𝑢1	𝑢1	NOUN
iajs-2809	231	37	,	,	PUNCT
iajs-2809	231	38	𝑢1	𝑢1	PROPN
iajs-2809	231	39	,	,	PUNCT
iajs-2809	231	40	𝑡	𝑡	X
iajs-2809	231	41	≥	≥	NOUN
iajs-2809	231	42	0	0	NUM
iajs-2809	231	43	.	.	PUNCT
iajs-2809	232	1	now	now	ADV
iajs-2809	232	2	we	we	PRON
iajs-2809	232	3	can	can	AUX
iajs-2809	232	4	solve	solve	VERB
iajs-2809	232	5	problems	problem	NOUN
iajs-2809	232	6	(	(	PUNCT
iajs-2809	232	7	24	24	NUM
iajs-2809	232	8	)	)	PUNCT
iajs-2809	232	9	,	,	PUNCT
iajs-2809	232	10	(	(	PUNCT
iajs-2809	232	11	25	25	NUM
iajs-2809	232	12	)	)	PUNCT
iajs-2809	232	13	,	,	PUNCT
iajs-2809	232	14	(	(	PUNCT
iajs-2809	232	15	26	26	NUM
iajs-2809	232	16	)	)	PUNCT
iajs-2809	232	17	and	and	CCONJ
iajs-2809	232	18	(	(	PUNCT
iajs-2809	232	19	27	27	NUM
iajs-2809	232	20	)	)	PUNCT
iajs-2809	232	21	.	.	PUNCT
iajs-2809	233	1	using	use	VERB
iajs-2809	233	2	the	the	DET
iajs-2809	233	3	simplex	simplex	NOUN
iajs-2809	233	4	routine	routine	NOUN
iajs-2809	233	5	,	,	PUNCT
iajs-2809	233	6	we	we	PRON
iajs-2809	233	7	obtain	obtain	VERB
iajs-2809	233	8	tht	tht	NOUN
iajs-2809	233	9	0.7154	0.7154	NUM
iajs-2809	233	10	is	be	AUX
iajs-2809	233	11	surely	surely	ADV
iajs-2809	233	12	the	the	DET
iajs-2809	233	13	worst	bad	ADJ
iajs-2809	233	14	optimum	optimum	NOUN
iajs-2809	233	15	,	,	PUNCT
iajs-2809	233	16	5.17	5.17	NUM
iajs-2809	233	17	is	be	AUX
iajs-2809	233	18	surely	surely	ADV
iajs-2809	233	19	the	the	DET
iajs-2809	233	20	best	good	ADJ
iajs-2809	233	21	optimum	optimum	NOUN
iajs-2809	233	22	,	,	PUNCT
iajs-2809	233	23	0.2767	0.2767	NUM
iajs-2809	233	24	is	be	AUX
iajs-2809	233	25	possibly	possibly	ADV
iajs-2809	233	26	the	the	DET
iajs-2809	233	27	worst	bad	ADJ
iajs-2809	233	28	optimum	optimum	NOUN
iajs-2809	233	29	,	,	PUNCT
iajs-2809	233	30	and	and	CCONJ
iajs-2809	233	31	13.85	13.85	NUM
iajs-2809	233	32	is	be	AUX
iajs-2809	233	33	possibly	possibly	ADV
iajs-2809	233	34	the	the	DET
iajs-2809	233	35	best	good	ADJ
iajs-2809	233	36	optimum	optimum	NOUN
iajs-2809	233	37	.	.	PUNCT
iajs-2809	234	1	the	the	DET
iajs-2809	234	2	optimal	optimal	ADJ
iajs-2809	234	3	solution	solution	NOUN
iajs-2809	234	4	of	of	ADP
iajs-2809	234	5	the	the	DET
iajs-2809	234	6	original	original	ADJ
iajs-2809	234	7	problem	problem	NOUN
iajs-2809	234	8	is	be	AUX
iajs-2809	234	9	(	(	PUNCT
iajs-2809	234	10	3.6	3.6	NUM
iajs-2809	234	11	,	,	PUNCT
iajs-2809	234	12	2.6	2.6	NUM
iajs-2809	234	13	)	)	PUNCT
iajs-2809	234	14	with	with	ADP
iajs-2809	234	15	the	the	DET
iajs-2809	234	16	objective	objective	ADJ
iajs-2809	234	17	value	value	NOUN
iajs-2809	234	18	(	(	PUNCT
iajs-2809	234	19	[	[	X
iajs-2809	234	20	0.7154	0.7154	NUM
iajs-2809	234	21	,	,	PUNCT
iajs-2809	234	22	5.17	5.17	NUM
iajs-2809	234	23	]	]	PUNCT
iajs-2809	234	24	,	,	PUNCT
iajs-2809	234	25	[	[	X
iajs-2809	234	26	0.2767	0.2767	NUM
iajs-2809	234	27	,	,	PUNCT
iajs-2809	234	28	13.85	13.85	NUM
iajs-2809	234	29	]	]	PUNCT
iajs-2809	234	30	)	)	PUNCT
iajs-2809	234	31	example2	example2	PROPN
iajs-2809	234	32	:	:	PUNCT
iajs-2809	234	33	consider	consider	VERB
iajs-2809	234	34	the	the	DET
iajs-2809	234	35	following	follow	VERB
iajs-2809	234	36	lfpp	lfpp	NOUN
iajs-2809	234	37	with	with	ADP
iajs-2809	234	38	rics	ric	NOUN
iajs-2809	234	39	in	in	ADP
iajs-2809	234	40	the	the	DET
iajs-2809	234	41	objective	objective	ADJ
iajs-2809	234	42	function	function	NOUN
iajs-2809	234	43	:	:	PUNCT
iajs-2809	234	44	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	234	45	𝑍	𝑍	PROPN
iajs-2809	234	46	=	=	SYM
iajs-2809	234	47	(	(	PUNCT
iajs-2809	234	48	[	[	X
iajs-2809	234	49	3.5,4.5	3.5,4.5	PROPN
iajs-2809	234	50	]	]	X
iajs-2809	234	51	,	,	PUNCT
iajs-2809	235	1	[	[	X
iajs-2809	235	2	3,5])𝑥1	3,5])𝑥1	NUM
iajs-2809	235	3	+	+	CCONJ
iajs-2809	235	4	(	(	PUNCT
iajs-2809	235	5	[	[	X
iajs-2809	235	6	2,3	2,3	NUM
iajs-2809	235	7	]	]	PUNCT
iajs-2809	235	8	,	,	PUNCT
iajs-2809	235	9	[	[	X
iajs-2809	235	10	1,4])𝑥2	1,4])𝑥2	NUM
iajs-2809	235	11	+	+	SYM
iajs-2809	235	12	(	(	PUNCT
iajs-2809	235	13	[	[	X
iajs-2809	235	14	8,10	8,10	NOUN
iajs-2809	235	15	]	]	PUNCT
iajs-2809	235	16	,	,	PUNCT
iajs-2809	235	17	[	[	X
iajs-2809	235	18	7,11	7,11	X
iajs-2809	235	19	]	]	X
iajs-2809	235	20	)	)	PUNCT
iajs-2809	235	21	(	(	PUNCT
iajs-2809	235	22	[	[	X
iajs-2809	235	23	1,1.5	1,1.5	X
iajs-2809	235	24	]	]	X
iajs-2809	235	25	,	,	PUNCT
iajs-2809	236	1	[	[	X
iajs-2809	236	2	0.5,2])𝑥1	0.5,2])𝑥1	X
iajs-2809	236	3	+	+	CCONJ
iajs-2809	236	4	(	(	PUNCT
iajs-2809	236	5	[	[	X
iajs-2809	236	6	1.5,1.75	1.5,1.75	NUM
iajs-2809	236	7	]	]	PUNCT
iajs-2809	236	8	,	,	PUNCT
iajs-2809	236	9	[	[	X
iajs-2809	236	10	1,2])𝑥2	1,2])𝑥2	NUM
iajs-2809	236	11	+	+	CCONJ
iajs-2809	236	12	(	(	PUNCT
iajs-2809	236	13	[	[	X
iajs-2809	236	14	4.5,5.5	4.5,5.5	NOUN
iajs-2809	236	15	]	]	X
iajs-2809	236	16	,	,	PUNCT
iajs-2809	237	1	[	[	X
iajs-2809	237	2	4,6	4,6	NOUN
iajs-2809	237	3	]	]	PUNCT
iajs-2809	237	4	)	)	PUNCT
iajs-2809	237	5	subject	subject	ADJ
iajs-2809	237	6	to	to	ADP
iajs-2809	237	7	:	:	PUNCT
iajs-2809	237	8	(	(	PUNCT
iajs-2809	237	9	28	28	NUM
iajs-2809	237	10	)	)	PUNCT
iajs-2809	237	11	ibn	ibn	PROPN
iajs-2809	237	12	al	al	PROPN
iajs-2809	237	13	-	-	PUNCT
iajs-2809	237	14	haitham	haitham	PROPN
iajs-2809	237	15	jour	jour	X
iajs-2809	237	16	.	.	PROPN
iajs-2809	238	1	for	for	ADP
iajs-2809	238	2	pure	pure	ADJ
iajs-2809	238	3	&	&	CCONJ
iajs-2809	238	4	appl	appl	PROPN
iajs-2809	238	5	.	.	PUNCT
iajs-2809	239	1	sci	sci	PROPN
iajs-2809	239	2	.	.	PROPN
iajs-2809	240	1	53	53	NUM
iajs-2809	240	2	(	(	PUNCT
iajs-2809	240	3	2)2022	2)2022	NUM
iajs-2809	240	4	81	81	NUM
iajs-2809	240	5	𝑥1	𝑥1	NOUN
iajs-2809	240	6	+	+	CCONJ
iajs-2809	240	7	3𝑥2	3𝑥2	NUM
iajs-2809	240	8	≤	≤	NUM
iajs-2809	240	9	30	30	NUM
iajs-2809	240	10	,	,	PUNCT
iajs-2809	240	11	−𝑥1	−𝑥1	PROPN
iajs-2809	240	12	+	+	CCONJ
iajs-2809	240	13	2𝑥2	2𝑥2	NUM
iajs-2809	240	14	≤	≤	NUM
iajs-2809	240	15	5	5	NUM
iajs-2809	240	16	,	,	PUNCT
iajs-2809	240	17	𝑥1	𝑥1	NOUN
iajs-2809	240	18	,	,	PUNCT
iajs-2809	240	19	𝑥2	𝑥2	PROPN
iajs-2809	240	20	≥	≥	NOUN
iajs-2809	240	21	0	0	NUM
iajs-2809	240	22	.	.	PUNCT
iajs-2809	240	23	solution	solution	NOUN
iajs-2809	240	24	problem	problem	NOUN
iajs-2809	240	25	(	(	PUNCT
iajs-2809	240	26	28	28	NUM
iajs-2809	240	27	)	)	PUNCT
iajs-2809	240	28	is	be	AUX
iajs-2809	240	29	transformed	transform	VERB
iajs-2809	240	30	into	into	ADP
iajs-2809	240	31	a	a	DET
iajs-2809	240	32	rough	rough	ADJ
iajs-2809	240	33	interval	interval	NOUN
iajs-2809	240	34	linear	linear	NOUN
iajs-2809	240	35	programming	programming	NOUN
iajs-2809	240	36	problem	problem	NOUN
iajs-2809	240	37	by	by	ADP
iajs-2809	240	38	charnes	charne	NOUN
iajs-2809	240	39	&	&	CCONJ
iajs-2809	240	40	cooper	cooper	PROPN
iajs-2809	240	41	method	method	VERB
iajs-2809	240	42	.	.	PUNCT
iajs-2809	241	1	we	we	PRON
iajs-2809	241	2	introduce	introduce	VERB
iajs-2809	241	3	𝑡	𝑡	NOUN
iajs-2809	241	4	=	=	NOUN
iajs-2809	241	5	1	1	NUM
iajs-2809	241	6	(	(	PUNCT
iajs-2809	241	7	[	[	X
iajs-2809	241	8	1,1.5],[0.5,2])𝑥1+([1.5,1.75],[1,2])𝑥2+([4.5,5.5],[4,6	1,1.5],[0.5,2])𝑥1+([1.5,1.75],[1,2])𝑥2+([4.5,5.5],[4,6	NUM
iajs-2809	241	9	]	]	PUNCT
iajs-2809	241	10	)	)	PUNCT
iajs-2809	241	11	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	241	12	𝑍	𝑍	PROPN
iajs-2809	241	13	=	=	SYM
iajs-2809	241	14	(	(	PUNCT
iajs-2809	241	15	[	[	X
iajs-2809	241	16	3.5,4.5	3.5,4.5	PROPN
iajs-2809	241	17	]	]	X
iajs-2809	241	18	,	,	PUNCT
iajs-2809	241	19	[	[	X
iajs-2809	241	20	3,5])𝑥1𝑡	3,5])𝑥1𝑡	NUM
iajs-2809	241	21	+	+	SYM
iajs-2809	241	22	(	(	PUNCT
iajs-2809	241	23	[	[	X
iajs-2809	241	24	2,3	2,3	NUM
iajs-2809	241	25	]	]	PUNCT
iajs-2809	241	26	,	,	PUNCT
iajs-2809	241	27	[	[	X
iajs-2809	241	28	1,4])𝑥2𝑡	1,4])𝑥2𝑡	NUM
iajs-2809	241	29	+	+	SYM
iajs-2809	241	30	(	(	PUNCT
iajs-2809	241	31	[	[	X
iajs-2809	241	32	8,10	8,10	NOUN
iajs-2809	241	33	]	]	PUNCT
iajs-2809	241	34	,	,	PUNCT
iajs-2809	241	35	[	[	X
iajs-2809	241	36	7,11])𝑡	7,11])𝑡	NUM
iajs-2809	241	37	subject	subject	NOUN
iajs-2809	241	38	to	to	ADP
iajs-2809	241	39	:	:	PUNCT
iajs-2809	241	40	(	(	PUNCT
iajs-2809	241	41	[	[	X
iajs-2809	241	42	1,1.5	1,1.5	X
iajs-2809	241	43	]	]	X
iajs-2809	241	44	,	,	PUNCT
iajs-2809	241	45	[	[	X
iajs-2809	241	46	0.5,2])𝑥1𝑡	0.5,2])𝑥1𝑡	NOUN
iajs-2809	241	47	+	+	CCONJ
iajs-2809	241	48	(	(	PUNCT
iajs-2809	241	49	[	[	X
iajs-2809	241	50	1.5,1.75	1.5,1.75	NUM
iajs-2809	241	51	]	]	PUNCT
iajs-2809	241	52	,	,	PUNCT
iajs-2809	241	53	[	[	X
iajs-2809	241	54	1,2])𝑥2𝑡	1,2])𝑥2𝑡	NUM
iajs-2809	241	55	+	+	SYM
iajs-2809	241	56	(	(	PUNCT
iajs-2809	241	57	[	[	X
iajs-2809	241	58	4.5,5.5	4.5,5.5	NOUN
iajs-2809	241	59	]	]	X
iajs-2809	241	60	,	,	PUNCT
iajs-2809	242	1	[	[	X
iajs-2809	242	2	4,6])𝑡	4,6])𝑡	NOUN
iajs-2809	242	3	=	=	SYM
iajs-2809	242	4	1	1	NUM
iajs-2809	242	5	𝑥1𝑡	𝑥1𝑡	NOUN
iajs-2809	242	6	+	+	NUM
iajs-2809	242	7	3𝑥2𝑡	3𝑥2𝑡	NOUN
iajs-2809	242	8	−	−	NOUN
iajs-2809	242	9	30𝑡	30𝑡	NUM
iajs-2809	242	10	≤	≤	ADV
iajs-2809	242	11	0	0	NUM
iajs-2809	242	12	,	,	PUNCT
iajs-2809	242	13	−𝑥1𝑡	−𝑥1𝑡	PUNCT
iajs-2809	242	14	+	+	CCONJ
iajs-2809	242	15	2𝑥2𝑡	2𝑥2𝑡	NUM
iajs-2809	242	16	−	−	NOUN
iajs-2809	242	17	5𝑡	5𝑡	ADJ
iajs-2809	242	18	≤	≤	NOUN
iajs-2809	242	19	0	0	NUM
iajs-2809	242	20	,	,	PUNCT
iajs-2809	242	21	𝑥1	𝑥1	NOUN
iajs-2809	242	22	,	,	PUNCT
iajs-2809	242	23	𝑥2	𝑥2	NOUN
iajs-2809	242	24	,	,	PUNCT
iajs-2809	242	25	𝑡	𝑡	X
iajs-2809	242	26	≥	≥	NOUN
iajs-2809	242	27	0	0	NUM
iajs-2809	242	28	.	.	PUNCT
iajs-2809	243	1	let	let	VERB
iajs-2809	243	2	𝑢𝑖	𝑢𝑖	INTJ
iajs-2809	243	3	=	=	SYM
iajs-2809	243	4	𝑥𝑖𝑡	𝑥𝑖𝑡	PROPN
iajs-2809	243	5	,	,	PUNCT
iajs-2809	243	6	𝑖	𝑖	X
iajs-2809	243	7	=	=	SYM
iajs-2809	243	8	1,2	1,2	NUM
iajs-2809	243	9	.	.	PUNCT
iajs-2809	244	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	244	2	𝑍	𝑍	PROPN
iajs-2809	244	3	=	=	SYM
iajs-2809	244	4	(	(	PUNCT
iajs-2809	244	5	[	[	X
iajs-2809	244	6	3.5,4.5	3.5,4.5	PROPN
iajs-2809	244	7	]	]	X
iajs-2809	244	8	,	,	PUNCT
iajs-2809	245	1	[	[	X
iajs-2809	245	2	3,5])𝑢1	3,5])𝑢1	NUM
iajs-2809	245	3	+	+	CCONJ
iajs-2809	245	4	(	(	PUNCT
iajs-2809	245	5	[	[	X
iajs-2809	245	6	2,3	2,3	NUM
iajs-2809	245	7	]	]	PUNCT
iajs-2809	245	8	,	,	PUNCT
iajs-2809	245	9	[	[	X
iajs-2809	245	10	1,4])𝑢2	1,4])𝑢2	NUM
iajs-2809	245	11	+	+	CCONJ
iajs-2809	245	12	(	(	PUNCT
iajs-2809	245	13	[	[	X
iajs-2809	245	14	8,10	8,10	NOUN
iajs-2809	245	15	]	]	PUNCT
iajs-2809	245	16	,	,	PUNCT
iajs-2809	245	17	[	[	X
iajs-2809	245	18	7,11])𝑡	7,11])𝑡	NUM
iajs-2809	245	19	subject	subject	NOUN
iajs-2809	245	20	to	to	ADP
iajs-2809	245	21	:	:	PUNCT
iajs-2809	245	22	(	(	PUNCT
iajs-2809	245	23	29	29	NUM
iajs-2809	245	24	)	)	PUNCT
iajs-2809	245	25	(	(	PUNCT
iajs-2809	246	1	[	[	X
iajs-2809	246	2	1,1.5	1,1.5	X
iajs-2809	246	3	]	]	X
iajs-2809	246	4	,	,	PUNCT
iajs-2809	246	5	[	[	X
iajs-2809	246	6	0.5,2])𝑢1	0.5,2])𝑢1	X
iajs-2809	246	7	+	+	CCONJ
iajs-2809	246	8	(	(	PUNCT
iajs-2809	246	9	[	[	X
iajs-2809	246	10	1.5,1.75	1.5,1.75	NUM
iajs-2809	246	11	]	]	PUNCT
iajs-2809	246	12	,	,	PUNCT
iajs-2809	246	13	[	[	X
iajs-2809	246	14	1,2])𝑢2	1,2])𝑢2	NUM
iajs-2809	246	15	+	+	CCONJ
iajs-2809	246	16	(	(	PUNCT
iajs-2809	246	17	[	[	X
iajs-2809	246	18	4.5,5.5	4.5,5.5	NOUN
iajs-2809	246	19	]	]	X
iajs-2809	246	20	,	,	PUNCT
iajs-2809	246	21	[	[	X
iajs-2809	246	22	4,6])𝑡	4,6])𝑡	NOUN
iajs-2809	246	23	=	=	SYM
iajs-2809	246	24	1	1	NUM
iajs-2809	246	25	𝑢1	𝑢1	NOUN
iajs-2809	246	26	+	+	CCONJ
iajs-2809	246	27	3𝑢2	3𝑢2	NUM
iajs-2809	246	28	−	−	NOUN
iajs-2809	246	29	30𝑡	30𝑡	NUM
iajs-2809	247	1	≤	≤	ADV
iajs-2809	247	2	0	0	NUM
iajs-2809	247	3	,	,	PUNCT
iajs-2809	247	4	−𝑢1	−𝑢1	PROPN
iajs-2809	247	5	+	+	CCONJ
iajs-2809	247	6	2𝑢2	2𝑢2	NUM
iajs-2809	247	7	−	−	NOUN
iajs-2809	247	8	5𝑡	5𝑡	ADJ
iajs-2809	247	9	≤	≤	NOUN
iajs-2809	247	10	0	0	NUM
iajs-2809	247	11	,	,	PUNCT
iajs-2809	247	12	𝑢1	𝑢1	NOUN
iajs-2809	247	13	,	,	PUNCT
iajs-2809	247	14	𝑢1	𝑢1	PROPN
iajs-2809	247	15	,	,	PUNCT
iajs-2809	247	16	𝑡	𝑡	X
iajs-2809	247	17	≥	≥	NOUN
iajs-2809	247	18	0	0	NUM
iajs-2809	247	19	.	.	PUNCT
iajs-2809	248	1	now	now	ADV
iajs-2809	248	2	by	by	ADP
iajs-2809	248	3	hamzehee	hamzehee	PROPN
iajs-2809	248	4	et	et	PROPN
iajs-2809	248	5	al	al	PROPN
iajs-2809	248	6	.	.	PUNCT
iajs-2809	249	1	[	[	X
iajs-2809	249	2	7	7	NUM
iajs-2809	249	3	]	]	X
iajs-2809	249	4	problem	problem	NOUN
iajs-2809	249	5	(	(	PUNCT
iajs-2809	249	6	29	29	NUM
iajs-2809	249	7	)	)	PUNCT
iajs-2809	249	8	can	can	AUX
iajs-2809	249	9	be	be	AUX
iajs-2809	249	10	changed	change	VERB
iajs-2809	249	11	into	into	ADP
iajs-2809	249	12	two	two	NUM
iajs-2809	249	13	linear	linear	ADJ
iajs-2809	249	14	programming	programming	NOUN
iajs-2809	249	15	with	with	ADP
iajs-2809	249	16	interval	interval	NOUN
iajs-2809	249	17	coefficients	coefficient	NOUN
iajs-2809	249	18	(	(	PUNCT
iajs-2809	249	19	30	30	NUM
iajs-2809	249	20	)	)	PUNCT
iajs-2809	249	21	&	&	CCONJ
iajs-2809	249	22	(	(	PUNCT
iajs-2809	249	23	31	31	NUM
iajs-2809	249	24	)	)	PUNCT
iajs-2809	249	25	.	.	PUNCT
iajs-2809	250	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	250	2	𝑍	𝑍	PROPN
iajs-2809	250	3	=	=	SYM
iajs-2809	250	4	(	(	PUNCT
iajs-2809	250	5	[	[	X
iajs-2809	250	6	3.5,4.5])𝑢1	3.5,4.5])𝑢1	NUM
iajs-2809	250	7	+	+	CCONJ
iajs-2809	250	8	(	(	PUNCT
iajs-2809	250	9	[	[	X
iajs-2809	250	10	2,3])𝑢2	2,3])𝑢2	NUM
iajs-2809	250	11	+	+	SYM
iajs-2809	250	12	(	(	PUNCT
iajs-2809	250	13	[	[	X
iajs-2809	250	14	8,10])𝑡	8,10])𝑡	NUM
iajs-2809	250	15	subject	subject	NOUN
iajs-2809	250	16	to	to	ADP
iajs-2809	250	17	:	:	PUNCT
iajs-2809	250	18	(	(	PUNCT
iajs-2809	250	19	30	30	NUM
iajs-2809	250	20	)	)	PUNCT
iajs-2809	250	21	(	(	PUNCT
iajs-2809	251	1	[	[	X
iajs-2809	251	2	1,1.5])𝑢1	1,1.5])𝑢1	NUM
iajs-2809	251	3	+	+	CCONJ
iajs-2809	251	4	(	(	PUNCT
iajs-2809	251	5	[	[	X
iajs-2809	251	6	1.5,1.75])𝑢2	1.5,1.75])𝑢2	NUM
iajs-2809	251	7	+	+	NOUN
iajs-2809	251	8	(	(	PUNCT
iajs-2809	251	9	[	[	X
iajs-2809	251	10	4.5,5.5])𝑡	4.5,5.5])𝑡	NOUN
iajs-2809	251	11	=	=	SYM
iajs-2809	251	12	1	1	NUM
iajs-2809	251	13	𝑢1	𝑢1	NOUN
iajs-2809	251	14	+	+	CCONJ
iajs-2809	251	15	3𝑢2	3𝑢2	NUM
iajs-2809	251	16	−	−	NOUN
iajs-2809	251	17	30𝑡	30𝑡	NUM
iajs-2809	251	18	≤	≤	ADV
iajs-2809	251	19	0	0	NUM
iajs-2809	251	20	,	,	PUNCT
iajs-2809	251	21	−𝑢1	−𝑢1	PROPN
iajs-2809	251	22	+	+	CCONJ
iajs-2809	251	23	2𝑢2	2𝑢2	NUM
iajs-2809	252	1	−	−	NOUN
iajs-2809	252	2	5𝑡	5𝑡	ADJ
iajs-2809	252	3	≤	≤	NOUN
iajs-2809	252	4	0	0	NUM
iajs-2809	252	5	,	,	PUNCT
iajs-2809	252	6	𝑢1	𝑢1	NOUN
iajs-2809	252	7	,	,	PUNCT
iajs-2809	252	8	𝑢1	𝑢1	PROPN
iajs-2809	252	9	,	,	PUNCT
iajs-2809	252	10	𝑡	𝑡	X
iajs-2809	252	11	≥	≥	NOUN
iajs-2809	252	12	0	0	NUM
iajs-2809	252	13	.	.	PUNCT
iajs-2809	253	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	253	2	𝑍	𝑍	PROPN
iajs-2809	253	3	=	=	SYM
iajs-2809	253	4	(	(	PUNCT
iajs-2809	253	5	[	[	X
iajs-2809	253	6	3,5])𝑢1	3,5])𝑢1	NUM
iajs-2809	253	7	+	+	CCONJ
iajs-2809	253	8	(	(	PUNCT
iajs-2809	253	9	[	[	X
iajs-2809	253	10	1,4])𝑢2	1,4])𝑢2	NUM
iajs-2809	253	11	+	+	CCONJ
iajs-2809	253	12	(	(	PUNCT
iajs-2809	253	13	[	[	X
iajs-2809	253	14	7,11])𝑡	7,11])𝑡	NUM
iajs-2809	253	15	subject	subject	NOUN
iajs-2809	253	16	to	to	ADP
iajs-2809	253	17	:	:	PUNCT
iajs-2809	253	18	(	(	PUNCT
iajs-2809	253	19	31	31	NUM
iajs-2809	253	20	)	)	PUNCT
iajs-2809	253	21	(	(	PUNCT
iajs-2809	254	1	[	[	X
iajs-2809	254	2	0.5,2])𝑢1	0.5,2])𝑢1	X
iajs-2809	254	3	+	+	CCONJ
iajs-2809	254	4	(	(	PUNCT
iajs-2809	254	5	[	[	X
iajs-2809	254	6	1,2])𝑢2	1,2])𝑢2	NUM
iajs-2809	254	7	+	+	CCONJ
iajs-2809	254	8	(	(	PUNCT
iajs-2809	254	9	[	[	X
iajs-2809	254	10	4,6])𝑡	4,6])𝑡	NOUN
iajs-2809	254	11	=	=	SYM
iajs-2809	254	12	1	1	NUM
iajs-2809	254	13	𝑢1	𝑢1	NOUN
iajs-2809	254	14	+	+	CCONJ
iajs-2809	254	15	3𝑢2	3𝑢2	NUM
iajs-2809	254	16	−	−	NOUN
iajs-2809	254	17	30𝑡	30𝑡	NUM
iajs-2809	254	18	≤	≤	ADV
iajs-2809	254	19	0	0	NUM
iajs-2809	254	20	,	,	PUNCT
iajs-2809	254	21	−𝑢1	−𝑢1	PROPN
iajs-2809	254	22	+	+	CCONJ
iajs-2809	254	23	2𝑢2	2𝑢2	NUM
iajs-2809	254	24	−	−	NOUN
iajs-2809	254	25	5𝑡	5𝑡	ADJ
iajs-2809	254	26	≤	≤	NOUN
iajs-2809	254	27	0	0	NUM
iajs-2809	254	28	,	,	PUNCT
iajs-2809	254	29	𝑢1	𝑢1	NOUN
iajs-2809	254	30	,	,	PUNCT
iajs-2809	254	31	𝑢1	𝑢1	PROPN
iajs-2809	254	32	,	,	PUNCT
iajs-2809	254	33	𝑡	𝑡	X
iajs-2809	254	34	≥	≥	NOUN
iajs-2809	254	35	0	0	NUM
iajs-2809	254	36	.	.	PUNCT
iajs-2809	254	37	problem	problem	NOUN
iajs-2809	254	38	(	(	PUNCT
iajs-2809	254	39	30	30	NUM
iajs-2809	254	40	)	)	PUNCT
iajs-2809	254	41	is	be	AUX
iajs-2809	254	42	converted	convert	VERB
iajs-2809	254	43	to	to	ADP
iajs-2809	254	44	linear	linear	ADJ
iajs-2809	254	45	programming	programming	NOUN
iajs-2809	254	46	problems	problem	NOUN
iajs-2809	254	47	(	(	PUNCT
iajs-2809	254	48	32	32	NUM
iajs-2809	254	49	)	)	PUNCT
iajs-2809	254	50	&	&	CCONJ
iajs-2809	254	51	(	(	PUNCT
iajs-2809	254	52	33	33	NUM
iajs-2809	254	53	)	)	PUNCT
iajs-2809	254	54	.	.	PUNCT
iajs-2809	255	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	255	2	𝑍	𝑍	PROPN
iajs-2809	255	3	=	=	SYM
iajs-2809	255	4	3.5𝑢1	3.5𝑢1	NUM
iajs-2809	255	5	+	+	NUM
iajs-2809	255	6	2𝑢2	2𝑢2	NUM
iajs-2809	255	7	+	+	CCONJ
iajs-2809	255	8	8𝑡	8𝑡	NOUN
iajs-2809	255	9	subject	subject	ADJ
iajs-2809	255	10	to	to	ADP
iajs-2809	255	11	:	:	PUNCT
iajs-2809	255	12	(	(	PUNCT
iajs-2809	255	13	32	32	NUM
iajs-2809	255	14	)	)	PUNCT
iajs-2809	255	15	ibn	ibn	PROPN
iajs-2809	255	16	al	al	PROPN
iajs-2809	255	17	-	-	PUNCT
iajs-2809	255	18	haitham	haitham	PROPN
iajs-2809	255	19	jour	jour	X
iajs-2809	255	20	.	.	PROPN
iajs-2809	256	1	for	for	ADP
iajs-2809	256	2	pure	pure	ADJ
iajs-2809	256	3	&	&	CCONJ
iajs-2809	256	4	appl	appl	PROPN
iajs-2809	256	5	.	.	PUNCT
iajs-2809	257	1	sci	sci	PROPN
iajs-2809	257	2	.	.	PROPN
iajs-2809	258	1	53	53	NUM
iajs-2809	258	2	(	(	PUNCT
iajs-2809	258	3	2)2022	2)2022	VERB
iajs-2809	258	4	82	82	NUM
iajs-2809	258	5	𝑢1	𝑢1	NOUN
iajs-2809	258	6	+	+	CCONJ
iajs-2809	258	7	1.5𝑢2	1.5𝑢2	NUM
iajs-2809	258	8	+	+	CCONJ
iajs-2809	258	9	4.5𝑡	4.5𝑡	NUM
iajs-2809	258	10	≤	≤	NUM
iajs-2809	258	11	1	1	NUM
iajs-2809	258	12	1.5𝑢1	1.5𝑢1	NUM
iajs-2809	258	13	+	+	SYM
iajs-2809	258	14	1.75𝑢2	1.75𝑢2	NUM
iajs-2809	258	15	+	+	CCONJ
iajs-2809	258	16	5.5𝑡	5.5𝑡	NUM
iajs-2809	258	17	≤	≤	NUM
iajs-2809	258	18	1	1	NUM
iajs-2809	258	19	𝑢1	𝑢1	NOUN
iajs-2809	258	20	+	+	CCONJ
iajs-2809	258	21	3𝑢2	3𝑢2	NUM
iajs-2809	258	22	−	−	NOUN
iajs-2809	258	23	30𝑡	30𝑡	NUM
iajs-2809	258	24	≤	≤	ADV
iajs-2809	258	25	0	0	NUM
iajs-2809	258	26	,	,	PUNCT
iajs-2809	258	27	−𝑢1	−𝑢1	PROPN
iajs-2809	258	28	+	+	CCONJ
iajs-2809	258	29	2𝑢2	2𝑢2	NUM
iajs-2809	258	30	−	−	NOUN
iajs-2809	258	31	5𝑡	5𝑡	ADJ
iajs-2809	258	32	≤	≤	NOUN
iajs-2809	258	33	0	0	NUM
iajs-2809	258	34	,	,	PUNCT
iajs-2809	258	35	𝑢1	𝑢1	NOUN
iajs-2809	258	36	,	,	PUNCT
iajs-2809	258	37	𝑢1	𝑢1	PROPN
iajs-2809	258	38	,	,	PUNCT
iajs-2809	258	39	𝑡	𝑡	X
iajs-2809	258	40	≥	≥	NOUN
iajs-2809	258	41	0	0	NUM
iajs-2809	258	42	.	.	PUNCT
iajs-2809	259	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	259	2	𝑍	𝑍	PROPN
iajs-2809	259	3	=	=	SYM
iajs-2809	259	4	4.5𝑢1	4.5𝑢1	NUM
iajs-2809	259	5	+	+	SYM
iajs-2809	259	6	3𝑢2	3𝑢2	NUM
iajs-2809	259	7	+	+	SYM
iajs-2809	259	8	10𝑡	10𝑡	NOUN
iajs-2809	259	9	subject	subject	ADJ
iajs-2809	259	10	to	to	ADP
iajs-2809	259	11	:	:	PUNCT
iajs-2809	259	12	(	(	PUNCT
iajs-2809	259	13	33	33	NUM
iajs-2809	259	14	)	)	PUNCT
iajs-2809	259	15	𝑢1	𝑢1	NOUN
iajs-2809	259	16	+	+	CCONJ
iajs-2809	259	17	1.5𝑢2	1.5𝑢2	NUM
iajs-2809	260	1	+	+	CCONJ
iajs-2809	260	2	4.5𝑡	4.5𝑡	NUM
iajs-2809	260	3	≤	≤	NUM
iajs-2809	260	4	1	1	NUM
iajs-2809	260	5	1.5𝑢1	1.5𝑢1	NUM
iajs-2809	260	6	+	+	SYM
iajs-2809	260	7	1.75𝑢2	1.75𝑢2	NUM
iajs-2809	260	8	+	+	CCONJ
iajs-2809	260	9	5.5𝑡	5.5𝑡	NUM
iajs-2809	260	10	≥	≥	NOUN
iajs-2809	260	11	1	1	NUM
iajs-2809	260	12	𝑢1	𝑢1	PROPN
iajs-2809	260	13	+	+	CCONJ
iajs-2809	260	14	3𝑢2	3𝑢2	NUM
iajs-2809	260	15	−	−	NOUN
iajs-2809	260	16	30𝑡	30𝑡	NUM
iajs-2809	260	17	≤	≤	ADV
iajs-2809	260	18	0	0	NUM
iajs-2809	260	19	,	,	PUNCT
iajs-2809	260	20	−𝑢1	−𝑢1	PROPN
iajs-2809	260	21	+	+	CCONJ
iajs-2809	260	22	2𝑢2	2𝑢2	NUM
iajs-2809	260	23	−	−	NOUN
iajs-2809	260	24	5𝑡	5𝑡	ADJ
iajs-2809	260	25	≤	≤	NOUN
iajs-2809	260	26	0	0	NUM
iajs-2809	260	27	,	,	PUNCT
iajs-2809	260	28	𝑢1	𝑢1	NOUN
iajs-2809	260	29	,	,	PUNCT
iajs-2809	260	30	𝑢1	𝑢1	PROPN
iajs-2809	260	31	,	,	PUNCT
iajs-2809	260	32	𝑡	𝑡	X
iajs-2809	260	33	≥	≥	NOUN
iajs-2809	260	34	0	0	NUM
iajs-2809	260	35	.	.	PUNCT
iajs-2809	261	1	and	and	CCONJ
iajs-2809	261	2	the	the	DET
iajs-2809	261	3	problem	problem	NOUN
iajs-2809	261	4	(	(	PUNCT
iajs-2809	261	5	31	31	NUM
iajs-2809	261	6	)	)	PUNCT
iajs-2809	261	7	is	be	AUX
iajs-2809	261	8	converted	convert	VERB
iajs-2809	261	9	to	to	ADP
iajs-2809	261	10	linear	linear	ADJ
iajs-2809	261	11	programming	programming	NOUN
iajs-2809	261	12	problems	problem	NOUN
iajs-2809	261	13	(	(	PUNCT
iajs-2809	261	14	34	34	NUM
iajs-2809	261	15	)	)	PUNCT
iajs-2809	261	16	&	&	CCONJ
iajs-2809	261	17	(	(	PUNCT
iajs-2809	261	18	35	35	NUM
iajs-2809	261	19	)	)	PUNCT
iajs-2809	261	20	.	.	PUNCT
iajs-2809	262	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	262	2	𝑍	𝑍	PROPN
iajs-2809	262	3	=	=	SYM
iajs-2809	262	4	3𝑢1	3𝑢1	PROPN
iajs-2809	262	5	+	+	CCONJ
iajs-2809	262	6	𝑢2	𝑢2	PROPN
iajs-2809	262	7	+	+	CCONJ
iajs-2809	262	8	7𝑡	7𝑡	NOUN
iajs-2809	262	9	subject	subject	ADJ
iajs-2809	262	10	to	to	ADP
iajs-2809	262	11	:	:	PUNCT
iajs-2809	262	12	(	(	PUNCT
iajs-2809	262	13	34	34	NUM
iajs-2809	262	14	)	)	PUNCT
iajs-2809	262	15	0.5𝑢1	0.5𝑢1	NUM
iajs-2809	263	1	+	+	CCONJ
iajs-2809	263	2	𝑢2	𝑢2	PROPN
iajs-2809	263	3	+	+	CCONJ
iajs-2809	263	4	4𝑡	4𝑡	NUM
iajs-2809	263	5	≤	≤	NUM
iajs-2809	263	6	1	1	NUM
iajs-2809	263	7	2𝑢1	2𝑢1	NUM
iajs-2809	263	8	+	+	CCONJ
iajs-2809	263	9	2𝑢2	2𝑢2	NUM
iajs-2809	263	10	+	+	CCONJ
iajs-2809	263	11	6𝑡	6𝑡	NUM
iajs-2809	263	12	≤	≤	NUM
iajs-2809	263	13	1	1	NUM
iajs-2809	263	14	𝑢1	𝑢1	NOUN
iajs-2809	263	15	+	+	CCONJ
iajs-2809	263	16	3𝑢2	3𝑢2	NUM
iajs-2809	263	17	−	−	NOUN
iajs-2809	263	18	30𝑡	30𝑡	NUM
iajs-2809	263	19	≤	≤	ADV
iajs-2809	263	20	0	0	NUM
iajs-2809	263	21	,	,	PUNCT
iajs-2809	263	22	−𝑢1	−𝑢1	PROPN
iajs-2809	263	23	+	+	CCONJ
iajs-2809	263	24	2𝑢2	2𝑢2	NUM
iajs-2809	263	25	−	−	NOUN
iajs-2809	263	26	5𝑡	5𝑡	ADJ
iajs-2809	263	27	≤	≤	NOUN
iajs-2809	263	28	0	0	NUM
iajs-2809	263	29	,	,	PUNCT
iajs-2809	263	30	𝑢1	𝑢1	NOUN
iajs-2809	263	31	,	,	PUNCT
iajs-2809	263	32	𝑢1	𝑢1	PROPN
iajs-2809	263	33	,	,	PUNCT
iajs-2809	263	34	𝑡	𝑡	X
iajs-2809	263	35	≥	≥	NOUN
iajs-2809	263	36	0	0	NUM
iajs-2809	263	37	.	.	PUNCT
iajs-2809	264	1	𝑀𝑎𝑥.	𝑀𝑎𝑥.	PROPN
iajs-2809	264	2	𝑍	𝑍	PROPN
iajs-2809	264	3	=	=	SYM
iajs-2809	264	4	5𝑢1	5𝑢1	NUM
iajs-2809	264	5	+	+	NUM
iajs-2809	264	6	4𝑢2	4𝑢2	NUM
iajs-2809	264	7	+	+	SYM
iajs-2809	264	8	11𝑡	11𝑡	NOUN
iajs-2809	264	9	subject	subject	VERB
iajs-2809	264	10	to	to	ADP
iajs-2809	264	11	:	:	PUNCT
iajs-2809	264	12	(	(	PUNCT
iajs-2809	264	13	35	35	NUM
iajs-2809	264	14	)	)	PUNCT
iajs-2809	264	15	0.5𝑢1	0.5𝑢1	NUM
iajs-2809	265	1	+	+	CCONJ
iajs-2809	265	2	𝑢2	𝑢2	PROPN
iajs-2809	265	3	+	+	CCONJ
iajs-2809	265	4	4𝑡	4𝑡	NUM
iajs-2809	265	5	≤	≤	NUM
iajs-2809	265	6	1	1	NUM
iajs-2809	265	7	2𝑢1	2𝑢1	NUM
iajs-2809	265	8	+	+	CCONJ
iajs-2809	265	9	2𝑢2	2𝑢2	NUM
iajs-2809	265	10	+	+	CCONJ
iajs-2809	265	11	6𝑡	6𝑡	NUM
iajs-2809	265	12	≥	≥	NOUN
iajs-2809	265	13	1	1	NUM
iajs-2809	265	14	𝑢1	𝑢1	NOUN
iajs-2809	265	15	+	+	CCONJ
iajs-2809	265	16	3𝑢2	3𝑢2	NUM
iajs-2809	265	17	−	−	NOUN
iajs-2809	265	18	30𝑡	30𝑡	NUM
iajs-2809	265	19	≤	≤	ADV
iajs-2809	265	20	0	0	NUM
iajs-2809	265	21	,	,	PUNCT
iajs-2809	265	22	−𝑢1	−𝑢1	PROPN
iajs-2809	265	23	+	+	CCONJ
iajs-2809	265	24	2𝑢2	2𝑢2	NUM
iajs-2809	265	25	−	−	NOUN
iajs-2809	265	26	5𝑡	5𝑡	ADJ
iajs-2809	265	27	≤	≤	NOUN
iajs-2809	265	28	0	0	NUM
iajs-2809	265	29	,	,	PUNCT
iajs-2809	265	30	𝑢1	𝑢1	NOUN
iajs-2809	265	31	,	,	PUNCT
iajs-2809	265	32	𝑢1	𝑢1	PROPN
iajs-2809	265	33	,	,	PUNCT
iajs-2809	265	34	𝑡	𝑡	X
iajs-2809	265	35	≥	≥	NOUN
iajs-2809	265	36	0	0	NUM
iajs-2809	265	37	.	.	PUNCT
iajs-2809	266	1	now	now	ADV
iajs-2809	266	2	,	,	PUNCT
iajs-2809	266	3	we	we	PRON
iajs-2809	266	4	can	can	AUX
iajs-2809	266	5	solve	solve	VERB
iajs-2809	266	6	problems	problem	NOUN
iajs-2809	266	7	(	(	PUNCT
iajs-2809	266	8	32	32	NUM
iajs-2809	266	9	)	)	PUNCT
iajs-2809	266	10	,	,	PUNCT
iajs-2809	266	11	(	(	PUNCT
iajs-2809	266	12	33	33	NUM
iajs-2809	266	13	)	)	PUNCT
iajs-2809	266	14	,	,	PUNCT
iajs-2809	266	15	(	(	PUNCT
iajs-2809	266	16	34	34	NUM
iajs-2809	266	17	)	)	PUNCT
iajs-2809	266	18	,	,	PUNCT
iajs-2809	266	19	and	and	CCONJ
iajs-2809	266	20	(	(	PUNCT
iajs-2809	266	21	35	35	NUM
iajs-2809	266	22	)	)	PUNCT
iajs-2809	266	23	.	.	PUNCT
iajs-2809	267	1	by	by	ADP
iajs-2809	267	2	using	use	VERB
iajs-2809	267	3	the	the	DET
iajs-2809	267	4	simplex	simplex	NOUN
iajs-2809	267	5	routine	routine	NOUN
iajs-2809	267	6	,	,	PUNCT
iajs-2809	267	7	we	we	PRON
iajs-2809	267	8	obtain	obtain	VERB
iajs-2809	267	9	that	that	SCONJ
iajs-2809	267	10	2.2376	2.2376	NUM
iajs-2809	267	11	is	be	AUX
iajs-2809	267	12	surely	surely	ADV
iajs-2809	267	13	the	the	DET
iajs-2809	267	14	worst	bad	ADJ
iajs-2809	267	15	optimum	optimum	NOUN
iajs-2809	267	16	,	,	PUNCT
iajs-2809	267	17	4.2028	4.2028	NUM
iajs-2809	267	18	is	be	AUX
iajs-2809	267	19	surely	surely	ADV
iajs-2809	267	20	the	the	DET
iajs-2809	267	21	best	good	ADJ
iajs-2809	267	22	optimum	optimum	NOUN
iajs-2809	267	23	,	,	PUNCT
iajs-2809	267	24	1.469	1.469	NUM
iajs-2809	267	25	is	be	AUX
iajs-2809	267	26	possibly	possibly	ADV
iajs-2809	267	27	the	the	DET
iajs-2809	267	28	worst	bad	ADJ
iajs-2809	267	29	optimum	optimum	ADJ
iajs-2809	267	30	and	and	CCONJ
iajs-2809	267	31	8.47	8.47	NUM
iajs-2809	267	32	is	be	AUX
iajs-2809	267	33	possibly	possibly	ADV
iajs-2809	267	34	the	the	DET
iajs-2809	267	35	best	good	ADJ
iajs-2809	267	36	optimum	optimum	NOUN
iajs-2809	267	37	.	.	PUNCT
iajs-2809	268	1	the	the	DET
iajs-2809	268	2	optimal	optimal	ADJ
iajs-2809	268	3	solution	solution	NOUN
iajs-2809	268	4	of	of	ADP
iajs-2809	268	5	the	the	DET
iajs-2809	268	6	original	original	ADJ
iajs-2809	268	7	problem	problem	NOUN
iajs-2809	268	8	is	be	AUX
iajs-2809	268	9	(	(	PUNCT
iajs-2809	268	10	30	30	NUM
iajs-2809	268	11	,	,	PUNCT
iajs-2809	268	12	0	0	NUM
iajs-2809	268	13	)	)	PUNCT
iajs-2809	268	14	with	with	ADP
iajs-2809	268	15	the	the	DET
iajs-2809	268	16	objective	objective	ADJ
iajs-2809	268	17	value	value	NOUN
iajs-2809	268	18	(	(	PUNCT
iajs-2809	268	19	[	[	X
iajs-2809	268	20	2.2376	2.2376	NUM
iajs-2809	268	21	,	,	PUNCT
iajs-2809	268	22	4.2028	4.2028	NUM
iajs-2809	268	23	]	]	PUNCT
iajs-2809	268	24	,	,	PUNCT
iajs-2809	268	25	[	[	X
iajs-2809	268	26	1.469	1.469	NUM
iajs-2809	268	27	,	,	PUNCT
iajs-2809	268	28	8.47	8.47	NUM
iajs-2809	268	29	]	]	PUNCT
iajs-2809	268	30	)	)	PUNCT
iajs-2809	268	31	5	5	NUM
iajs-2809	268	32	.	.	X
iajs-2809	268	33	conclusion	conclusion	NOUN
iajs-2809	268	34	in	in	ADP
iajs-2809	268	35	this	this	DET
iajs-2809	268	36	study	study	NOUN
iajs-2809	268	37	,	,	PUNCT
iajs-2809	268	38	we	we	PRON
iajs-2809	268	39	proposed	propose	VERB
iajs-2809	268	40	a	a	DET
iajs-2809	268	41	new	new	ADJ
iajs-2809	268	42	approach	approach	NOUN
iajs-2809	268	43	to	to	ADP
iajs-2809	268	44	solving	solve	VERB
iajs-2809	268	45	a	a	DET
iajs-2809	268	46	linear	linear	ADJ
iajs-2809	268	47	fractional	fractional	ADJ
iajs-2809	268	48	programming	programming	NOUN
iajs-2809	268	49	problem	problem	NOUN
iajs-2809	268	50	with	with	ADP
iajs-2809	268	51	rough	rough	ADJ
iajs-2809	268	52	interval	interval	NOUN
iajs-2809	268	53	coefficients	coefficient	NOUN
iajs-2809	268	54	in	in	ADP
iajs-2809	268	55	the	the	DET
iajs-2809	268	56	objective	objective	ADJ
iajs-2809	268	57	function	function	NOUN
iajs-2809	268	58	.	.	PUNCT
iajs-2809	269	1	in	in	ADP
iajs-2809	269	2	this	this	DET
iajs-2809	269	3	technique	technique	NOUN
iajs-2809	269	4	,	,	PUNCT
iajs-2809	269	5	we	we	PRON
iajs-2809	269	6	changed	change	VERB
iajs-2809	269	7	rough	rough	ADJ
iajs-2809	269	8	intervals	interval	NOUN
iajs-2809	269	9	into	into	ADP
iajs-2809	269	10	ordinary	ordinary	ADJ
iajs-2809	269	11	intervals	interval	NOUN
iajs-2809	269	12	and	and	CCONJ
iajs-2809	269	13	utilized	utilize	VERB
iajs-2809	269	14	convex	convex	NOUN
iajs-2809	269	15	combinations	combination	NOUN
iajs-2809	269	16	of	of	ADP
iajs-2809	269	17	points	point	NOUN
iajs-2809	269	18	instead	instead	ADV
iajs-2809	269	19	of	of	ADP
iajs-2809	269	20	intervals	interval	NOUN
iajs-2809	269	21	.	.	PUNCT
iajs-2809	270	1	the	the	DET
iajs-2809	270	2	basic	basic	ADJ
iajs-2809	270	3	problem	problem	NOUN
iajs-2809	270	4	is	be	AUX
iajs-2809	270	5	converted	convert	VERB
iajs-2809	270	6	into	into	ADP
iajs-2809	270	7	linear	linear	ADJ
iajs-2809	270	8	programming	programming	NOUN
iajs-2809	270	9	using	use	VERB
iajs-2809	270	10	some	some	DET
iajs-2809	270	11	techniques	technique	NOUN
iajs-2809	270	12	.	.	PUNCT
iajs-2809	271	1	we	we	PRON
iajs-2809	271	2	configure	configure	NOUN
iajs-2809	271	3	series	series	PROPN
iajs-2809	271	4	linear	linear	PROPN
iajs-2809	271	5	programming	programming	NOUN
iajs-2809	271	6	problems	problem	NOUN
iajs-2809	271	7	and	and	CCONJ
iajs-2809	271	8	obtain	obtain	VERB
iajs-2809	271	9	optimal	optimal	ADJ
iajs-2809	271	10	solutions	solution	NOUN
iajs-2809	271	11	and	and	CCONJ
iajs-2809	271	12	values	value	NOUN
iajs-2809	271	13	by	by	ADP
iajs-2809	271	14	using	use	VERB
iajs-2809	271	15	the	the	DET
iajs-2809	271	16	simplex	simplex	NOUN
iajs-2809	271	17	method	method	NOUN
iajs-2809	271	18	.	.	PUNCT
iajs-2809	272	1	for	for	ADP
iajs-2809	272	2	future	future	ADJ
iajs-2809	272	3	studies	study	NOUN
iajs-2809	272	4	,	,	PUNCT
iajs-2809	272	5	we	we	PRON
iajs-2809	272	6	can	can	AUX
iajs-2809	272	7	try	try	VERB
iajs-2809	272	8	to	to	PART
iajs-2809	272	9	perform	perform	VERB
iajs-2809	272	10	this	this	DET
iajs-2809	272	11	technique	technique	NOUN
iajs-2809	272	12	and	and	CCONJ
iajs-2809	272	13	notion	notion	NOUN
iajs-2809	272	14	in	in	ADP
iajs-2809	272	15	any	any	DET
iajs-2809	272	16	fractional	fractional	ADJ
iajs-2809	272	17	programming	programming	NOUN
iajs-2809	272	18	.	.	PUNCT
iajs-2809	273	1	ibn	ibn	PROPN
iajs-2809	273	2	al	al	PROPN
iajs-2809	273	3	-	-	PUNCT
iajs-2809	273	4	haitham	haitham	PROPN
iajs-2809	273	5	jour	jour	X
iajs-2809	273	6	.	.	PROPN
iajs-2809	273	7	for	for	ADP
iajs-2809	273	8	pure	pure	ADJ
iajs-2809	273	9	&	&	CCONJ
iajs-2809	273	10	appl	appl	PROPN
iajs-2809	273	11	.	.	PUNCT
iajs-2809	274	1	sci	sci	PROPN
iajs-2809	274	2	.	.	PROPN
iajs-2809	275	1	53	53	NUM
iajs-2809	275	2	(	(	PUNCT
iajs-2809	275	3	2)2022	2)2022	NUM
iajs-2809	275	4	83	83	NUM
iajs-2809	275	5	references	reference	NOUN
iajs-2809	275	6	1	1	NUM
iajs-2809	275	7	.	.	PUNCT
iajs-2809	275	8	charnes	charne	NOUN
iajs-2809	275	9	,	,	PUNCT
iajs-2809	275	10	a.	a.	NOUN
iajs-2809	275	11	;	;	PUNCT
iajs-2809	275	12	cooper	cooper	PROPN
iajs-2809	275	13	,	,	PUNCT
iajs-2809	275	14	w.	w.	PROPN
iajs-2809	275	15	w.	w.	PROPN
iajs-2809	275	16	programming	programming	PROPN
iajs-2809	275	17	with	with	ADP
iajs-2809	275	18	linear	linear	ADJ
iajs-2809	275	19	fractional	fractional	ADJ
iajs-2809	275	20	functionals	functional	NOUN
iajs-2809	275	21	,	,	PUNCT
iajs-2809	275	22	naval	naval	ADJ
iajs-2809	275	23	research	research	NOUN
iajs-2809	275	24	logistics	logistic	NOUN
iajs-2809	275	25	quarterly	quarterly	ADV
iajs-2809	275	26	,	,	PUNCT
iajs-2809	275	27	1962	1962	NUM
iajs-2809	275	28	,	,	PUNCT
iajs-2809	275	29	9	9	NUM
iajs-2809	275	30	,	,	PUNCT
iajs-2809	275	31	3‐4	3‐4	PROPN
iajs-2809	275	32	,	,	PUNCT
iajs-2809	275	33	181	181	NUM
iajs-2809	275	34	-	-	SYM
iajs-2809	275	35	186	186	NUM
iajs-2809	275	36	,	,	PUNCT
iajs-2809	275	37	2	2	NUM
iajs-2809	275	38	.	.	X
iajs-2809	275	39	lu	lu	PROPN
iajs-2809	275	40	,	,	PUNCT
iajs-2809	275	41	h.	h.	PROPN
iajs-2809	275	42	;	;	PUNCT
iajs-2809	275	43	huang	huang	PROPN
iajs-2809	275	44	,	,	PUNCT
iajs-2809	275	45	g.	g.	PROPN
iajs-2809	275	46	;	;	PUNCT
iajs-2809	275	47	he	he	PRON
iajs-2809	275	48	,	,	PUNCT
iajs-2809	275	49	l.	l.	PROPN
iajs-2809	275	50	an	an	DET
iajs-2809	275	51	inexact	inexact	ADJ
iajs-2809	275	52	rough	rough	ADJ
iajs-2809	275	53	-	-	PUNCT
iajs-2809	275	54	interval	interval	NOUN
iajs-2809	275	55	fuzzy	fuzzy	ADJ
iajs-2809	275	56	linear	linear	ADJ
iajs-2809	275	57	programming	programming	NOUN
iajs-2809	275	58	method	method	NOUN
iajs-2809	275	59	for	for	ADP
iajs-2809	275	60	generating	generate	VERB
iajs-2809	275	61	conjunctive	conjunctive	ADJ
iajs-2809	275	62	water	water	NOUN
iajs-2809	275	63	-	-	PUNCT
iajs-2809	275	64	allocation	allocation	NOUN
iajs-2809	275	65	strategies	strategy	NOUN
iajs-2809	275	66	to	to	ADP
iajs-2809	275	67	agricultural	agricultural	ADJ
iajs-2809	275	68	irrigation	irrigation	NOUN
iajs-2809	275	69	systems	system	NOUN
iajs-2809	275	70	,	,	PUNCT
iajs-2809	275	71	applied	apply	VERB
iajs-2809	275	72	mathematical	mathematical	ADJ
iajs-2809	275	73	modelling	modelling	NOUN
iajs-2809	275	74	,	,	PUNCT
iajs-2809	275	75	2011	2011	NUM
iajs-2809	275	76	,	,	PUNCT
iajs-2809	275	77	35	35	NUM
iajs-2809	275	78	,	,	PUNCT
iajs-2809	275	79	9	9	NUM
iajs-2809	275	80	,	,	PUNCT
iajs-2809	275	81	4330	4330	NUM
iajs-2809	275	82	-	-	SYM
iajs-2809	275	83	4340	4340	NUM
iajs-2809	275	84	,	,	PUNCT
iajs-2809	275	85	.	.	PUNCT
iajs-2809	276	1	3	3	X
iajs-2809	276	2	.	.	X
iajs-2809	276	3	pawlak	pawlak	ADJ
iajs-2809	276	4	,	,	PUNCT
iajs-2809	276	5	z.	z.	PROPN
iajs-2809	276	6	rough	rough	ADJ
iajs-2809	276	7	sets	set	NOUN
iajs-2809	276	8	.	.	PUNCT
iajs-2809	277	1	international	international	ADJ
iajs-2809	277	2	journal	journal	NOUN
iajs-2809	277	3	of	of	ADP
iajs-2809	277	4	information	information	NOUN
iajs-2809	277	5	and	and	CCONJ
iajs-2809	277	6	computer	computer	NOUN
iajs-2809	277	7	science	science	NOUN
iajs-2809	277	8	,	,	PUNCT
iajs-2809	277	9	1982	1982	NUM
iajs-2809	277	10	.	.	PUNCT
iajs-2809	278	1	4	4	X
iajs-2809	278	2	.	.	X
iajs-2809	278	3	youness	youness	NOUN
iajs-2809	278	4	,	,	PUNCT
iajs-2809	279	1	e.	e.	PROPN
iajs-2809	279	2	a.	a.	PROPN
iajs-2809	279	3	characterizing	characterize	VERB
iajs-2809	279	4	solutions	solution	NOUN
iajs-2809	279	5	of	of	ADP
iajs-2809	279	6	rough	rough	ADJ
iajs-2809	279	7	programming	programming	NOUN
iajs-2809	279	8	problems	problem	NOUN
iajs-2809	279	9	,	,	PUNCT
iajs-2809	279	10	european	european	PROPN
iajs-2809	279	11	journal	journal	PROPN
iajs-2809	279	12	of	of	ADP
iajs-2809	279	13	operational	operational	ADJ
iajs-2809	279	14	research	research	NOUN
iajs-2809	279	15	,	,	PUNCT
iajs-2809	279	16	2006	2006	NUM
iajs-2809	279	17	,	,	PUNCT
iajs-2809	279	18	168	168	NUM
iajs-2809	279	19	,	,	PUNCT
iajs-2809	279	20	3,1019	3,1019	NUM
iajs-2809	279	21	-	-	SYM
iajs-2809	279	22	1029	1029	NUM
iajs-2809	279	23	,	,	PUNCT
iajs-2809	279	24	.	.	PUNCT
iajs-2809	280	1	5	5	X
iajs-2809	280	2	.	.	X
iajs-2809	280	3	osman	osman	PROPN
iajs-2809	280	4	,	,	PUNCT
iajs-2809	280	5	m.	m.	NOUN
iajs-2809	280	6	;	;	PUNCT
iajs-2809	280	7	lashein	lashein	NOUN
iajs-2809	280	8	,	,	PUNCT
iajs-2809	280	9	e.	e.	PROPN
iajs-2809	280	10	;	;	PUNCT
iajs-2809	280	11	youness	youness	NOUN
iajs-2809	280	12	,	,	PUNCT
iajs-2809	280	13	e.	e.	PROPN
iajs-2809	280	14	;	;	PUNCT
iajs-2809	280	15	atteya	atteya	VERB
iajs-2809	280	16	,	,	PUNCT
iajs-2809	280	17	t.	t.	PROPN
iajs-2809	280	18	mathematical	mathematical	ADJ
iajs-2809	280	19	programming	programming	NOUN
iajs-2809	280	20	in	in	ADP
iajs-2809	280	21	rough	rough	ADJ
iajs-2809	280	22	environment	environment	NOUN
iajs-2809	280	23	,	,	PUNCT
iajs-2809	280	24	optimisation	optimisation	NOUN
iajs-2809	280	25	,	,	PUNCT
iajs-2809	280	26	2011	2011	NUM
iajs-2809	280	27	,	,	PUNCT
iajs-2809	280	28	60	60	NUM
iajs-2809	280	29	,	,	PUNCT
iajs-2809	280	30	5	5	NUM
iajs-2809	280	31	,	,	PUNCT
iajs-2809	280	32	603	603	NUM
iajs-2809	280	33	-	-	SYM
iajs-2809	280	34	611	611	NUM
iajs-2809	280	35	.	.	PUNCT
iajs-2809	281	1	6	6	NUM
iajs-2809	281	2	.	.	X
iajs-2809	281	3	rebolledo	rebolledo	PROPN
iajs-2809	281	4	,	,	PUNCT
iajs-2809	281	5	m.	m.	NOUN
iajs-2809	281	6	rough	rough	ADJ
iajs-2809	281	7	intervals	interval	NOUN
iajs-2809	281	8	—	—	PUNCT
iajs-2809	281	9	enhancing	enhance	VERB
iajs-2809	281	10	intervals	interval	NOUN
iajs-2809	281	11	for	for	ADP
iajs-2809	281	12	qualitative	qualitative	ADJ
iajs-2809	281	13	modeling	modeling	NOUN
iajs-2809	281	14	of	of	ADP
iajs-2809	281	15	technical	technical	ADJ
iajs-2809	281	16	systems	system	NOUN
iajs-2809	281	17	,	,	PUNCT
iajs-2809	281	18	artificial	artificial	ADJ
iajs-2809	281	19	intelligence	intelligence	NOUN
iajs-2809	281	20	,	,	PUNCT
iajs-2809	281	21	2006	2006	NUM
iajs-2809	281	22	,	,	PUNCT
iajs-2809	281	23	170	170	NUM
iajs-2809	281	24	,	,	PUNCT
iajs-2809	281	25	8	8	NUM
iajs-2809	281	26	-	-	SYM
iajs-2809	281	27	9	9	NUM
iajs-2809	281	28	,	,	PUNCT
iajs-2809	281	29	667	667	NUM
iajs-2809	281	30	-	-	SYM
iajs-2809	281	31	685	685	NUM
iajs-2809	281	32	,	,	PUNCT
iajs-2809	281	33	.	.	PUNCT
iajs-2809	282	1	7	7	X
iajs-2809	282	2	.	.	X
iajs-2809	282	3	hamzehee	hamzehee	PROPN
iajs-2809	282	4	,	,	PUNCT
iajs-2809	282	5	a.	a.	NOUN
iajs-2809	282	6	;	;	PUNCT
iajs-2809	282	7	yaghoobi	yaghoobi	NOUN
iajs-2809	282	8	,	,	PUNCT
iajs-2809	282	9	m.	m.	NOUN
iajs-2809	282	10	a.	a.	NOUN
iajs-2809	282	11	;	;	PUNCT
iajs-2809	282	12	mashinchi	mashinchi	PROPN
iajs-2809	282	13	,	,	PUNCT
iajs-2809	282	14	m.	m.	NOUN
iajs-2809	282	15	linear	linear	PROPN
iajs-2809	282	16	programming	programming	NOUN
iajs-2809	282	17	with	with	ADP
iajs-2809	282	18	rough	rough	ADJ
iajs-2809	282	19	interval	interval	NOUN
iajs-2809	282	20	coefficients	coefficient	NOUN
iajs-2809	282	21	,	,	PUNCT
iajs-2809	282	22	journal	journal	NOUN
iajs-2809	282	23	of	of	ADP
iajs-2809	282	24	intelligent	intelligent	ADJ
iajs-2809	282	25	&	&	CCONJ
iajs-2809	282	26	fuzzy	fuzzy	ADJ
iajs-2809	282	27	systems	system	NOUN
iajs-2809	282	28	,	,	PUNCT
iajs-2809	282	29	2014	2014	NUM
iajs-2809	282	30	,	,	PUNCT
iajs-2809	282	31	26	26	NUM
iajs-2809	282	32	,	,	PUNCT
iajs-2809	282	33	3	3	NUM
iajs-2809	282	34	,	,	PUNCT
iajs-2809	282	35	1179	1179	NUM
iajs-2809	282	36	-	-	SYM
iajs-2809	282	37	1189	1189	NUM
iajs-2809	282	38	,	,	PUNCT
iajs-2809	282	39	8	8	NUM
iajs-2809	282	40	.	.	PUNCT
iajs-2809	283	1	borza	borza	NOUN
iajs-2809	283	2	,	,	PUNCT
iajs-2809	283	3	m.	m.	NOUN
iajs-2809	283	4	;	;	PUNCT
iajs-2809	283	5	rambely	rambely	ADV
iajs-2809	283	6	,	,	PUNCT
iajs-2809	283	7	a.	a.	PROPN
iajs-2809	283	8	s.	s.	PROPN
iajs-2809	283	9	;	;	PUNCT
iajs-2809	283	10	saraj	saraj	PROPN
iajs-2809	283	11	,	,	PUNCT
iajs-2809	283	12	m.	m.	NOUN
iajs-2809	283	13	solving	solve	VERB
iajs-2809	283	14	linear	linear	ADJ
iajs-2809	283	15	fractional	fractional	ADJ
iajs-2809	283	16	programming	programming	NOUN
iajs-2809	283	17	problems	problem	NOUN
iajs-2809	283	18	with	with	ADP
iajs-2809	283	19	interval	interval	NOUN
iajs-2809	283	20	coefficients	coefficient	NOUN
iajs-2809	283	21	in	in	ADP
iajs-2809	283	22	the	the	DET
iajs-2809	283	23	objective	objective	NOUN
iajs-2809	283	24	function.a	function.a	PROPN
iajs-2809	283	25	new	new	ADJ
iajs-2809	283	26	approach	approach	NOUN
iajs-2809	283	27	,	,	PUNCT
iajs-2809	283	28	applied	apply	VERB
iajs-2809	283	29	mathematical	mathematical	ADJ
iajs-2809	283	30	sciences,2012	sciences,2012	NOUN
iajs-2809	283	31	,	,	PUNCT
iajs-2809	283	32	6	6	NUM
iajs-2809	283	33	,	,	PUNCT
iajs-2809	283	34	69	69	NUM
iajs-2809	283	35	,	,	PUNCT
iajs-2809	283	36	3443	3443	NUM
iajs-2809	283	37	-	-	SYM
iajs-2809	283	38	3452	3452	NUM
iajs-2809	283	39	,	,	PUNCT
iajs-2809	283	40	2012	2012	NUM
iajs-2809	283	41	.	.	PUNCT
iajs-2809	284	1	9	9	X
iajs-2809	284	2	.	.	X
iajs-2809	284	3	ammar	ammar	PROPN
iajs-2809	284	4	,	,	PUNCT
iajs-2809	284	5	e.	e.	PROPN
iajs-2809	284	6	;	;	PUNCT
iajs-2809	284	7	muamer	muamer	PROPN
iajs-2809	284	8	,	,	PUNCT
iajs-2809	284	9	m.solving	m.solve	VERB
iajs-2809	284	10	a	a	DET
iajs-2809	284	11	rough	rough	ADJ
iajs-2809	284	12	interval	interval	NOUN
iajs-2809	284	13	linear	linear	NOUN
iajs-2809	284	14	fractional	fractional	ADJ
iajs-2809	284	15	programming	programming	NOUN
iajs-2809	284	16	problem	problem	NOUN
iajs-2809	284	17	,	,	PUNCT
iajs-2809	284	18	journal	journal	NOUN
iajs-2809	284	19	:	:	PUNCT
iajs-2809	284	20	journal	journal	NOUN
iajs-2809	284	21	of	of	ADP
iajs-2809	284	22	advances	advance	NOUN
iajs-2809	284	23	in	in	ADP
iajs-2809	284	24	mathematics	mathematic	NOUN
iajs-2809	284	25	,	,	PUNCT
iajs-2809	284	26	2015	2015	NUM
iajs-2809	284	27	,	,	PUNCT
iajs-2809	284	28	10	10	NUM
iajs-2809	284	29	,	,	PUNCT
iajs-2809	284	30	4	4	NUM
iajs-2809	284	31	.	.	NOUN
iajs-2809	284	32	10	10	NUM
iajs-2809	284	33	.	.	PUNCT
iajs-2809	285	1	khalifa	khalifa	PROPN
iajs-2809	285	2	,	,	PUNCT
iajs-2809	285	3	a.	a.	NOUN
iajs-2809	285	4	on	on	ADP
iajs-2809	285	5	solutions	solution	NOUN
iajs-2809	285	6	of	of	ADP
iajs-2809	285	7	linear	linear	ADJ
iajs-2809	285	8	fractional	fractional	ADJ
iajs-2809	285	9	programming	programming	NOUN
iajs-2809	285	10	problems	problem	NOUN
iajs-2809	285	11	with	with	ADP
iajs-2809	285	12	roughinterval	roughinterval	ADJ
iajs-2809	285	13	coefficients	coefficient	NOUN
iajs-2809	285	14	in	in	ADP
iajs-2809	285	15	the	the	DET
iajs-2809	285	16	objective	objective	ADJ
iajs-2809	285	17	functions	function	NOUN
iajs-2809	285	18	,	,	PUNCT
iajs-2809	285	19	journal	journal	NOUN
iajs-2809	285	20	of	of	ADP
iajs-2809	285	21	fuzzy	fuzzy	ADJ
iajs-2809	285	22	mathematics,2018	mathematics,2018	NOUN
iajs-2809	285	23	,	,	PUNCT
iajs-2809	285	24	26	26	NUM
iajs-2809	285	25	,	,	PUNCT
iajs-2809	285	26	2	2	NUM
iajs-2809	285	27	,	,	PUNCT
iajs-2809	285	28	415	415	NUM
iajs-2809	285	29	-	-	SYM
iajs-2809	285	30	422	422	NUM
iajs-2809	285	31	,	,	PUNCT
iajs-2809	285	32	.	.	PUNCT
iajs-2809	286	1	11	11	NUM
iajs-2809	286	2	.	.	X
iajs-2809	287	1	mustafa	mustafa	PROPN
iajs-2809	287	2	,	,	PUNCT
iajs-2809	287	3	r.	r.	PROPN
iajs-2809	287	4	;	;	PUNCT
iajs-2809	287	5	sulaiman	sulaiman	PROPN
iajs-2809	287	6	,	,	PUNCT
iajs-2809	287	7	n.	n.	PROPN
iajs-2809	287	8	a.	a.	PROPN
iajs-2809	288	1	a	a	DET
iajs-2809	288	2	new	new	ADJ
iajs-2809	288	3	mean	mean	ADJ
iajs-2809	288	4	deviation	deviation	NOUN
iajs-2809	288	5	and	and	CCONJ
iajs-2809	288	6	advanced	advanced	ADJ
iajs-2809	288	7	mean	mean	NOUN
iajs-2809	288	8	deviation	deviation	NOUN
iajs-2809	288	9	techniques	technique	NOUN
iajs-2809	288	10	to	to	PART
iajs-2809	288	11	solve	solve	VERB
iajs-2809	288	12	multi	multi	ADJ
iajs-2809	288	13	-	-	ADJ
iajs-2809	288	14	objective	objective	ADJ
iajs-2809	288	15	fractional	fractional	ADJ
iajs-2809	288	16	programming	programming	NOUN
iajs-2809	288	17	problem	problem	NOUN
iajs-2809	288	18	via	via	ADP
iajs-2809	288	19	pointslopes	pointslope	NOUN
iajs-2809	288	20	formula	formula	NOUN
iajs-2809	288	21	,	,	PUNCT
iajs-2809	288	22	pakistan	pakistan	PROPN
iajs-2809	288	23	journal	journal	PROPN
iajs-2809	288	24	of	of	ADP
iajs-2809	288	25	statistics	statistic	NOUN
iajs-2809	288	26	and	and	CCONJ
iajs-2809	288	27	operation	operation	NOUN
iajs-2809	288	28	research	research	NOUN
iajs-2809	288	29	,	,	PUNCT
iajs-2809	288	30	2021	2021	NUM
iajs-2809	288	31	,	,	PUNCT
iajs-2809	288	32	10511064	10511064	NUM
iajs-2809	288	33	,	,	PUNCT
iajs-2809	288	34	.	.	PUNCT
iajs-2809	289	1	12	12	NUM
iajs-2809	289	2	.	.	PUNCT
iajs-2809	290	1	sulaiman	sulaiman	PROPN
iajs-2809	290	2	,	,	PUNCT
iajs-2809	290	3	n.	n.	PROPN
iajs-2809	290	4	a.	a.	NOUN
iajs-2809	290	5	;	;	PUNCT
iajs-2809	290	6	mustafa	mustafa	PROPN
iajs-2809	290	7	,	,	PUNCT
iajs-2809	290	8	r.	r.	PROPN
iajs-2809	290	9	b.	b.	PROPN
iajs-2809	290	10	using	use	VERB
iajs-2809	290	11	harmonic	harmonic	ADJ
iajs-2809	290	12	mean	mean	NOUN
iajs-2809	290	13	to	to	PART
iajs-2809	290	14	solve	solve	VERB
iajs-2809	290	15	multi	multi	ADJ
iajs-2809	290	16	-	-	ADJ
iajs-2809	290	17	objective	objective	ADJ
iajs-2809	290	18	linear	linear	ADJ
iajs-2809	290	19	programming	programming	NOUN
iajs-2809	290	20	problems	problem	NOUN
iajs-2809	290	21	,	,	PUNCT
iajs-2809	290	22	american	american	ADJ
iajs-2809	290	23	journal	journal	PROPN
iajs-2809	290	24	of	of	ADP
iajs-2809	290	25	operations	operation	NOUN
iajs-2809	290	26	research	research	NOUN
iajs-2809	290	27	,	,	PUNCT
iajs-2809	290	28	2016	2016	NUM
iajs-2809	290	29	,	,	PUNCT
iajs-2809	290	30	6	6	NUM
iajs-2809	290	31	,	,	PUNCT
iajs-2809	290	32	1	1	NUM
iajs-2809	290	33	,	,	PUNCT
iajs-2809	290	34	25	25	NUM
iajs-2809	290	35	-	-	SYM
iajs-2809	290	36	30	30	NUM
iajs-2809	290	37	,	,	PUNCT
iajs-2809	290	38	.	.	PUNCT
iajs-2809	291	1	13	13	NUM
iajs-2809	291	2	.	.	PUNCT
iajs-2809	292	1	sulaiman	sulaiman	PROPN
iajs-2809	292	2	,	,	PUNCT
iajs-2809	292	3	n.	n.	PROPN
iajs-2809	292	4	;	;	PUNCT
iajs-2809	292	5	sadiq	sadiq	PROPN
iajs-2809	292	6	,	,	PUNCT
iajs-2809	292	7	g.	g.	PROPN
iajs-2809	292	8	;	;	PUNCT
iajs-2809	292	9	abdulrahim	abdulrahim	PROPN
iajs-2809	292	10	,	,	PUNCT
iajs-2809	292	11	b.used	b.use	VERB
iajs-2809	292	12	a	a	DET
iajs-2809	292	13	new	new	ADJ
iajs-2809	292	14	transformation	transformation	NOUN
iajs-2809	292	15	technique	technique	NOUN
iajs-2809	292	16	for	for	ADP
iajs-2809	292	17	solving	solve	VERB
iajs-2809	292	18	multiobjective	multiobjective	ADJ
iajs-2809	292	19	linear	linear	ADJ
iajs-2809	292	20	fractional	fractional	ADJ
iajs-2809	292	21	programming	programming	NOUN
iajs-2809	292	22	problem	problem	NOUN
iajs-2809	292	23	,	,	PUNCT
iajs-2809	292	24	ijrras	ijrras	NOUN
iajs-2809	292	25	(	(	PUNCT
iajs-2809	292	26	18	18	NUM
iajs-2809	292	27	)	)	PUNCT
iajs-2809	292	28	,	,	PUNCT
iajs-2809	292	29	2014,2	2014,2	NUM
iajs-2809	292	30	,	,	PUNCT
iajs-2809	292	31	122	122	NUM
iajs-2809	292	32	-	-	SYM
iajs-2809	292	33	131	131	NUM
iajs-2809	292	34	.	.	PUNCT