


































Advances in Business Research 2010 Volume 1.pdf


26

Advances in Business Research

2010, Vol. 1, No. 1, 26-35

Jolayemi

 !"#$%"&%#%'()!)*#)+",-.%/"0-'"1/2!!)!3"1'-.4+#)-!"542!#)#)%*")!"2",4/#)61'-.4+#7",4/#)6

1/2!#7"2!.",4/#)682'%$-4*%"9!:)'-!(%!#
 

Joel Jolayemi, Tennessee State University   

 !"#$%&"'(')*+",)"*)&#*-.#-*)"(!/"&%0'1%23"451(3)0%"(!/"615*-!!%,57&"05/)1"8496"05/)1:"#5"0(;)"%#"05*)"(/5'#(<1)"

and applicable in industries. The New model obtained from the restructuring process has only 7 constraints while the 

496"05/)1"$(&"==>" !")(.$"52"#$)")?(0'1)&+"#$)"!-0<)*"52"%#)*(#%5!&"<)25*)"*)(.$%!@"5'#%0(1%#3",(&"&0(11)*"25*"#$)"!),"

05/)1"#$(!"25*"#$)"496"05/)1>"A$)")?(0'1)&"&$5,"#$(#"(&"<5#$"05/)1&"%!.*)(&)"%!"&%B)+"#$)"496"05/)1"<).50)&"05*)"

(!/"05*)"/%2C.-1#"#5"&51D)"#$(!"#$)"!),"05/)1>" !"05&#"52"#$)")?(0'1)&+"#$)"5'#%0(1"&51-#%5!&"'*5/-.)/"<3"#$)"E?.)1"

FGGH"&51D)*"(*)"D)*3".15&)"#5"#$5&)"'*5/-.)/"<3"I JK6>"

 !"#$%$!&"'$(#)*"&+$#$"+(,$"-$$!").(#/!0"/!&$#$)&)"/!"&+$"(112/%(&/.!)".3")4112'"%+(/!"5678"9(!(0$9$!&"9$&+.:)"

(!:"&$%+!/;4$)"/!"/!:4)&#/$)<"=+$)$"+(,$"!$%$))/&(&$:"&+$"!$$:"3.#"&+$":$,$2.19$!&)".3"9.#$"(!:"!$>"67"9$&+.:)"

(!:"&$%+!/;4$)*"1(#&/%42(#2'".1&/9/?(&/.!"9.:$2)".#"&$%+!/;4$)"&+(&"/!&$0#(&$".#").2,$").9$".#"9(!'"67"1#.-2$9)"

simultaneously for best decision outcomes. Unlike before, industries are now beginning to realize that modeling and 

).2,/!0"67"1#.-2$9)")$1(#(&$2'"2$(:"&.")4-@.1&/9(2").24&/.!)"(!:*"%.!)$;4$!&2'*"A2$))@&+(!@-$)&B":$%/)/.!)".4&%.9$)<

C$)1/&$" &+$)$*" 14-2/)+$:" >.#D)" .!" &+$" :$,$2.19$!&)" .3" 9.:$2)E&$%+!/;4$)" 3.#" /!&$0#(&/!0" (!:" ).2,/!0" 67"

1#.-2$9)")&/22"3(22"-$2.>"!$$:)<"=+$"3$>"9.)&"/!&$0#(&$:"9.:$2)"/!"&+$)$"14-2/%(&/.!)"/!&$0#(&$"(-.4&"&+#$$"&."F,$"

.3"&+$)$"1#.-2$9)<"6.9$".3"&+$)$"9.:$2)"%(!"-$")$$!"/!"G#9&?$!"$&"(2<"5HIIJ8*"G&&(#(!"K"G&&(#(!"5LMMN8*"O.2('$9/"

K"P2.#4!!/>."5LMMQ8*"R.4&#'"$&"(2<"5LMMI8*"6(#04&"K"R.9$/S!"5LMMN8*"=/>(#/"$&"(2<"5LMHM8*"=)/(D/)"K"T(1(0$.#0/.4"

5LMMN8*""(!:"U.4"K""V#.)9(!!"5LMMI8<

G#9&?$!"$&"(2<"5HIIJ8":$,$2.1$:"("9/W$:"/!&$0$#"2/!$(#"1#.0#(99/!0"5X YT8"02.-(2"67"9.:$2"3.#":$&$#9/!/!0Z"

5H8" &+$"!49-$#" (!:" 2.%(&/.!".3"C7)*" 5L8" %4)&.9$#" @" :/)&#/-4&/.!" %$!&#$" ())/0!9$!&)*" 5[8" &+$"!49-$#".3" $%+$2.!)*"

(!:"5Q8"1#.:4%&"@"12(!&"())/0!9$!&)< G&&(#(!"K"G&&(#(!"5LMMN8"1#.,/:$"(!".,$#,/$>".3"%.!&$91.#(#'")4112'"%+(/!"

5678"9(!(0$9$!&")')&$9)"&+#.40+"(!"/!@:$1&+".,$#,/$>"(!:"(!(2')/)".3"7.22(-.#(&/,$"T2(!!/!0*"\.#$%()&/!0*"(!:"

R$12$!/)+9$!&"57T\R8<"7T\R"+$21)")4112'"%+(/!"1(#&!$#)" &." /!&$0#(&$":$9(!:"(!:" 2.0/)&/%"12(!!/!0*"1#.:4%&/.!"

scheduling, and new product design.

O.2('$9/" K" P2.#4!!/>." 5LMMQ8" 3.#942(&$" (" &>.@)&(0$" 67" 9.:$2" &+(&" :$&$#9/!$)" &+$" .1&/9(2" ;4(!&/&/$)" .3"

products to be produced at each plant, transported from each plant to each WH, subcontracted at each WH, and kept 

in inventory at each WH. The model also determines the optimal amount of extensions needed at each WH. It is one 

of the few most highly integrated models. However, due to the large numbers of its constraints and binary variables, 

its size increases rapidly as the numbers of products, plants, and WHs increase.

=+$" .1&/9/?(&/.!" .3" (" &>.@$%+$2.!" 67" /!" >+/%+" 1#.:4%&/.!*" /!,$!&.#'*" &#(!)1.#&(&/.!*" -(%D2.00/!0*" (!:"

)4-%.!&#(%&/!0":$%/)/.!)"(#$"/!&$0#(&$:"/)"&+$"3.%4)".3"&+$"(#&/%2$"-'"6(#04&"K"R.9$/S!"5LMMN8<"=)/(D/)"K"T(1(0$.#0/.4"

5LMMN8":$,$2.1$:"("9.:$2"&+(&"/!&$0#(&$)"1#.:4%&/.!*"3(%/2/&'"2.%(&/.!*"(!:":/)&#/-4&/.!"(2.!0)/:$">/&+".&+$#"-4)/!$))"

issues like import duties, plant utilization and maintenance, and exchange rates.

G"+'-#/:"=(04%+/@ 994!$"(11#.(%+"/)"(112/$:"-'"=/>(#/"$&"(2<" 5LMHM8" &.".1&/9/?$"(!" /!&$0#(&$:")4112'"%+(/!"

:$)/0!"1#.-2$9">/&+"942&/12$")+/11/!0".1&/.!)*":/)&#/-4&$:"%4)&.9$#":$9(!:)*"(!:"FW$:"2$(:"&/9$)<"R.4&#'"$&"(2<"

5LMMI8"(!:"U.4"K"V#.))9(!!"5LMMI8":$,$2.1"942&/@$%+$2.!"/!,$!&.#'"12(!!/!0"9.:$2)">/&+"2$(:"&/9$"(!:":$9(!:"

uncertainties. You and Grossmann’s model differs from the model by Routry et al. by the addition of transportation 

component.

X.#$"$W(912$)".3"&+$)$"&'1$)".3"/!&$0#(&$:"9.:$2)"%(!"(2)."-$")$$!"/!"7.D$2$?"K"]49"5HI^I8*"74!+("K"X4&(#$22/"

5LMMN8*"Y/(!0"5LMM^8*"R/?D"$&"(2<"5LMM^8*"_/:'(&+/"$&"(2<"5LMMN8*"(!:"U4!0"$&"(2<"5LMM`8*"&."9$!&/.!"("3$><""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

 Due to their integrated nature, these models are usually very large and structurally complex. These can make 

&+$9":/3F%42&"&."(112'<"a$!%$*"-$)/:$)"&+$"!$$:"3.#":$,$2.1/!0"9.#$"/!&$0#(&$:")4112'"%+(/!".1&/9/?(&/.!"9.:$2)E

&$%+!/;4$)*" &+$#$" /)" (2)." (" 0#$(&" !$$:" 3.#" #$@$W(9/!/!0" ).9$".3" &+$" $W/)&/!0".!$)" @" 1(#&/%42(#2'" &+$"9.)&" +/0+2'"

integrated among them - for the possibility of modifying, simplifying, or restructuring them to make them more 

$33$%&/,$*"9.#$"$3F%/$!&*"$()/$#"&.").2,$"(!:*"%.!)$;4$!&2'*"9.#$"(:.1&(-2$"(!:"(112/%(-2$"/!"/!:4)&#/$)<

G)"%(!"-$")$$!"3#.9".4#"$(#2/$#":/)%4))/.!)"(-.,$*"&+$"9.:$2"-'"O.2('$9/"K"P2.#4!!/>."5LMMQ8"/)".!$".3"&+$"

few highly-integrated models. The model has many constraints and many decision variables - including many binary 

variables. This makes its size to increase rapidly as the numbers of products, plants, and warehouses increase.



27

Advances in Business Research

2010, Vol. 1, No. 1, 26-35

Jolayemi

Although, the model is applicable in industries in its present form, after a careful examination, we have observed 
that its adoptability and applicability will be much more enhanced if its size can be reduced much further. We believe 
that this can be achieved by carefully examining it and identifying where and how it can be restructured.

=+$#$3.#$*"&+$"3.%4)".3"&+/)"#$)$(#%+"/)"&."#$)&#4%&4#$"(!:")/912/3'"O.2('$9/"(!:"P2.#4!!/>.b)"9.:$2"5O@P"9.:$28"

&."9(D$"/&"9.#$"$33$%&/,$*"9.#$"$3F%/$!&*"9.#$"(:.1&(-2$"(!:"9.#$"(112/%(-2$<"G3&$#"("%(#$342"2..D"(&"&+$"9.:$2*">$"

F!:"&+(&").9$")1$%/(2"3$(&4#$)"(!:")&#4%&4#$)".3"&+$"%.!)&#(/!&)"%(!"-$"$W12./&$:"&."#$:4%$"/&)")/?$",$#'")4-)&(!&/(22'"

(!:"&.")/912/3'"/&"5&+$"9.:$28"/!".#:$#"&."9(D$"/&"$()/$#"&.").2,$<

=+$"#$)&#4%&4#/!0".3"&+$"9.:$2">/22"!.&"-$"3.%4)$:".!2'".!"/&)"%.!)&#(/!&)<"c$">/22"(2)."#$:$F!$"(!:"#$)&#4%&4#$"

some terms of the model’s objective function. In the way the objective function is presently formulated, it is easy to 
commit errors in the expansions of some of the terms that have multiple summations. We will clearly illustrate the 
procedures involved in the restructuring process.

Numerical examples will be given to illustrate the restructured model and to compare it with the old model. The 
%.91(#/).!)">/22"-$":.!$">/&+"#$)1$%&"&."9.:$2")/?$*").24&/.!"#$)42&)*"(!:"%.914&(&/.!(2"$3F%/$!%'<"=+$"Y dCP"(!:"

Excel 2007 solvers will be used for this purpose.
One of the major assumptions underlying the developments of the J-O model is that a producer will like to use his 

capacities to the maximum before subcontracting. While we believe that this is a good assumption that will suit the 
operations strategies of many producers, we also believe that there are many producers whose decision to produce or 
)4-%.!&#(%&">/22"-$"-()$:").2$2'".!"&+$"%.)&"()).%/(&$:">/&+"$(%+<"c$">/22")+.>"+.>"&+/)"9.:$2"%(!"-$"(112/$:";4/&$"

easily when the decision to produce or subcontract is mainly based on cost consideration.

;$%"<6 ",-.%/"2!."=#*">%*#'4+#4')!3"1'-+%**

 !" .#:$#" 3.#" &+$" #$)&#4%&4#/!0" 1#.%$))" &." -$" %2$(#2'" .#">$22@/224)&#(&$:*">$" F#)&" 1#$)$!&" &+$" O@P"9.:$2" (!:" /&)"

/!&$#1#$&(&/.!)"-$3.#$"$9-(#D/!0".!"&+$"1#.%$))<"=+$"O@P"9.:$2*"()":$,$2.1$:"-'"O.2('$9/"K"P2.#4!!/>."5LMMQ8*"/)"

()"3.22.>)Z
 

ijt

T

1t

J

1j

N

1i

ijtijt 1-ijt

T

1t

J

1j

N

1i

ijtijt

T

1t

J

1j

P

1p

N

1i

iptipjtijt q)g - s(  v)h - (s  ) v- y(s Maximize           
= = == = == = = =

++  

 jit

T

1t

 J

1j

jtipjt 

T

1t

J

1j

P

1p

N

1i

ipjtiptiptipt

T

1t

P

1p

N

1i

ipt we - yk - )xc  zf( -          
= == = = == = =

+  

 

subject to 
 

(1.1) ....   T. ...., 2, 1,  t P; ...., 2, 1,  p R; ...., 2, 1, r      ,b  xa rpt

N

1i

iptript === !
=

 

 

(1.2) ....   T. ...., 2, 1,  t P; ...., 2, 1,  p N; ...., 2, 1,  i ,u  x iptipt ===  

(1.3) ....   T. ...., 2, 1,  t J; ...., 2, 1,  j N; ...., 2, 1,  i    ,d   v- q   v y ijtijtijt1-ijt

P

1p

ipjt ====++ 
=

 

(1.4) ....   T. ...., 2, 1,  t P; ...., 2, 1,  p N; ...., 2, 1,  i 0,   x- zL iptipt1 ===  

 

(1.5) ....   T.  t0,  v

J

1j

N

1

ijt ==  
= =i

 

 

(1.6) ....   1. t0,  v

J

1j

N

1

1-ijt ==  
= =i

 

 

 

(1.7) ....   T. ...., 2, 1,  t N; ...., 2, 1,  t 0,  u  - d 

J

1j

P

1p

iptijtit == 
!
!
!

"

#

$
$
$

%

&

' '
= =

 



28

Advances in Business Research

Model Interpretation

 !"# $%&'# '"%(# )*# '!"# )+,"-'./"# *01-'.)1# .&# '!"# ')'23# %"/"10"# *%)(# '!"# &23"&# )*# 233# 4%)50-'&# &!.44"5# ')# '!"#

62%"!)0&"&#789&:#7";-305.1<#'!"#=021'.'."&#>"4'#.1#.1/"1')%?#2'#'!"#+"<.11.1<#)*#"2-!#4"%.)5:#*%)(#233#4321'&#.1#

all the T periods. The second term is the total net revenue from the sales of inventory kept in all WHs over the T 
periods after all the total inventory costs have been subtracted. The third term is the net revenue realized from all 
products subcontracted during the planning horizon after deducting the total cost of subcontracting. The fourth 

 
=

==!

J

1j

ijtit2 (1.8) ....   T....., 2, 1,  t N; ...., 2, 1,  i 0,  q - L  

(1.9) ....   T. ...., 2, 1,  t N; ...., 2, 1,  i ,u - d  q

J

1j

P

1p

iptijtit

J

1j

ijt ==
 
 
 

!

"

#
#
#

$

%

& ' ''
= ==

 

(1.10) ....   T ...., 2, 1,  t P; ...., 2, 1,  p N; ...., 2, 1,  i 0,   x- y ipt

J

1j

ipjt ==== 
=

 

 

(1.11) ....   T. ...., 2, 1,  t J, ...., 2, 1,  j   w w - q  v  y ojt

t

1m

jmijt

N

1i

i

N

1i

1-ijtiipjt

P

1p

N

1i

i == ++ !!!!!
==== =

 

 tp, j, i,  1or  0   and 1or  0  z 0;  y , x, w, v,q itiptipjtiptjtijtijt  ==!  

 

where: 
 

:a ript   amount of resource r required to produce a unit of product i in plant p in period t. 

:brpt   the total amount of resource r available in plant p in period t. 

: i   conversion factor in m
3
 per ton of product i. 

 :cipt  
the production cost per unit of product i in plant p in period t. 

:dijt  
demand for product i in warehouse j in period t. 

:e jt  
the cost of construction/extension per m

3
 of warehouse j in period t. 

:fipt  
the setup cost with respect to product i in plant p in period t. 

:gijt  
the unit cost of subcontracting product i in warehouse j in period t. 

 :hijt  
the holding cost per unit of product i in warehouse j in period t. 

ijptk :  the cost of transporting a unit of product i from plant p to warehouse j in period t. 

1L  :  a very large number. 

2L :  a very large number. 

it :     a binary variable which is 1 if the quantities of product i produced in all P plants in period t cannot meet 

customers’ demands and zero otherwise.  

ijtq :  the quantity of product i subcontracted in warehouse j in period t.  

ijts :  the selling price of product i in warehouse j in period t. 

iptu :  the maximum possible capacity of plant p in period t with respect to product i. 

ijtv :    the amount of inventory of product i in warehouse j in period t. It is the portion of ipjty  that is set aside to 

be kept in warehouse j in period t. 

jtw :  the amount of extension in m
3
 that is needed in warehouse j in period t. 

ojtw :  the initial capacity of warehouse j in period t. 

 xipt :  the quantity of product i produced in plant p in period t. 

ipjty :  the quantity of product i (in tons) shipped from plant p to warehouse j in period t. 

iptz :  a binary variable which is 1 if product i is produced in plant p in period t and zero otherwise.  

 

2010, Vol. 1, No. 1, 26-35

Jolayemi

 that is set aside to 



29

Advances in Business Research

 !" #$%" #&#'(")&!#"&*"+,&-.)# &/0"  /)(.- /1" #$%"!%#.+")&!#!0"&2%," #$%"3"+%, &-!4"3$%"5*#$" #%,6" !" #$%" #&#'(")&!#"&*"

transporting all products from all plants to all WHs over the T periods. The last term is the total cost of extension 
/%%-%-" /"'(("#$%"7"89!" /"'(("#$%"3"+%, &-!4"3$.!0"#$%"&:;%)# 2%"*./)# &/"6'< 6 =%!"#$%"#&#'("+,&5#"&:#' /%-"'*#%,"

subtracting the production, transportation, inventory, WH constructions, and setup costs from total revenue.

>&/!#,' /#" ?@4@A" %/!.,%!" #$'#" '" 6'/.*')#.,%," -&%!" /&#" +('/" :%B&/-" #$%" ,%!&.,)%!" #$'#" ',%" '2' (':(%" '#" %')$"

+('/#"  /"%')$"+%, &-4">&/!#,' /#" ?@4CA" %<+,%!!%!" #$%" *')#" #$'#"/&"+('/#" )'/"+,&-.)%":%B&/-"  #!"6'< 6.6")'+') #B4" 

>&/!#,' /#"?@4DA"%/':(%!"'"6'/.*')#.,%,"#&"!'# !*B"-%6'/-!"*&,"%')$"+,&-.)#4">&/!#,' /#"?@4EA"%/!.,%!"#$'#" *"+,&-.)#" "

 !"+,&-.)%-" /"+('/#"+" /"'/B"+%, &-"#0"#$%,%"F ((":%"'"!%#.+")&!#")$',1%-"'1' /!#"$ 6"F #$",%!+%)#"#&" 4">&/!#,' /#"?@4GA"

%/!.,%!"#$'#"/&" /2%/#&,B",%6' /!" /"!#&)H"'#"#$%"%/-"&*"#$%"+('// /1"$&, =&/4">&/!#,' /#"?@4IA"%<+,%!!%!"#$%")&/- # &/"

that inventory should not be carried from one planning horizon to the next. In many situations, a producer would 

( H%"#&"&+%,'#%"$ !"+('/#!"#&"6'< 6.6")'+') #B":%*&,%"!.:)&/#,')# /14">&/!#,' /#!"?@4JA"'/-"?@4KA"6'H%"#$ !"+&!! :(%4"

>&/!#,' /#"?@4LA"%/!.,%!"#$'#"#$%"#&#'("M.'/# #B"&*" #%6" "!.:)&/#,')#%-" /"'(("#$%"7"89!" /"+%, &-"#" !"/&#"1,%'#%,"#$'/"

the difference between the total demand for i from all the J WHs and the total capacity of all the P plants with respect 

to item i in period t. This ensures that subcontracting is done only when demand cannot be met after all plant capacities 

have been fully utilized. 

N2%,B"+,&-.)#"+,&-.)%-"'#"%2%,B"+('/#"$'!"#&":%"!$ ++%-"#&"#$%"89!4">&/!#,' /#"?@4@OA"6'H%!"#$ !"+&!! :(%4"P"

6'/.*')#.,%,"F&.(-"/&#"( H%"89"!+')%"#&":%"'")&/!#,' /#"#&"$ !"&+%,'# &/!4">&/!#,' /#"?@4@@A"#'H%!")',%"&*"#$ !4"3$%"

constraint ensures that extension is made at any WH at any period whenever necessary.

 !"#$%&#'(&)*()&+,-%.&!*#''

(/0%1023145341678%9:%3/0%9;<0536=0%:4753697

""" Q/" #$%" 5,!#" #%,6" &*" #$%" &:;%)# 2%" *./)# &/0" vijt is supposed to be the portion of yipjt kept in inventory in 

warehouse j in plant p. However, the exclusion of subscript p among the subscripts of vijt  does not show that vijt is a 

portion of or that there is any relationship or connection between yipjt and vijt.

      Additionally, since both yipjt and vijt are being summed over i, p, j, and t, the omission of p among the 

subscripts of vijt may lead to errors in the expansion of the multiple summation term, if great care is not taken.

   To correct for these, we replace vijt with vipjt" /"#$%"5,!#"'/-"!%)&/-"#%,6!"&*"#$%"&:;%)# 2%"*./)# &/"'/-"!.6"

#$%"!%)&/-"#%,6"&2%," 0"+0";0"'/-"#0"( H%"#$%"5,!#"#%,60"#&"&:#' /R

   

 

    where vipjt is the amount of product i from plant p kept in inventory in warehouse j in period t and vipjt is a  

portion of yipjt 

%&023145341678%9:%3/0%>9?0@A2%597231B6732

  In restructuring the model’s constraints, we identify and eliminate variables and constraints that are redundant. 

8%" '(!&" ,%-.)%0" 6%,1%0" '/-S&," ,%*&,6.('#%" !&6%" )&/!#,' /#!4"3$%" )&/!#,' /#T,%!#,.)#., /1" +,&)%!!"  !" '!" +,%!%/#%-"

below.

"""" >&/!#,' /#!"?@4GA"'/-"?@4IA"',%"/&#$ /1"6&,%"#$'/":&./-")&/!#,' /#!4"3$%,%*&,%0"#$%B")'/":%"F, ##%/"'!":&./-"

!#'#%6%/#!"'#"#$%"%/-"&*"#$%"6&-%("'!"*&((&F!R

 

   ijt

T

1t

J

1j

N

1i

ijtijt

T

1t

J

1j

P

1p

N

1i

1-ipjtijtijt

T

1t

J

1j

P

1p

N

1i

ipjtipjtijt q)g - s(  v)h - s(  ) v- y(s Maximize            
= = == = = == = = =

++  

 jt

T

1t

 J

1j

jtipjt 

T

1t

J

1j

P

1p

N

1i

ipjtiptiptipt

T

1t

P

1p

N

1i

ipt we - yk - )xc  zf( -          
= == = = == = =

+  

 

 

ijtv = 0 for t = T and  v 1-ijt = 0 for T = 1 and for all i, and j (i = 1, 2, .…, N; j = 1, 2, …, J) …. (i). This reduces 

the number of the constraints of the model. 

In constraint (1.7), given that  d

J

1j

ijt 
=

>  
=

P

1p

iptu and that the values of  d

J

1j

ijt 
=

and  
=

P

1p

iptu are known, on 

substituting these known values into the constraint, it simplifies to: 0  it   for all i and t (i = 1, 2, …., N; t = 1, 2, …, 

2010, Vol. 1, No. 1, 26-35

Jolayemi



30

Advances in Business Research

T), …. (ii). Similarly, if  d

J

1j

ijt 
=

<   
=

P

1p

iptu , constraint (1.7) will simplify to: 0   it   
for all i and t (i = 1, 2, ….,N; t = 

1, 2, …., T), .… (iii). By definition,  it can only be 0 or 1. Therefore, 0  it    it is 0 or 1, and 0   it   0  it = .  

Therefore, if necessary, the constraint can be written as a bound constraint at the end of the model as:  it = 0 or 

1. for all i and all t (i = 1, 2, ….,N; t = 1, 2, …., T), .... (iv). Otherwise, the constraint can be eliminated completely. 

Either way, the number of the model’s constraints is reduced by NT. In constraint (1.8),  

 
=

!

J

1j

ijtit2  0  q - L   
=

!

J

1j

ijtit2   q  L   
=

J

1j

ijt   q    L it2 .... (v). In constraint (1.9), it is given that: 

  u - d  q

J

1j

P

1p

iptijtit

J

1j

ijt
 
 
 

!

"

#
#
#

$

%

& ' ''
= ==

…. (vi). Since 
2
L  is defined to be a very large number,  L

it2
in (v) must be, at 

least, equal to or greater than  u - d 

J

1j

P

1p

iptijtit
 
 
 

!

"

#
#
#

$

%

& &
= =

in (vi) above. Hence, if constraint (vi) is satisfied, constraint (v) 

must be satisfied. Therefore, constraint (1.8) is redundant and must be dropped. This further reduces the number of 

the model’s constraints by another NT. 

Recall from the definition of  it that  it = 0 for  d

J

1j

ijt 
=

   
=

P

1p

iptu . This implies that, in constraint (1.9), 

 
=

J

1j

ijt  q   0 whenever  d

J

1j

ijt 
=

    
=

P

1p

iptu . Since ijtq must be equal to or greater than zero for all i, j, and t,  
=

J

1j

ijt  q   

0  ijtq = 0 for all i, j, and t for which  d

J

1j

ijt 
=

    
=

P

1p

ipt.u Thus, in any application of this model, all constraints in 

(1.9) for which  d

J

1j

ijt 
=

  
=

P

1p

iptu should be dropped from the model together with their associated variables s'qijt  

This will reduce the size of the model much further. 

As defined earlier in the first sub-section,  it = 1 for  d

J

1j

ijt 
=

!   
=

P

1p

iptu . Hence, 1 can be substituted for any it

for which  d

J

1j

ijt 
=

!   
=

P

1p

iptu  in constraint (1.9) during the model’s application  

Therefore, with 1 substituted for  it whenever  d

J

1j

ijt 
=

!  
=

P

1p

iptu and with all constraints for which  d

J

1j

ijt 
=

   

 
=

P

1p

iptu completely eliminated in constraint (1.9), all the binary variables s'
it

will be eliminated from constraint 

(1.9) during the model’s application. Furthermore, since constraints (1.7) and (1.8) have been deleted from the 

model, all the binary variables  s'it will be completely eliminated from the model during its application. This will 

reduce the model’s size drastically. Putting everything together, the restructured or the New model is given as:  

ijt

T

1t

J

1j

N

1i

ijtijt

T

1t

J

1j

P

1p

N

1i

1-ipjtijtijt

T

1t

J

1j

P

1p

N

1i

ipjtipjtijt q)g - s(  v)h - s(  ) v- y(s Maximize            
= = == = = == = = =

++  

jt

T

1t

 J

1j

jtipjt 

T

1t

J

1j

P

1p

N

1i

ipjtiptiptipt

T

1t

P

1p

N

1i

ipt we - yk - )xc  zf( -          
= == = = == = =

+  

 

2010, Vol. 1, No. 1, 26-35

Jolayemi



31

Advances in Business Research

     subject to

As can be seen from the above results, the restructuring of the model has reduced the number of its main constraints 

from 11 to 7 through the elimination of four major constraints. This will amount to a very big reduction in any industry 

application in which many products, many plants, many WHs, and many periods are involved. 

        The restructuring has lead to the complete elimination of the binary variables from the model, leaving only the 

binary variables This will prevent the model’s size from increasing rapidly with increases in the numbers of products 

and plants. We believe that this new model will be much easier to solve than the old model, due to its smaller size. Data 

 !" #!#$%&'()#'*)"'$!+),%--)#-(&).")/012)"#(%"!),%$2)%$)$2#'),%$2)$2")&-*)/&*"-3)40!$2"!/&!"5)%$()(&-0$%&'),%--)!"60%!")

less computer memory to process. 

 !"#$%&'()#*'"+(#,)' -)&."+'$%,. ,

7')$2")8!($)(0.("1$%&')&9)$2%()("1$%&'5),"):%;")9&0!)'0/"!%1#-)"<#/ -"()$&)%--0($!#$")$2")!"($!01$0!"*)/&*"-)#'*)$&)

compare it with the J-O model. In the second subsection, we present the results of further examples given to illustrate 

the new model and compare it with the J-O model. The LINDO and Excel 2007 solutions to the examples are also 

1&/ #!"*)%')$2")(0.("1$%&')=%3"3)$2")("1&'*)(0.("1$%&'>3))

 /0123456)#7508619:)";<16)%66/9=25=3;>9:)5><)&;085239;>9

?2")8!($)"<#/ -")=("")"<#/ -")@>)%')$#.-")@)%()&')#)/0-$%A "!%&*)#'*)/0-$%A !&*01$) !&*01$%&'A*%($!%.0$%&')#'*)

warehouse capacity problem involving 2 products, 2 plants, 2 warehouses, and 2 periods.

In the numerical example, the total demands for products 1 and 2 in period 2 are each less than plant capacity 

for each product. This made two constraints to be dropped from each of the model’s subcontract constraints (see 

1&'($!#%'$()=@3B>)#'*)=@3C>)&9) $2")DAE)/&*"-)#'*)1&'($!#%'$)=F3G>)&9) $2")H",)/&*"-) %') $2") !";%&0()("1$%&'>) %') $2")

example. This reduced the number of constraints of the new model in the example from 56 to 54 and of the J-O model 

from 60 to 56.  It also reduced the number of the J-O model’s binary variables from 12 to 10. The application of the 

results in the last subsection of the previous section)!"-#$%':)$&)1&'($!#%'$)=F3G>)!"*01"*)$2")'0/."!)&9)$2")'",)/&*"-I()

binary variables from 10 to 8.

As shown in the table, the numbers of the New and the J-O models’ continuous variables in the example are 40 

and 36 respectively. This is after dropping all subcontract variables for which total demand is less than total plant 

 

(2.1) ....    T. ...., 2, 1,  t P; ...., 2, 1,  p R; ...., 2, 1, r  ,b  xa rpt

N

1i

iptript === !
=

 

(2.2) ....   T. ...., 2, 1,  t P; ...., 2, 1,  p N; .... 2, 1,  i ,u  x iptipt ===  

....(2.3)   T. ...., 2, 1,  t J; ...., 2, 1,  j N; ...., 2, 1,  i ,d v - q  v y ijt

P

1p

ijt ijt

P

1p

P

1p

1-ijtipjt ====++    
== =

 

(2.4) ....    T. ...., 2, 1,  t N; ...., 2, 1,  i ,u - d  q

J

1j

P

1p

iptijtit

J

1j

ijt ==
 
 
 

!

"

#
#
#

$

%

& ' ''
= ==

 

(2.5) ....   T. ...., 2, 1,  t P; ...., 2, 1,  p N; ...., 2, 1,  i 0,   x- zL iptipt1 ===  

(2.6) ....    T ...., 2, 1,  t P; ...., 2, 1,  p N; ...., 2, 1,  i 0,   x- y ipt

J

1j

ipjt ==== 
=

 

                                                                                           

....(2.7)   T. ...., 2, 1,  t J; ...., 2, 1,  j  , w w - q  v  y ojt

t

1m

jmijt

N

1i

i

P

1p

N

1i

1-ijtiipjt

P

1p

N

1i

i == ++ !!!!!!
=== == =

 

Where it = 1 for  d

J

1j

ijt 
=

>  
=

P

1p

iptu and 0 otherwise, ijt q = 0 for  d

J

1j

ijt 
=

   
=

P

1p

iptu , ijtv = 0 for t = T, 1-ijtv = 0 for t 

= 1, and  1.or  0  z  t;and j, p, i, allfor  0  y , x, w, v,q iptipjtiptjtijtijt =  

         

2010, Vol. 1, No. 1, 26-35

Jolayemi



32

Advances in Business Research

capacities for the two products. The number of the new model’s continuous variables is greater than that of the J-O 

 !"#$%&#'()*#%+(*%'(,%&#%*##,%-,%./#%01*.%(,"%$(*.%*)&*#'.-!,*%!2%./#%31#4-!)*%*#'.-!,5%./#%-,4#,.!16%'! 3!,#,.%!2%./#%

former has more variables than those of the latter.

The LINDO solver was used to obtain optimal solution to each problem. The solver was run on a HP personal 

'! 3).#1%7-./%(%8#,.-) %+95%:;<<=>?
3
%31!'#**!1%),"#1%@-,"!7*%A8;%B/#%*!$).-!,*%.!%./#%.7!% !"#$*%*/!7#"%C!!"%

results.

The solutions to the New and J-O models were obtained after 111 and 141 iterations respectively. This amounts to a 

difference of 30 iterations. This shows that, with smaller number of binary variables, the new model is easier to solve.

The optimal value of each of the model’s objective function is $3,536,800.00. This shows that the imperfections 

-,%./#%*.1)'.)1#*%!2%./#%DEF% !"#$%"!%,!.%(22#'.%-.*%(&-$-.6%.!%31!")'#%C!!"%1#*)$.*;%B/#6%!,$6% (G#%-.% !1#%"-20')$.%.!%

solve.

The numbers of products, plants, warehouses, and periods involved in example 2 is the same with those in example 

:%+*##%.(&$#%:5;%?!7#4#1H%),$-G#%-,%#I( 3$#%:H%,!,#%!2%./#%*)&'!,.1('.%4(1-(&$#*%(,"%'!,*.1(-,.*%!2%./#%DEF% !"#$%-*%

dropped in example 2. This is because total demand for each product in each period is greater than plant capacity in 

the example. As a result, the number of its constraints is 60 and the numbers of its continuous and binary variables are 

40 and 12 respectively.

Also, since total demand for each product in each period is greater than plant capacity, none of the New model’s 

subcontract constraints and variables is dropped. Hence, the number of the model’s constraints is 56 while the number 

of its continuous variables is now 44 (see%.(&$#%J5;%?!7#4#1H%./#%(33$-'(.-!,%!2%./#%1#*)$.*%-,%the last subsection of the 

previous section%1#$(.-,C%.!%'!,*.1(-,.%+J;K5% (G#*%./#%,) &#1%!2%./#%,#7% !"#$L*%&-,(16%4(1-(&$#*%.!%&#%M%N%O)*.%(*%-,%

example 1.

The outputs of the LINDO solutions to the two models in the example show that the number of iterations before 

obtaining optimal solutions to the new and J-O models are 120 and 144 respectively, resulting in a difference of 24 

iterations. This result corroborates our conclusion in example 1 that the new model is easier to solve. Again, the two 

models produce the same optimal value of the objective function.

In example 3, the number of products is 3 and the total demand for each product in period 2 is less than plant 

capacity for each product. Like in example 1, this made three constraints to be dropped from each of the model’s 

*)&'!,.1('.% '!,*.1(-,.*% +-;#;% 21! %'!,*.1(-,.*% +:;M5% (,"% +:;<5%!2% ./#% DEF% !"#$% (,"% +J;K5%!2% ./#%P#7% !"#$5;%B/-*%

reduced the number of constraints of the J-O model from 80 to 74 and that of the new model from 74 to 71 respectively. 

The number of J-O model’s binary variables is also reduced from 18 to 14. The applications of the results in the last 

subsection of the previous section%1#$(.-,C%.!%'!,*.1(-,.%+J;K5%1#")'#"%./#%,) &#1%!2%./#%,#7% !"#$L*%&-,(16%4(1-(&$#*%

from 15 to 12.

Table 1 shows that the LINDO solutions to the two models were obtained after 185 and 215 iterations respectively, 

7/-'/%-*%(%"-22#1#,'#%!2%=Q%-.#1(.-!,*;%B/-*%*)33!1.*%./#%1#*)$.*%-,%./#%01*.%.7!%#I( 3$#*%7/-'/%*/!7%./(.%./#%,#7%

model is easier to solve.

Unlike in example 3, none of the two plants has enough capacity to satisfy demand for each product in each period 

in example 4. Therefore, due to the reasons explained for the similar case of example 2, the numbers of the new and 

J-O models’ constraints are 74 and 80 respectively and the numbers of their binary variables are respectively 12 and 

18. The number of each model’s continuous variables is now 65 and 60 respectively.

 !"#$%&'%($)*#+)%,-%+.$%/*0$123!#%45!06#$)%,7%+.$%/$8%!79%+.$%:;<%=,9$#)>

As can be seen in table 1 above, the LINDO solver produced optimal solutions to the two models (the new and the 

DEF% !"#$*5%(2.#1%:R<%(,"%JMS%-.#1(.-!,*%1#*3#'.-4#$6;%B/-*%( !),.*%.!%(%"-22#1#,'#%!2%:K<%-.#1(.-!,*H%7/-'/%-*%(%4#16%

 
Model details and values  

of solution parameters 

 

Examples 

1 2 3 4 

New J-O New J-O New J-O New J-O 

Number of products 2 2 2 2 3 3 3 3 

Number of plants 2 2 2 2 2 2 2 2 

Number of warehouses 2 2 2 2 2 2 2 2 

Number of periods 2 2 2 2 2 2 2 2 

Number of constraints 54 56 56 60 71 74 74 80 

Number of continuous variables 40 36 44 40 57 52 65 60 

Number of binary variables 8 10 8 12 12 14 12 18 

Number of iterations 111 141 120 144 185 215 179 285 

Objective function value 35360800 35360800 35661800 35661800 25020320 25020320 30580720 30580720 

 

2010, Vol. 1, No. 1, 26-35

Jolayemi



33

Advances in Business Research

big difference. This remarkable result could be due to the fact that the difference between the new and J-O models’ 

binary variables is larger in this example than in example 2.

@#%0,"%./#%TUPVF%*!$4#1%.!%&#%4#16%C!!"%2!1%*!$4-,C%./#%.7!% !"#$*;%W2.#1%#,.#1-,C%./#%-,3).%"(.(%2!1%#('/% !"#$H%

./#%*!$4#1%31!")'#"%!3.- ($%*!$).-!,%-,*.(,.(,#!)*$6%(.%(%'$-'G%!,%./#%X!$4#%'!  (,";%B/#%*!$4#1L*%1#3!1.*Y*.(.-*.-'*%

!,%#('/%*!$).-!,%*/!7#"%./#%#$(3*#%.- #%.!%&#%Z#1!;%B/-*%*/!7*%./(.%./#%[8\%.- #%2!1%#('/%*!$).-!,%-*%4-1.)($$6%Z#1!;

?*1+.$1%45!06#$)%!79%@,06!12),7)

In this subsection, we use the Excel solver to solve each of the four numerical examples in the previous subsection 
+./#%31#4-!)*%*)&*#'.-!,%!2%./-*%*#'.-!,5;%B/#%1#(*!,*%2!1%./#*#%(1#%.!%#,(&$#%)*]%+:5%"#.#1 -,#%./#%*!$).-!,%.- #%2!1%

#('/%!2%./#%,) #1-'($%#I( 3$#*H%+J5%'! 3(1#%./#%^I'#$%*!$).-!,*%(,"%*!$).-!,%.- #*%2!1%./#%,#7% !"#$%7-./%./!*#%!2%

./#%DEF% !"#$%-,%#('/%,) #1-'($%#I( 3$#H%(,"%+=5%'! 3(1#%./#%TUPVF%(,"%^I'#$%JQQR%*!$).-!,*;%

Additionally, we use a much bigger numerical example than any of the four numerical examples in the previous 
subsection to compare the two models. The purpose of this is to give the illustrations and comparisons of the two 
 !"#$*%*! #% #(*)1#%!2%31('.-'($%1#($-.6;%B(&$#%J%*/!7*%./#%1#*)$.*%!2%./#%04#%,) #1-'($%#I( 3$#*%2!1%#('/%!2%./#%

two models.
U,%#I( 3$#%:%+*##%.(&$#%J5%./#%!3.- ($%!&O#'.-4#E2),'.-!,%4($)#*%31!")'#"%&6%./#%#I'#$%*!$4#1%2!1%./#%.7!% !"#$*%

are the same. These optimal objective function values are also the same with the objective-function values produced 
2!1%./#%.7!% !"#$*%&6%./#%TUPVF%*!$4#1;%B/-*%*/!7*%./(.%./#%^I'#$%JQQR%(,"%./#%TUPVF%*!$4#1%(1#%#_)($$6%C!!"%2!1%

solving LP problems of this size. The Excel solver produced optimal solution to each model in 2 seconds under the 
#I( 3$#;%B/-*% #(,*%./(.%./#%.7!% !"#$*%(1#%#_)($$6%#(*6%.!%*!$4#%),"#1%./-*%#I( 3$#;

Like in example 1, the optimal objective-function values produced by the Excel solver for the two models in 
example 2 are the same. However, each of the values deviates from the corresponding value produced by the LINDO 
*!$4#1%&6%Q;M`%+*##%.(&$#%J5;%X-,'#%./-*%-*%(%4#16%* ($$%"#4-(.-!,H%-.%'(,%&#%-,2#11#"%./(.%./#%^I'#$%*!$4#1%-*%*.-$$%(*%

good as the LINDO solver in solving LP problems of these sizes. As can be seen in the table, the sizes of the LP 
problems for the New and the J-O models in this example are slightly larger than their respective sizes in example1. 
The numbers of their binary variables are also slightly larger. 

The Excel-solution times for the New and the J-O models are one and two seconds respectively. This shows that 
the new model is relatively much easier to solve with Excel than the J-O model.

Again, the Excel solver produced the same objective-function values for the two models in example 3. Each of 
the objective-function values deviates from the corresponding value produced by LINDO by 0.00012%, which is 
practically zero. This shows that the Excel solver is still as good as LINDO in solving the problems, even though they 
(1#%$(1C#1%./(,%./#%31!&$# *%-,%./#%01*.%.7!%#I( 3$#*;%

The Excel solution times for the new and J-O models are 3 and 4 seconds respectively. This again shows that the 
new model is easier to solve than the J-O model.

The results in example 4 are similar to those in the three earlier examples. The Excel solutions to the new and 
the J-O models are the same. The solutions deviate from the LINDO solutions by only 0.0001%. The Excel solution 
times to the new and the J-O models are 4 and 7 seconds respectively, showing that the new model is relatively much 
easier to solve than the J-O model. 
B/#% #I( 3$#*% +#I( 3$#*% :% .!% K5% */!7% ./(.% (*% #('/% !"#$L*% *-Z#% (,"% ,) &#1% !2% &-,(16% 4(1-(&$#*% -,'1#(*#H% -.%

&#'! #*% !1#%(,"% !1#%"-20')$.%.!%*!$4#a%(,"%./(.%./#%DEF% !"#$%&#'! #*% )'/% !1#%"-20')$.%.!%*!$4#%./(,%./#%,#7%

model.       
To inject some practical realities into the illustrations and comparisons of the two models and of the Excel and 

LINDO solutions, we increase the sizes of the problems in example 5. The LP problem for the new model has 150 
constraints and 144 variables, 27 of which are binary variables. The one for the J-O model has 159 constraints and 
153 variables, 36 of which are binary.   
W*%'(,%&#%*##,%-,%./#%.(&$#%+.(&$#%J5H%./#%^I'#$%*!$4#1%31!")'#"%./#%*( #%*!$).-!,%.!%./#%.7!% !"#$*;%B/#%!3.- ($%

objective-function value for each of the two models deviates from the corresponding value produced by LINDO by 
1.82%. This is a small deviation for LP problems of these sizes. This shows that Excel 2007 LP solver is almost as 
good as the LINDO solver in solving big new and J-O models of these sizes.

The Excel-solution times for the new and the J-O models in this example are 26 and 42 seconds respectively. This 
strongly validates our observation in example 4 that as each model’s size and number of binary variables increase, it 
&#'! #*% !1#%(,"% !1#%"-20')$.%.!%*!$4#a%(,"%./(.%./#%DEF% !"#$%&#'! #*% )'/% !1#%"-20')$.%.!%*!$4#%./(,%./#%,#7%

model. This means that the adoptions and applications of the new model in industries will result in much bigger time 
and cost savings than the applications of the J-O model. This can make the new model to be more appealing, more 
applicable, and more adoptable than the J-O model in industries. 

2010, Vol. 1, No. 1, 26-35

Jolayemi



34

Advances in Business Research

 !"#$%A'% .$%1$)*#+)%,-%+.$%$53$#%!79%BC/D<%),#*+2,7)%+,%+.$%/$8%!79%:;<%0,9$#)

E66#23!+2,7%27%F1,9*3$;,1;G*"3,7+1!3+%D$32)2,7)

One of the major assumptions underlining the development of our model is that a producer will like to use his 
plants to maximum capacities before subcontracting. However, there are situations in which produce-or-subcontract 
"#'-*-!,%(1#%&(*#"% (-,$6%!,%'!*.%'!,*-"#1(.-!,*;%B/-*% !"#$%'(,%&#%4#16%#(*-$6% !"-0#"%.!% (G#%-.%(33$-'(&$#%),"#1%

such situations.
V1!33-,C%'!,*.1(-,.%+J;K5%21! %./#%P#7% !"#$%($$!7*%./#%31!")'#1%.!%*)&'!,.1('.%-2%-.%-*%'/#(3#1%./(,%.!%31!")'#%

and vice versa, irrespective of whether he has used his plant to maximum capacity or not. The fact that the model 
can be easily applied under this type of situations further enhances its adoptability and applicability in industries.

We would like to recommend here that in applying the model to make produce-or-subcontract decision, the cost 
that may be incurred by the producer due to unused capacities must be factored in.

Concluding Remarks

We have examined and restructured the J-O model to correct some imperfections in its formulation. The new 
 !"#$%!&.(-,#"%21! %./#%1#*.1)'.)1-,C%31!'#**%/(*%!,$6%R%'!,*.1(-,.*%7/-$#%./#%!1-C-,($% !"#$%+./#%DEF% !"#$5%/(*%

11. The restructuring process leads to the complete elimination of some binary variables from the new model.
 We used numerical examples to illustrate and compare the two models. The examples were solved with LINDO 

and Excel 2007 solvers. In each example, the number of iterations before obtaining optimal solution with LINDO 
was smaller for the new model than for the J-O model. In nearly all the examples, the Excel-solution times for the 
new model were less than those for the J-O model. In some of the examples, the differences between the numbers of 
iterations for the two models were large.
W*%./#%.7!% !"#$*L%*-Z#*%-,'1#(*#]%+:5%./#%,) &#1%!2%./#%DEF% !"#$L*%&-,(16%4(1-(&$#*%&#'! #*% )'/%$(1C#1%./(,%

./(.%!2%./#%,#7% !"#$H%+J5%./#%^I'#$E*!$).-!,%.- #%-,'1#(*#*H%(,"%+=5%./#%,) &#1%!2%-.#1(.-!,*%&#2!1#%!3.- ($%*!$).-!,*%

7#1#%!&.(-,#"%+7-./%TUPVF5%-,'1#(*#*;%%B/#%!&4-!)*%'!,'$)*-!,%21! %./#*#%-*%./(.%./#%DEF% !"#$%&#'! #*% !1#%(,"%

 !1#%"-20')$.%.!%*!$4#%./(,%./#%,#7% !"#$%(*%./#%.7!% !"#$*L%*-Z#*%-,'1#(*#;

  The Excel solutions to the two models are the same in each of the numerical examples. The LINDO solutions 
to the two models are also the same in each example. These show that the imperfections in the structures of the J-O 
model do not actually affect its ability to produce good solutions. They only affect its processing and computational 
.- #H%(,"%./-*%'(,%&#%(%&-C%31!&$# %-2%-.%+./#%DEF% !"#$5%-*%$(1C#;

  It is very obvious from the results of the numerical examples that the applications of the New model will lead 
.!%*-C,-0'(,.%.- #%(,"%'!*.%*(4-,C*%!4#1%./#%(33$-'(.-!,%!2%./#%DEF% !"#$;%W3(1.%21! %./#%2('.%./(.%./#%,#7% !"#$%

1#_)-1#*%$#**%*!$).-!,Y'! 3).(.-!,($%.- #%(,"%'!*.H%"(.(%31#3(1(.-!,*%(,"%#,.1-#*%7-$$%&#% )'/%#(*-#1%7-./%-.% ./(,%

7-./%./#%DEF% !"#$;%W""-.-!,($$6H%7-./%./#%,#7% !"#$L*%* ($$#1%*-Z#H%$#**%'! 3).#1% # !16%7-$$%&#%1#_)-1#"%2!1%"(.(%

storage and processing.
W%4#16%C!!"%("4(,.(C#%!2%./#% !"#$%-*%./(.%-.%'(,%&#%#(*-$6% !"-0#"%2!1%(33$-'(.-!,*%-,%*-.)(.-!,*%7/#1#%31!")'#E

or-subcontract decisions can be based mainly on cost considerations, irrespective of whether there are enough 
capacities or not.  

   The examples show that Excel 2007 solver produces good solutions to the two models. In most of the examples, 
the optimal value of the objective function produced by the Excel solver are very close to those produced by LINDO. 
The only obvious drawback is that the former is much slower than the latter. After entering the in-put data in each 
example, optimal solution was instantaneously produced by LINDO at a click on the Solve command. It took some 
couple of seconds to solve each example with Excel 2007 solver.

 
Examples Model No. of  

binary  

variables 

No. of  

continuous  

variables 

Total No. 

of 

variables 

No.  

of  

constraints 

Objective  

function- 

value  

(Excel) 

Objective  

function- 

value  

(LINDO)  

Deviation  

from the  

LINDO  

value 

Excel- 

solution  

time  

 

Example 1 New  Model 8 40 48 54 35360800 35360800 0.0% 2 seconds 

J-O Model 10 36 46 56 35360800 35360800 0.0% 2 seconds 

Example 2 New  Model 8 44 52 56 35360800 35661800 0.8% 1 second 

J-O Model 12 40 52 60 35360800 35661800 0.8% 2 seconds 

Example 3 New  Model 12 57 69 70 25020288 25020320 0.00012% 3 seconds 

J-O Model 14 52 66 72 25020288 25020320 0.00012% 4 seconds 

Example 4 New  Model 12 65 77 74 30580688 30580720 0.00010% 4 seconds 

J-O Model 18 60 78 80 30580688 30580720 0.00010% 7 seconds 

Example 5 New  Model 27 117 144 150 74676055 76060300 1.82% 26 seconds 

J-O Model 36 117 153 159 74676055 76060300 1.82% 42 seconds 

 

2010, Vol. 1, No. 1, 26-35

Jolayemi



35

Advances in Business Research

 !"! !#$!

 !"#$%&'()*'()!+,&'(-*'(./!!01+&'(2*'(3(2!/4#+&'(5*(6778*(-9+:/9(1;<<9=(>?/0&("/&/@%"%&#(/#(A0@0#/9(%B;0<"%&#(

corporation. Interfaces'(C8D(E7F7G*

 ##/!/&'(H*'(3( ##/!/&'(I*(CJJK*(L+99/:+!/#0M%(1;<<9=(>?/0&("/&/@%"%&#*(Business Process Management Journal, 

6GD(G7JFNJN*

L+O%9%$'(I*'(3();"'(P*(67Q7*(R01#!0:;#0+&(1=1#%"1(F(,/!%?+;1%(9+>/#0+&(/&A(>/</>0#=*(%&'()'(6KD(N8F86*

L;&?/'(L*'(3(H;#/!%990'(S*(CJJ8*( (1<!%/A1?%%#F:/1%A(+<#0"0$/#0+&("+A%9(4+!(#?%(0&#%@!/#%A(<!+:9%"(+4(<!+A;>0&@(/&A(

distributing a major weekly news magazine. !*+,-')./0,*+.)1/,2/%-'+)34,.5/ '5')+67'(6KED(7C8F7NJ*

P+9/=%"0'(P*'(3(T9+!;&&0,+'(S*(CJJN*( (A%#%!"0&01#0>("+A%9(4+!(<9/&&0&@(<!+A;>#0+&(B;/&#0#0%1(0&(/(";9#0F<9/&#'(";9#0F

warehouse environment with extensible capacities. 8.3'+.)34,.)1/0,*+.)1/,2/9+,:*634,./!6,.,&465'(QKD(77F66G*

Liang, T. 2008. Integrating production-transportation decision with multiple goals in supply chains. International 

Journal of Production Research'(NED(6NKKF6NQ7*

Rizk, N., Martel, A., & D’Amours, S. 2008. Synchronized production-distribution planning in a single-plant multi-

distribution network. ;7'/0,*+.)1/,2/37'/%-'+)34,.)1/ '5')+67/<,64'3='(87D(7JF6JN*

Routry, S., & Maddala, K. 2009. Multi-echelon supply chain inventory planning with demand and lead time uncertainty. 

8.3'+.)34,.)1/0,*+.)1/,2/%-'+)34,.5/ '5')+67'(8D(C86FC8N*

I/!@;#'(S*'(3(U+"%0V&'(.*(CJJK*(L/</>0#/#%A(<!+A;>#0+&(/&A(1;:>+&#!/>#0&@(0&(/(1%!0/9(1;<<9=(>?/0&*(88!/;+).5)634,., 

G7D(6JG6F6JNG*

Tiwari, M., Raghavendra, N., Argrawal, S., & Goyal, S. 2010. A hybrid Taguchi-immune approach to optimize an 

integrated supply chain design problem with multiple shipping. !*+,-')./0,*+.)1/,2/%-'+)34,.5/ '5')+67'(CJGD(

95-106.

Tsiakis, P., & Papageorgiou, L. 2007. Optimal production allocation and distribution supply chain networks. 

8.3'+.)34,.)1/0,*+.)1/,2/9+,:*634,./!6,.,&465'(666D(NEQFNQG*

W0A=/#?0'(X*'(L%9%:0'(Y*'(Y9?%A?90'(I*'(3(P%,O%1'(Y*(CJJK*(Z&#%@!/#%A(<!+A;>#0+&(0&M%&#+!=FA01#!0:;#0+&(1=1#%"(A%10@&(

,0#?(!01OD(H+A%9(4+!";9/#0+&(/&A(?%;!01#0>(1+9;#0+&*(Transportation Science'(N6D(G7CFNJQ*

[+;'( S*'( 3( -!+11"/&'( Z*( CJJ7*( H;9#0F%>?%9+&( 1;<<9=( >?/0&( ,0#?( 0&M%&#+!=( ;&A%!( ;&>%!#/0&#=D( HZX5\( "+A%91'(

computational strategies. >8$7!/0,*+.)1'(8ED(N67FNNJ*

Yung, K., Tang, J., lp, A., & Wang, D. 2006. Heuristics for joint decisions in production, transportation, and order 

B;/&#0#=*(Transportation Science'(NJD(77F66E*

Joel Jolayemi is a professor of operations and supply chain management at Tennessee State University. He received 
?01(\?*R*(0&(+<%!/#0+&1(!%1%/!>?(4!+"(L/1%(]%1#%!&(U%1%!M%(^&0M%!10#=*(.01(>;!!%&#(!%1%/!>?(0&#%!%1#1(0&>9;A%(1;<<9=(

>?/0&("/&/@%"%&#'( #+#/9(B;/90#=("/&/@%"%&#'(<!+V%>#(1>?%A;90&@'(A%>010+&(/&/9=101'(/&A("/#?%"/#0>/9("+A%90&@_

optimization. He has published in International Journal of Production Research, International Journal of Production 
Y>+&+"0>1'( <<90%A(H/#?%"/#0>1(/&A(L+"<;#/#0+&'( <<90%A(I#/#01#0>1(/&A(R/#/( &/9=101'(2+#/9(̀ ;/90#=(H/&/@%"%&#(

/&A();10&%11(Ya>%99%&>%'(T"%@/D(2?%(Z&#%!&/#0+&/9(P+;!&/9(+4(H/&/@%"%&#(I>0%&>%'(I/&O?=/D(2?%(Z&A0/&(P+;!&/9(+4(

Statistics, Ecological Modeling, and many others.

2010, Vol. 1, No. 1, 26-35

Jolayemi


