id sid eid entity type gojar-5993 1 1 gojar PERSON gojar-5993 1 2 4 CARDINAL gojar-5993 1 3 2025 CARDINAL gojar-5993 1 4 gojar PERSON gojar-5993 1 5 4 CARDINAL gojar-5993 1 6 1 february 2025 60 CARDINAL gojar-5993 1 7 mark laisin PERSON gojar-5993 1 8 collins edike & r. n. ujumadu ORG gojar-5993 3 1 integer hulls PERSON gojar-5993 7 1 linear programming ORG gojar-5993 8 1 i. introduction linear programming PERSON gojar-5993 9 1 the 1930s DATE gojar-5993 9 2 leonid kantorovich PERSON gojar-5993 10 1 kantorovich PERSON gojar-5993 10 2 gojar PERSON gojar-5993 10 3 4 CARDINAL gojar-5993 10 4 1 february 2025 61 CARDINAL gojar-5993 10 5 the nobel prize in economics WORK_OF_ART gojar-5993 11 1 world war ii EVENT gojar-5993 12 1 george dantzig PERSON gojar-5993 12 2 1947 DATE gojar-5993 16 1 the 1980s DATE gojar-5993 16 2 karmarkar PERSON gojar-5993 16 3 1984 DATE gojar-5993 18 1 linear inequalities ORG gojar-5993 21 1 1999 DATE gojar-5993 24 1 laisin et al PERSON gojar-5993 25 1 2024 CARDINAL gojar-5993 27 1 two CARDINAL gojar-5993 27 2 gojar PERSON gojar-5993 27 3 4 CARDINAL gojar-5993 27 4 1 february 2025 CARDINAL gojar-5993 27 5 62 CARDINAL gojar-5993 27 6 i. conditions PERSON gojar-5993 28 1 bounds PERSON gojar-5993 29 1 1998 DATE gojar-5993 31 1 2.1 CARDINAL gojar-5993 33 1 2.2 CARDINAL gojar-5993 33 2 polyhedron linear ORG gojar-5993 34 1 finitely many half CARDINAL gojar-5993 36 1 2.3 CARDINAL gojar-5993 37 1 ∈ ORG gojar-5993 41 1 βŠ† CARDINAL gojar-5993 41 2 𝐦𝐚𝐱 DATE gojar-5993 42 1 ∈ ORG gojar-5993 44 1 2011 DATE gojar-5993 45 1 gojar PERSON gojar-5993 45 2 4 CARDINAL gojar-5993 45 3 1 february 2025 CARDINAL gojar-5993 45 4 63 CARDINAL gojar-5993 45 5 2.1 CARDINAL gojar-5993 45 6 nonempty CARDINAL gojar-5993 46 1 ∢ PERSON gojar-5993 47 1 𝑨𝒙 ≀ ORG gojar-5993 47 2 2011 DATE gojar-5993 48 1 2.2 CARDINAL gojar-5993 48 2 ∢= ORG gojar-5993 49 1 𝑭 βŠ† ORG gojar-5993 50 1 𝑨′𝒙 ≀ 𝒃′ PERSON gojar-5993 50 2 𝑨𝒙 ≀ ORG gojar-5993 50 3 2011 DATE gojar-5993 51 1 2.1 CARDINAL gojar-5993 52 1 βŠ† CARDINAL gojar-5993 53 1 2011 DATE gojar-5993 54 1 2.3 CARDINAL gojar-5993 54 2 𝑨𝒙 ≀ ORG gojar-5993 55 1 nonempty ORG gojar-5993 55 2 βŠ† PRODUCT gojar-5993 56 1 𝑨′𝒙 ≀ 𝒃′ PERSON gojar-5993 56 2 𝑨𝒙 ≀ ORG gojar-5993 56 3 2008 DATE gojar-5993 56 4 2.4 CARDINAL gojar-5993 56 5 𝑠𝑖𝑧𝑒 PERSON gojar-5993 56 6 𝐴 ≀ 2𝑠𝑖𝑧𝑒(𝐴 ORG gojar-5993 56 7 2.5 CARDINAL gojar-5993 56 8 ∈ ORG gojar-5993 58 1 πŸπŸŽπŸπŸ’ DATE gojar-5993 59 1 gojar PERSON gojar-5993 59 2 4 CARDINAL gojar-5993 59 3 1 february 2025 DATE gojar-5993 59 4 64 CARDINAL gojar-5993 59 5 2.4 CARDINAL gojar-5993 59 6 linear programming ORG gojar-5993 59 7 lpp ORG gojar-5993 61 1 βˆ‘ π’‚π’Šπ’‹π’™π’Š PERSON gojar-5993 61 2 π’Š=𝟏 ≀ MONEY gojar-5993 61 3 𝟐 GPE gojar-5993 62 1 two CARDINAL gojar-5993 62 2 elmuti PERSON gojar-5993 62 3 2003 DATE gojar-5993 62 4 genova PERSON gojar-5993 62 5 2011 DATE gojar-5993 63 1 2.5 CARDINAL gojar-5993 64 1 2.6 CARDINAL gojar-5993 64 2 al. GPE gojar-5993 64 3 2014 DATE gojar-5993 72 1 elmuti PERSON gojar-5993 72 2 2003 DATE gojar-5993 72 3 genova PERSON gojar-5993 72 4 2011 DATE gojar-5993 73 1 gojar PERSON gojar-5993 73 2 4 CARDINAL gojar-5993 73 3 1 february 2025 DATE gojar-5993 73 4 65 iii QUANTITY gojar-5993 73 5 𝑨𝒙 ≀ ORG gojar-5993 75 1 𝐦𝐚𝐱 PERSON gojar-5993 75 2 𝑷}is PRODUCT gojar-5993 76 1 3.2.8 ORG gojar-5993 76 2 𝟎 LOC gojar-5993 77 1 3.26 CARDINAL gojar-5993 77 2 𝟎. CARDINAL gojar-5993 83 1 𝑳𝑷 PERSON gojar-5993 83 2 𝑨𝒙 ≀ ORG gojar-5993 88 1 gojar PERSON gojar-5993 89 1 4 CARDINAL gojar-5993 89 2 1 february 2025 CARDINAL gojar-5993 89 3 66 CARDINAL gojar-5993 89 4 4.3 CARDINAL gojar-5993 89 5 2.1 CARDINAL gojar-5993 89 6 2.2 CARDINAL gojar-5993 89 7 2.3 CARDINAL gojar-5993 89 8 2.4 DATE gojar-5993 89 9 2.5 CARDINAL gojar-5993 90 1 1 CARDINAL gojar-5993 91 1 linear programming ORG gojar-5993 91 2 𝑨𝒙 ≀ ORG gojar-5993 92 1 𝑨𝒙 ≀ ORG gojar-5993 93 1 2.1 CARDINAL gojar-5993 95 1 𝟏 CARDINAL gojar-5993 95 2 𝟐 GPE gojar-5993 95 3 𝑨𝑰 PRODUCT gojar-5993 96 1 𝒃𝑰is NORP gojar-5993 96 2 𝑨𝑰 CARDINAL gojar-5993 96 3 𝑨𝑰 FAC gojar-5993 96 4 o DATE gojar-5993 96 5 𝑨𝑰 CARDINAL gojar-5993 96 6 𝑨𝑰 PRODUCT gojar-5993 96 7 𝑨𝑰 FAC gojar-5993 98 1 gojar PERSON gojar-5993 98 2 4 CARDINAL gojar-5993 98 3 1 february 2025 CARDINAL gojar-5993 98 4 67 CARDINAL gojar-5993 98 5 2 CARDINAL gojar-5993 98 6 𝒄𝑻𝒙 ORG gojar-5993 98 7 1 CARDINAL gojar-5993 98 8 a. o PERSON gojar-5993 102 1 3 CARDINAL gojar-5993 102 2 βŠ† CARDINAL gojar-5993 103 1 𝑨𝒙 ≀ ORG gojar-5993 105 1 linear ORG gojar-5993 105 2 zero CARDINAL gojar-5993 106 1 zeros CARDINAL gojar-5993 107 1 cramer PERSON gojar-5993 107 2 𝑨′′ PERSON gojar-5993 107 3 𝒃′. ORG gojar-5993 107 4 2.4 CARDINAL gojar-5993 107 5 2.5 CARDINAL gojar-5993 108 1 + πŸπ’(π’”π’Šπ’›π’†(𝑨′′′ GPE gojar-5993 113 1 schrijver DATE gojar-5993 113 2 1998 DATE gojar-5993 113 3 1986 DATE gojar-5993 113 4 gojar PERSON gojar-5993 114 1 4 CARDINAL gojar-5993 114 2 1 february 2025 CARDINAL gojar-5993 114 3 68 CARDINAL gojar-5993 115 1 two CARDINAL gojar-5993 115 2 two CARDINAL gojar-5993 116 1 100 hours TIME gojar-5993 116 2 80 CARDINAL gojar-5993 117 1 one CARDINAL gojar-5993 117 2 50 MONEY gojar-5993 117 3 40 MONEY gojar-5993 122 1 ο‚· CARDINAL gojar-5993 123 1 ο‚· CARDINAL gojar-5993 124 1 integer ORG gojar-5993 124 2 𝑷𝑰 ORG gojar-5993 124 3 mip ORG gojar-5993 125 1 1 CARDINAL gojar-5993 126 1 π’™πŸ + πŸπ’™πŸ ≀ πŸ–πŸŽ DATE gojar-5993 127 1 2. CARDINAL gojar-5993 127 2 𝑷𝑰 ORG gojar-5993 127 3 integer ORG gojar-5993 127 4 𝑷𝑰 ORG gojar-5993 129 1 mip ORG gojar-5993 130 1 𝑷𝑰 ORG gojar-5993 130 2 gojar PERSON gojar-5993 131 1 4 CARDINAL gojar-5993 131 2 1 february 2025 69 DATE gojar-5993 132 1 bounded ORG gojar-5993 132 2 𝑷𝑰 ORG gojar-5993 134 1 graham PERSON gojar-5993 137 1 linear ORG gojar-5993 142 1 gojar PERSON gojar-5993 142 2 4 CARDINAL gojar-5993 142 3 1 february CARDINAL gojar-5993 142 4 70 CARDINAL gojar-5993 147 1 β„πŸ ORG gojar-5993 147 2 polygon ORG gojar-5993 148 1 π’™πŸ + πŸ‘π’™πŸ ≀ MONEY gojar-5993 149 1 integer ORG gojar-5993 149 2 𝟎, 𝟎 DATE gojar-5993 149 3 𝟐 NORP gojar-5993 150 1 ο‚· CARDINAL gojar-5993 151 1 polygon ORG gojar-5993 152 1 ο‚· CARDINAL gojar-5993 154 1 ο‚· CARDINAL gojar-5993 155 1 ο‚· CARDINAL gojar-5993 159 1 gojar PERSON gojar-5993 159 2 4 CARDINAL gojar-5993 159 3 1 february CARDINAL gojar-5993 159 4 71 CARDINAL gojar-5993 161 1 vi PERSON gojar-5993 166 1 2008 DATE gojar-5993 167 1 0 CARDINAL gojar-5993 168 1 hacettepe PERSON gojar-5993 168 2 37 CARDINAL gojar-5993 168 3 41-55 CARDINAL gojar-5993 169 1 w. GPE gojar-5993 169 2 w. h. PERSON gojar-5993 169 3 pulleyblank ORG gojar-5993 169 4 w. r., & schrijver PERSON gojar-5993 169 5 a. PERSON gojar-5993 169 6 1986 DATE gojar-5993 171 1 wiley PERSON gojar-5993 172 1 g. b. ( PERSON gojar-5993 172 2 1947 DATE gojar-5993 173 1 linear programming ORG gojar-5993 174 1 princeton university press ORG gojar-5993 175 1 g. b. ( PERSON gojar-5993 175 2 1947 DATE gojar-5993 177 1 t. c. koopmans GPE gojar-5993 177 2 339-347 CARDINAL gojar-5993 178 1 wiley PERSON gojar-5993 179 1 elmuti PERSON gojar-5993 179 2 2003 DATE gojar-5993 181 1 american journal of business ORG gojar-5993 181 2 18 CARDINAL gojar-5993 181 3 33 CARDINAL gojar-5993 181 4 genova PERSON gojar-5993 181 5 k. PERSON gojar-5993 181 6 guliashki PERSON gojar-5993 182 1 2011 DATE gojar-5993 183 1 linear ORG gojar-5993 184 1 journal of cybernetics ORG gojar-5993 185 1 11 CARDINAL gojar-5993 186 1 gupta PERSON gojar-5993 186 2 prem kumar PERSON gojar-5993 186 3 d. s. hira PERSON gojar-5993 186 4 2014 DATE gojar-5993 186 5 seventh ORDINAL gojar-5993 186 6 gojar PERSON gojar-5993 186 7 4 CARDINAL gojar-5993 186 8 1 february 2025 DATE gojar-5993 186 9 72 CARDINAL gojar-5993 187 1 rajendra ravindra pvt ltd PERSON gojar-5993 187 2 ram nagar ORG gojar-5993 187 3 delhi110055 GPE gojar-5993 187 4 s. chand & company pvt ltd ORG gojar-5993 187 5 india kantorovich GPE gojar-5993 187 6 l. v. ( PERSON gojar-5993 189 1 leningrad state university ORG gojar-5993 190 1 karmarkar PERSON gojar-5993 190 2 n. PERSON gojar-5993 190 3 1984 DATE gojar-5993 191 1 linear programming ORG gojar-5993 192 1 combinatorica PERSON gojar-5993 192 2 4(4 PRODUCT gojar-5993 192 3 373 CARDINAL gojar-5993 193 1 laisin PERSON gojar-5993 193 2 m. PERSON gojar-5993 193 3 edike PERSON gojar-5993 193 4 2024 CARDINAL gojar-5993 194 1 11 CARDINAL gojar-5993 194 2 11 CARDINAL gojar-5993 194 3 g. l. PERSON gojar-5993 194 4 l. a. PERSON gojar-5993 194 5 1999 DATE gojar-5993 196 1 wiley PERSON gojar-5993 197 1 schrijver GPE gojar-5993 197 2 a. PERSON gojar-5993 197 3 1998 DATE gojar-5993 198 1 linear ORG gojar-5993 198 2 integer programming ORG gojar-5993 199 1 wiley PERSON gojar-5993 202 1 mark laisin PERSON gojar-5993 202 2 chukwuemeka odumegwu ojukwu university ORG gojar-5993 202 3 uli GPE gojar-5993 202 4 anambra state GPE gojar-5993 202 5 nigeria GPE gojar-5993 204 1 collins edike PERSON gojar-5993 204 2 chukwuemeka odumegwu ojukwu university ORG gojar-5993 204 3 uli GPE gojar-5993 204 4 anambra state GPE gojar-5993 204 5 nigeria GPE gojar-5993 205 1 edikecollins505@gmail PERSON gojar-5993 205 2 dr r. n. PERSON gojar-5993 205 3 chukwuemeka odumegwu ojukwu university ORG gojar-5993 205 4 uli GPE gojar-5993 205 5 anambra state GPE gojar-5993 205 6 nigeria GPE gojar-5993 206 1 apa laisin PERSON gojar-5993 206 2 m. PERSON gojar-5993 206 3 edike GPE gojar-5993 206 4 r. n. PERSON gojar-5993 206 5 2025 DATE gojar-5993 208 1 4(1 CARDINAL gojar-5993 208 2 60-72 CARDINAL gojar-5993 209 1 mla laisin PERSON gojar-5993 209 2 mark PERSON gojar-5993 209 3 edike GPE gojar-5993 209 4 collins, & ujumadu ORG gojar-5993 209 5 r. n. PERSON gojar-5993 210 1 gojar PERSON gojar-5993 210 2 4 CARDINAL gojar-5993 210 3 1 CARDINAL gojar-5993 210 4 2025 DATE gojar-5993 211 1 60-72 CARDINAL