Using only the items given in the tables below, formulate an optimization problem in standard form...

 

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Using only the items given in the tables below, formulate an optimization problem in standard form to create a minimum cost pizza which satisfies both the nutritional requirements of Table 1 and bounds on item quantities given in Table 2. Use the nutritional data of Table 3 and the cost data of Table 4 in your model. Use the MATLAB linear optimization routines linprog to solve the problem. Interpret your results. Table 1 Nutrient Calcium Requirement Units 750.0 mg Iron 12.0 mg Protein 48.5 gram Vitamin A 4500.0 IU Thiamine 1.3 mg Niacin 16.0 mg Riboflavin 1.6 mg Vitamin C 30.0 mg Table 2 UPPER AND LOWER BOUNDS ON PIZZA ITEMS Item Upper Bounds* Lower Bound* cheese (x1) 2.270 1.703 sauce (x2) 1.986 1.140 dough (x3) 5.249 4.266 pepperoni (x4) 0.983 N/A ham (x5) 1.135 N/A bacon (x6) 0.993 N/A g.pepper (x7) 1.561 N/A onion (x8) 0.993 N/A celery (x9) mushroom(x10) 1.561 N/A 1.135 N/A tomato (x11) 1.703 N/A pineapple (x12) 1.703 N/A meat (x4, X5, X6) N/A 0.993 veg. (x7, x8, x9, x11) N/A 0.993 fungi (x10) N/A 0.922 Table 2 * Amount in hundreds of grams. * Amount in hundreds of grams. Item cheese (x1) sauce (x2) Innequality constraints Upper Constraint Lower Constraint x12.270 -x1-1.703 Table 2 X21.986 -x2 -1.140 dough (x3) X35.249 -x3-4.266 pepperoni (x4) 24 < 0.983 -x40 ham (x5) 5 < 1.135 -x50 bacon (x6) X6 0.993 -x60 g.pepper (x7) X71.561 -x70 onion (x8) X8 0.993 -x80 celery (x9) mushroom(x10) tomato (x11) X9 1.561 -x90 X101.135 -x100 X111.703 11 <0 pineapple (x12) X12 1.703 -x120 meat (x4, X5, X6) N/A -x4- X5- X6 -0.993 veg. (x7, x8, x9, X11) N/A -x7,-x8 x9 x11 -0.993 fungi (x10) N/A -x10 -0.922 Table 3 NUTRITIONAL DECOMPOSITION OF PIZZA ITEMS* 2 Item Calc Iron Prot cheese (x1) 517.700 .222 20.000 sauce (x2) dough (x3) pepperoni (x4) 10.000 2.500 14.000 1.800 2.000 18.233 3.826 14.224 15.000 Vit A 3000.000 800.000 Thia Niac Ribo Vit C .022 6.000 .244 0.100 1.400 .060 6.000 .586 8.852 .628 2.000 ham (x5) bacon (x6) 9.031 2.291 14.692 13.000 1.189 8.392 - .740 4.009 0.178 g.pepper (x7) onion (x8) 9.459 .675 27.273 .545 1.351 1.818 celery (x9) 40.000 .250 - 209.460 18.182 125.000 mushroom (x10) 6.000 .800 3.000 - tomato (x11) 13.333 .533 1.333 450.000 pineapple (x12) 12.016 .310 .387 .361 1.828 .114 .081 .540 .081 127.030 .363 .545 .036 10.000 .025 .500 .025 10.000 .100 4.300 .460 .066 .800 25.194 .081 .193 .019 3.000 .04 22.667 6.977 * Units as in Table 1 Table 4 COSTS OF PIZZA ITEMS Item Cost in cents/100 grams cheese (x1) 93.20 sauce (x2) 70.48 dough (x3) 19.26 pepperoni (x4) 88.22 ham (x5) 88.10 bacon (x6) 88.54 green pepper (x7) 50.20 onion (x8) 10.66 celery (x9) 28.28 mushrooms (x10) 62.40 tomatoes (x11) 44.06 pineapple (x12) 51.96 Using only the items given in the tables below, formulate an optimization problem in standard form to create a minimum cost pizza which satisfies both the nutritional requirements of Table 1 and bounds on item quantities given in Table 2. Use the nutritional data of Table 3 and the cost data of Table 4 in your model. Use the MATLAB linear optimization routines linprog to solve the problem. Interpret your results. Table 1 Nutrient Calcium Requirement Units 750.0 mg Iron 12.0 mg Protein 48.5 gram Vitamin A 4500.0 IU Thiamine 1.3 mg Niacin 16.0 mg Riboflavin 1.6 mg Vitamin C 30.0 mg Table 2 UPPER AND LOWER BOUNDS ON PIZZA ITEMS Item Upper Bounds* Lower Bound* cheese (x1) 2.270 1.703 sauce (x2) 1.986 1.140 dough (x3) 5.249 4.266 pepperoni (x4) 0.983 N/A ham (x5) 1.135 N/A bacon (x6) 0.993 N/A g.pepper (x7) 1.561 N/A onion (x8) 0.993 N/A celery (x9) mushroom(x10) 1.561 N/A 1.135 N/A tomato (x11) 1.703 N/A pineapple (x12) 1.703 N/A meat (x4, X5, X6) N/A 0.993 veg. (x7, x8, x9, x11) N/A 0.993 fungi (x10) N/A 0.922 Table 2 * Amount in hundreds of grams. * Amount in hundreds of grams. Item cheese (x1) sauce (x2) Innequality constraints Upper Constraint Lower Constraint x12.270 -x1-1.703 Table 2 X21.986 -x2 -1.140 dough (x3) X35.249 -x3-4.266 pepperoni (x4) 24 < 0.983 -x40 ham (x5) 5 < 1.135 -x50 bacon (x6) X6 0.993 -x60 g.pepper (x7) X71.561 -x70 onion (x8) X8 0.993 -x80 celery (x9) mushroom(x10) tomato (x11) X9 1.561 -x90 X101.135 -x100 X111.703 11 <0 pineapple (x12) X12 1.703 -x120 meat (x4, X5, X6) N/A -x4- X5- X6 -0.993 veg. (x7, x8, x9, X11) N/A -x7,-x8 x9 x11 -0.993 fungi (x10) N/A -x10 -0.922 Table 3 NUTRITIONAL DECOMPOSITION OF PIZZA ITEMS* 2 Item Calc Iron Prot cheese (x1) 517.700 .222 20.000 sauce (x2) dough (x3) pepperoni (x4) 10.000 2.500 14.000 1.800 2.000 18.233 3.826 14.224 15.000 Vit A 3000.000 800.000 Thia Niac Ribo Vit C .022 6.000 .244 0.100 1.400 .060 6.000 .586 8.852 .628 2.000 ham (x5) bacon (x6) 9.031 2.291 14.692 13.000 1.189 8.392 - .740 4.009 0.178 g.pepper (x7) onion (x8) 9.459 .675 27.273 .545 1.351 1.818 celery (x9) 40.000 .250 - 209.460 18.182 125.000 mushroom (x10) 6.000 .800 3.000 - tomato (x11) 13.333 .533 1.333 450.000 pineapple (x12) 12.016 .310 .387 .361 1.828 .114 .081 .540 .081 127.030 .363 .545 .036 10.000 .025 .500 .025 10.000 .100 4.300 .460 .066 .800 25.194 .081 .193 .019 3.000 .04 22.667 6.977 * Units as in Table 1 Table 4 COSTS OF PIZZA ITEMS Item Cost in cents/100 grams cheese (x1) 93.20 sauce (x2) 70.48 dough (x3) 19.26 pepperoni (x4) 88.22 ham (x5) 88.10 bacon (x6) 88.54 green pepper (x7) 50.20 onion (x8) 10.66 celery (x9) 28.28 mushrooms (x10) 62.40 tomatoes (x11) 44.06 pineapple (x12) 51.96