SMU OREM 4390 - A Linear Program for Optimizing Bologna Wiener Production

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Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 161981-03 Spring 1981 A Linear Program forOptimizing Bologna and WienerProduction Alma C. Evans A LINEAR PROGRAM FOROPTIMIZING BOLOGNA AND WIENER PRUDUCIION Aima C. EvansOREM 4390Spring 19810U Table of Contents Purpose............................................................1 Background........................................................1 Model & Solution Description ......................................2 Mathanatical formulations......................................3 Variables ...................................................7 Analysis..........................................................8 Comparisons......................................................12 Recommendations..................................................PURPOSE The purpose of the design project is to formulate a model to optimize a meat packers production of bologna and wieners. BACKGROUND The meat packing industry today uses computers for accounting and inventory purposes, but theystill use man-figured formula sheets and intuition to mix their products. These recipes meet USDA requirements but that is their limit. The reipes are supposedly minimum cost formula-tions, but this will be proven false. The capability to optimize product cost, volume, and profit is available. The data necessary is the meat packers cost of ingredients and his limit of ingredients available for mixing. The costs are available twice daily over the National Provisioner Beef and Pork Wires. Ingredient limits depend on the warehouse space and production capacity of the packing house. Since packing houses already own small computer systems it would not be unreasonable to assume a Optirniaztion Package would be easy to enter. The doubt and guesswork involved in the product would be lessened and the master mixer freed to supervise the mixing instead of formulating recipes. In talking to Carlo Werner, a master sausage maker, he feels the mixing is the critical part in making products. Therefore the ingredients should meet regulations and,if mixed properly, be a flavorful product. -1-MODEL & SOLUTION DESCRIPTION Six models were solved in order to make comparisons of the optimal solutions based on the objective functions. The Min Cost/lb and Max Volume models were run for two separate products. The Two Product models choose the optimal amount to make of each product according to the objective function, Max Volume or Max Profit. To find an optimal recipe, USDA standardI and ingredient limits were taken into account. The limits are input by the user. The model solved has a Beef & Pork label, therefore government standards for that label apply. The label restriction for Beef & Pork is at least 707 of the ingredients must be Beef & Pork, of that 7O%, 3O7 must be beef and 3O7 must be pork. This gives the producer a lO°L margin in selecting the meat most economical to use at production time. The user must input the costs associated with the ingredients. For the ingredients used, 807 lean beef, 507 lean beef and beef plates are considered beef meat. For pork, 807 lean pork, 50% lean pork and pork jowls are considered pork meat. The spices and preservatives depend on the amount of the mix. To formulate this problem, a basic linear program was used. To solve it, BLP was used, a program written be Dick Barr. -2-Min Cost/lb, All Meat, No Limits Objective Function tixi = z S. t. Fat '?307 Binding PXj 5570 ColorRX 5570 Weight xi=l 70/30 .Xi?707o 307 Xj.3070C = cost of ingredient i Xi = ingredient i Fl = fat % in ingredient i Pi bind rating for ingredient i RI. = color rating for ingredient i i = 1,2,3,4,7 i = 1,7 i = 2,3,4 Max Volume, All Meat, LimitsObjective Function .7 2 xi = Volume = A S. t. Limits Xis Li. FatFXi S 30%, (A) Uri-qXi5570(A) ColorRjX? 5570(A) continued....Xi = pounds of ingredient i Li = limit in lbs. of ingredient i -3-70/30 Label X1 + X6 ^:30%(A) X2 + X3 + X43070(A) X1+ X2 + X + X4 +70(A) Weight Xj=A Lj Min Cost/lb, Meat & Other Parts, No Limit Objective Function Cixi = z S. t. Fat.FXiS 30% Binding? 5570 ColorRjX 2:55% Lzi Weigit xi = 1 70/30 Label 2X M.Ci = cost of ingredient i Xi = amount of ingredient i i = 1,2,3,4,7,13 I = 1, 7,13 .X1^t3070i = 2,3,4 Max Volume, Meat & Other Parts, Limits Objective Function Xj = Volume = B S. t. Limits Xj L continued.Xj = pounds of ingredient i -4-FatM. (B) Binding5570 (B) Color55%(B) twi 70/30 Label IXi ^t 70%(B) Xi .30%(B) 30%(B)i12,3,47,13 i l,7l3 i =. 2,3,4 Weigit fti = B S. t. WghtA= A WghtB (3 ZX. = B ( 3 FatA3O7 (A) irk FatB330%(B) BindAPX ? 5570 (A)j = ingredient j of product B i = ingredient i of product A BindBè 5570(B) continued....Two Products Objective Functions A + B = VolumeA = lbs of product A (all meat). B = lbs of product B (meat & parts)- RevA + BevB - CostA CostB = Profit -5-CobrA iRX. 557 (A) Co1qB5570(B) Limits Xj +XLj 70/30 Label ZXi ^ 7070(A) ZXJ 707,(B)X3070(A)3070(B)I X3_ 30%(A)307(B)= lb, limit of ingredient i& or products A & B i =l,2,3,47 i =1,2,3,4,7,13 i =1,7 i =l,7l3 i =2,3,4 i =2,3,4Additional Constraints for Profit CostACjXj = CostA CostB= CostB RevA2.80A = RevA RevB2.20B = RevBA = lbs, of product A B = lbs. of product BVARIABLES HIBEEF, HIBEEFA & HIBEEFB = 807 lean Beef MEDPRK, NEDPBKA & NEDPRKB = 507 lean pork HIPRK,HIPRKA & HIPRKB = 8Cr/0 lean pork F7-JOWLS, FZJOWLA & FZJOWLB = frozen pork jowls HRTS, HRTSA & HRI'SB = beef hearts NAVELS, NAVELSA & NAVELSB = beef navels REGBEEF, RECBFA & BEGBFB = 5O, lean beef CHEEK, CHEEKB = beef cheek meat TRIPE, TRIPEB = beef tripe PRKEiRTS PRKHRTh = pork hearts PRKFAT, PRKFATh = prok backfat BELLIES, BEI.LYB = pork bellies BPLATEB BFLATEB = beef platesANALYSIS Min Cost, All Meat, No Limits 'Recipe" 7.6% lean beef 80% 40.0% lean pork, 80% 20.3% beef hearts 977 beef navels 22.4% lean beef, 50%Min Cost/lb = $.56081 *50 Pork would have to cost $.104 to be included in the solution *Frozen pork jowls would have to cost $.189 to be included in the solution *Fat, Weight, Binding, Beef & Pork Requirement and beef Requirement are all binding constraints *The recipe has a 10% surplus of pork becuase it was cheaper to use than beef in making up the extra 10% to reach the 70% requirement of Beef & Pork *If the fat constraint is increased by one, cost decreases by $.04: If the binding constraint were increased by one, you would increase cost by $1.83 *Fat Constraint has a very


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