Red Brand Canners

Case Study: RED BRAND CANNERS Vice President of Operations Mr. Michell Gorden Controller Mr. William Copper Sale Manager Mr. Charles Myers Production Manager Mr. Dan Tucker Purpose: Decide the amount of tomato products to pack at this season. Tomato Products Whole Tomato Tomato Juice Tomato Paste Information: 1. Amount of Tomato: 3,000,000 pounds to be delivered. Tomato quality: 20% (grade A) ? 3,000,000 = 600,000 pounds 80% (grade B) ? 3,000,000 = 2,400,000 pounds (provided by production manager) 2. Demand forecasts & selling prices (provided by sale manager): Products Demand Whole canned tomato no limitation

Others Refer Exhibit 1 1 lbs. correction (800,000/18) = 44444. 5 Cases Selling prices has been set in light of the long-term marketing strategy of the company. Potential sales have been forecasted at these prices. 3. Purchasing price & product profitability (provided by controller) Purchasing price 6cents/pound Net profit Refer Exhibit 2 Grade A 9 points Grade B 5 points Product Minimum requirement Whole tomato 8 points Tomato juice 6 points Tomato paste 5 points (without grade A) 2 3. 8 -(0. 54+0. 26+0. 38+0. 77) = 1. 85 4. 0-(1. 18+0. 24+0. 4+0. 7) = 1. 48 4. 5 – (1. 32+0. 36+0. 85+0. 65) = 1. 32 0. 3 5. 0,000 pounds of grade “A” tomatoes are available at 8. 5 cents per pound. (provided by the Vice president of operations) 6. Sale manager re-computes the marginal profits (Exhibit 3). Linear Programming Solutions (a) How to use the crop of 3,000,000 lbs. of tomatoes? (b) Whether to purchase an additional 80,000 lbs. of A-grade tomatoes? Part (a) Formulation: WA = lbs. of A-grade tomatoes in whole. WB = lbs. of B-grade tomatoes in whole. JA = lbs. of A-grade tomatoes in juice. JB = lbs. of B-grade tomatoes in juice. 3 ?N(1) (2) ¤§Ap?? ¤eµ{¦? 1 CASE = 0. 0518* 25= 1. 295 1 CASE = [(0. 0932*(3/4)+0. 0518*(1/4)]*18

PA = lbs. of A-grade tomatoes in paste. PB = lbs. of B-grade tomatoes in paste. 600,000 lbs. – 3WB ? U 0 WB ? O 600,000/3 = 200,000 600,000 + 200,000 = 800,000 lbs. Demand of whole tomatoes ? O 800,000 lbs. = 44,444. 5 ? 18 lbs Demand of tomatoes Juice ? O 50,000 cases = 50,000 ? 20 lbs = 1,000,000 lbs Demand of tomatoes paste ? O 80,000 cases = 80,000 ? 25 lbs = 2,000,000 lbs Grade “A” ? O 600,000 Grade “B” ? O 2,400,000 ( 3,000,000lbs ? 20% ) = 600,000 lbs. ( 3,000,000lbs ? 80% ) = 2,400,000 lbs. 4 Quality requirement for whole tomato: Quality requirement for whole tomato: (0. 9? WA + 0. 5? WB)/2 ? U 0. 8? WA + WB)/2 ? WA – 3WB ? U 0 Quality requirement for tomato juice: (0. 9? JA + 0. 5? JB)/2 ? U 0. 6? (JA + JB)/2 Constraints: WA CWA 1 WB CWB 1 JA CJA 1 JB CJB 1 1 1 1 1 -3 3 -1 1 1 1 1 1 PA CPA PB CPB ? O ? O ? O ? O ? O ? U ? U 14,400,000 1,000,000 2,000,000 600,000 2,400,000 0 0 ? 3JA – JB ? U 0 800,000 lbs. Coefficients of Objective Function: Both Cooper’s and Myers’ figures (Exhibits 2 and 3) are wrong. Contribution = selling price – variable cost (excluding tomato cost) Thus, CWA = CWB = 1. 48/18 = 0. 0822 CJA = CJB = 1. 32/20 = 0. 066 CPA = CPB = 1. 85/25 = 0. 074 The contribution = $225,340 – $180,000 = $45,340. Optimal primal solution WA 525,00 WB 175,000 JA 75,000 JB 225,000 PA 0 PB 2? 106 Optimal value = 225340 Optimal dual solution Column Constraint 7 1 0 8 2 0 9 3 0. 0161 10 4 0. 0903 11 5 0. 0579 12 6 8. 1? 10-3 13 7 8. 1? 10-3 Value Shadow price on constraint 4 = 0. 0903 Sensitivity on cost values variable Lower limit 1 0. 0606 0. 2336 0. 0822 2 0. 0606 0. 5454 0. 0822 3 -0. 0884 0. 0876 0. 066 4 1. 45333? 10-2 5 6 0. 0579 -? 0. 1064 0. 074 Upper limit Current value 0. 803111 ? 10-2 +? 0. 074 0. 066 6 Sensitivity on the right-hand sides constraint 1 700,000 2 300,000 3 1. 45333? 10-6 4 133,333 1. 2? 106 600,000 2. 2? 106 2. 8? 106 2. 4? 106 6 -600,000 7 -200,000 Lower limit Upper limit Current value +? 1. 44? 107 +? 106 2. 2? 106 2? 106 46,000 0 1. 4? 106 0 Parametric analysis on constraint 4 Shodow prices 0. 0903 0. 0876 0. 08493 0. 0822 0. 074 0. 066 0 Lower limit Upper limit Current value 600,000 1,200,000 1,200,000 7,200,000 12,000,000 14,000,000 15,000,000 1,200,000 1,200,000 7,200,000 12,000,000 14,000,000 15,000,000 +? 225,340 279,520 279,520 789,120 1,183,680 1,331,680 1,397,680 7 Parametric Analysis on the Right-hand side of Constraint 4 (availability of grade A tomato) const#4? 600,000 + , ? [0,? ) ? ? ,183,680 798,120 ±??? = ±??? =0. 074 (1,183,680 ? 789,120) 12,000,000 ? 7,200,00 394,560 = 4,800,000 =0. 0822 ±??? =0. 066 ±??? = 0 279,520 ±? = ?? (789,120? 279,520) 7,200,000 ? 1,200,000 509,600 = 6,000,000 = 0. 08493 225,340 ±??? = (279,520 ? 225,340) 1,200,000? 600,000 54180 = 600,000 = 0. 0903 133,333 600,000 1,200,000 7,200,000 12,000,000 14,000,000 15,000,000 Part (b) Solve the problem with 680,000 lbs of tomatoes. ( The same conclusion could be reached by inspecting the dual variable of the availability constraint of A-grade tomatoes (constraint 4) in the optimal solution. Since the dual variable $0. 903/lb. > $0. 08/lb. And this value is constant for an additional 600,000 lbs. of grade A tomatoes, purchasing 80,000 lbs. will result i a net increase of the contribution. Linear programming solution with 68,000lbs grade “A” tomatoes Optimal primal solution WA 615,000 WB 205,000 JA 65,000 JB 195,000 PA 0 PB 2? 106 Optimal value = 232564 8 Net profit of 80,000lbs. A-grade = (232564-22534)-80,000? 0. 085 = 7224-6800 = 424 Optimal dual solution Column Constraint Value 7 1 0 8 2 0 9 3 0. 0161 10 4 0. 0903 11 5 0. 0579 12 6 8? 10-3 13 7 8? 10-3 Part(c) Comparison of Results using Different Objective Coefficients

Correct objective function: CWA = CWB = 1. 48/18 = 0. 0822, CPA = CPB = 1. 85/25 = 0. 074 Net Profit = CWAWA + C WBWB + CJAJA + C JBJB + C PAPA + C PB PB – $180,000 CJA = CJB = 1. 32/20 = 0. 066, Myer’s objective function: CWA = CWB = 0. 01, CJA = CJB = 0. 08, CPA = CPB = 0. 55 Net Profit = CWAWA + C WBWB + CJAJA + C JBJB + C PAPA + C PB PB Cooper’s objective function: CWA = CWB = 0. 12/18, CJA = CJB = 0. 09/20, CPA = CPB = 0. 12/25 Net Profit = CWAWA + C WBWB + CJAJA + C JBJB + C PAPA + C PB PB 9 Myers 0 Cooper 800,000 lb 605,000 200,000 0 0 0 2,000,000 0 2,000,000 2,800,000 200,000 lb $45,778

Correct 700,000 lb 525,000 175,000 300,000 lb 75,000 225,000 2,000,000 0 2,000,000 3,000,000 0 lb $225,340 $180,000 $45,340 Whole WA WB JA JB PA PB Juice Paste 0 0 1,000,000 250,000 750,000 2,000,000 350,000 1,650,000 3,000,000 0 $48,000 Total Unused grade-B Objective function ( O ) Fruit cost ( F ) Net profit ( O – F ) Unallocated or uncovered tomatoes ( U ) $14,000 $34,000 $12,000 $33,778 0 $45,340 O–F-U °YAD¤@ ¦p whole tomato ? u? NA? ¤@¤j? L°O °a»u? IAE¶R¶q¦O©w ¦p¤U¤§Ao«Y : , , : AE¶R¶q x (? c) 0 < x ? 100,000 100,000< x ? 600,000 600,000 < x ? 800,000 ?? »u $5. 00/? c $4. 50/? c 4. 00/? c A E ¶ R A ` » u 3,550,000 2,750,000 500,000 A E ¶ R ¶/ q ? c 800,000 100,000 600,000 µu : ? O¦pAE¶R¶q 80,000? c®EA`»u¬° $5. 00 ? 80,000, AE¶R¶q 700,000? c«h A`»u¬° $5. 00? 100,000+$4. 50? 500,000+$4. 00? 100,000, 10 °YAD¤G : (a) ? H¤U¤T-O±o? o¦U¤O¦? ¤@-O-n¦??? , -«·s«O?? ¦? °YAD????? C? O ¦?? C (i) whole tomato? I?? ¶q??? ?? , (ii) tomato juice? I?? ¶q?? ] ?? , (iii) tomato paste ? I?? ¶q??? ???? °YAD¤T : (a) ¦p? H? «? o¦u¦?? v¦O? {, Charles Myers (sale manager) »{ ¬° tomato paste ¦U¤O-n? I?? P ? c(? A¤@Au? y tomato ), juice ?? ¦n¬OJ ? c(? A¤GAu? y), A`°a?? AB¤? §C©o I ($)(? A¤T Au? ), 11 °YAD? | : (6-a) ¦bRed Brand Canner®?? O¤¤, ¦pµf-X¤§AE¶R»u®?? IA, B¤G ? Oµ?? A¦O? §, A, B,? A»u®? ¤A§O¬° a, b (cents/lb. ), ? I??? N? l¤§ µf-X»u-E¬°0, ? A? oµf-X? I¤j? NA? ¶q¬° 3,000,000 lbs. (20%¬° A? A), ¦y? i¤??????? ±AAE , ¦b? a? L±o? o¤? AU?? ±?? p¤U -«·s , «O?? ¦? °YAD????? C? O¦? ?C (6-b) ¦bRed Brand Canner®?? O¤¤, ? I??? N? l¤§µf-X¤??? µ?? A? i Aa?? »u®? ¬° d (cents/lb. ), -«·s«O?? ¦? °YAD????? C? O¦?? C °YAD¤- : Red Brand Canner®?? O¤¤, ? C¤@? OAoAY¶·-n? g? L Y ¤G? O????? [¤u, X ???? X, (? A¤@¶? ¬q? B? z¦p? M¬~ ? NµNAI©Oµ? ), Y ???? (? A¤G¶? ¬q? B? z ¦p? ]? E , ? E? c , , , , µ? , ? IDan Tucker, production manager ¤A? R¤H¤O»P????? ]???? »Y? D¦p¤U : ?C? c¤§¤H¤O»Y? D X ???? ¤§»Y? D Y ???? ¤§»Y? D whole tomatoes tomato juice tomato paste a1 hours/case b1 hours/case c1 hours/case a2 hours/case b2 hours/case c2 hours/case a3 hours/case b3 hours/case c3 hours/case ?N©o¦P¤@©u? `¦? ¦h? OAoAY¦P®E? I??????? ]?? »P¤H¤O?? ¦? — ¤µ¤u? t¤A°t¤§ , , ¤H¤O»PX, Y ????? i? I¤u®E¤A§O¬° ? , ? , ? ?? >? ?a? L±o? o¤? AU?? ±?? p¤U -« , , ·s«O?? ¦? °YAD????? C? O¦?? C 12 (4-3) ? H¤W? O¦? A?? I inear Programming package ? D? N±o? i? H¤U? e°T L , ¤µ? A ? o? i¦A??? N 1,300,000 lbs. ¤§A? Aµf-X, ?? u¬O 8. 5cent/lb. , ±z??? M©w¬O§_¦A AE¶R? ¦AAE¶R¦h¤O A`¦@? i? W? [¦h¤O§Q? i (? C¤@¶µµ? ®??? -n»? ©u? z? N lbs? ? ) WA WB 175,000 JA 75,000 JB 225,000 PA 0 PB 2? 106 525,000 Optimal value = 225340 Optimal dual solution Column Constraint Value 7 1 8 2 0 9 3 0. 0161 10 4 0. 0903 11 5 0. 0579 12 6 8. 1? 10-3 13 7 8. 1? 10-3 0 “Shadow price on constraint 4” = 0. 0903 Sensitivity on cost values variable Lower limit Upper limit Current value 1 0. 0606 0. 2336 0. 0822 2 0. 0606 0. 5454 0. 0822 3 -0. 0884 0. 0876 0. 066 4 1. 45333? 10-2 0. 803111? 10-2 5 -? 0. 1064 0. 074 6 0. 0579 +? 0. 074 0. 066 13