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Management Science III

Management Science III. RED BRAND CANNERS CASE MEDIA SELECTION PROBLEM vehicle to introduce integer linear programming CASE: GLOBAL SOURCING OF MALT. RED BRAND CANNERS CASE. DECIDING ON THE PRODUCTION QUANTITIES OF: WHOLE CANNED TOMATOES TOMATO JUICE TOMATO PASTE.

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Management Science III

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  1. Management Science III • RED BRAND CANNERS CASE • MEDIA SELECTION PROBLEM • vehicle to introduce integer linear programming • CASE: GLOBAL SOURCING OF MALT

  2. RED BRAND CANNERS CASE DECIDING ON THE PRODUCTION QUANTITIES OF: WHOLE CANNED TOMATOES TOMATO JUICE TOMATO PASTE

  3. TOTAL CROP IS 3,000,000 POUNDS • BOUGHT AT 6 CENTS PER POUND • 20% OF CROP IS QUALITY GRADE A (9 points) • 80% OF CROP IS QUALITY GRADE B (5 points)

  4. QUESTION: WHAT PRODUCT MIX TO MAKE IN ORDER TO MAXIMIZE PROFIT?

  5. 1. WHY CAN NO MORE THAN 800,000 POUNDS OF CWT BE PRODUCED? • QA = # POUNDS OF GRADE A USED • QB = # POUNDS OF GRADE B USED • QA + QB = TOTAL PRODUCTION OF CWT • 9 QA + 5 QB = TOTAL POINTS USED CWT REQUIRES 8 POINTS PER POUND ON AVERAGE, AND HENCE 9 QA + 5 QB >= 8 , QA + QB WHICH IMPLIES THAT QA >= 3QB

  6. 2. COOPER: USE THE ENTIRE CROP FOR CWT, THEN FOR PASTE, AND THEN FOR JUICE …. STRATEGY BASED ON NET PROFIT PER CASE SOLD (SEE TABLE 2) RESULTS IN: • 800,000/18 = 44,444 cases CWT • 80,000 cases of PASTE (equals max demand) • 0 cases of JUICE • COOPER’s NET PROFIT: • $ 17,777.60 + $ 28,000.00 = $ 45,777.60 (44,444 x $0.40 + 80.000 x $0.35)

  7. So … is this any good? NO! COOPER IGNORES THE 200,000 POUNDS OF TOMATOES THAT CANNOT BE USED …... REAL PROFIT = $ 45,777.60 - 200,000 x $0.06 = $ 33,777.60

  8. 3. MYERS (Table 4): • DERIVES THE PRICE PER POUND OF “A” AND “B” TOMATOES IN CENTS • A PRODUCTION STRATEGY ON BASIS OF MARGINAL PROFIT PER POUND

  9. Cont’d • … 2,000,000 pounds “B” to produce PASTE, which gives 80,000 cases (equals maximum demand) • 400,000 pounds of “B” and 600,000 pounds of “A” to produce JUICE, giving 50,000 cases • PROFIT = 80.000 x $0,55 + 50.000 x $0.08 = $ 48,000

  10. IS MYERS CORRECT … • WELL ….. THE REAL PROFIT IS EQUAL TO 80,000 x $1.85 + 50,000 x $ 1.32 - 3,000,000 x $0.06 = $ 34,000!!!! Contribution before tomato costs (Table 4)

  11. WHAT ARE BOTH COOPER AND MYERS FUNDAMENTALLY DOING WRONG?

  12. THEY ARE STRUGGLING WITH THE TOMATO COST • COOPER: QUANTITY BASED INPUT PRICING • MYERS: QUALITY-BASED INPUT PRICING • BUT ….. THE TOMATO COSTS ARE SUNK COST …. AND HENCE SHOULD NOT AFFECT THE DECISION MAKING! • WHAT IS MORE, NEITHER IS ADDRESSING THE ISSUE WHETHER THE ADDITIONAL GRADE “A” TOMATOES SHOULD BE BOUGHT …..

  13. LINEAR PROGRAMMING! BY THE WAY ….. ARE ALL NECESSARY CONDITIONS FOR THE APPLICATIONS OF LP SATISFIED?

  14. LINEAR PROGRAMMING • DECISION VARIABLES • OBJECTIVE FUNCTION • CONSTRAINTS

  15. DECISION VARIABLES • AC := POUNDS OF GRADE “A” IN CWT • BC := POUNDS OF GRADE “B” IN CWT • AJ := POUNDS OF GRADE “A” IN JUICE • BJ := POUNDS OF GRADE “B” IN JUICE • AP := POUNDS OF GRADE “A” IN PASTE • BP := POUNDS OF GRADE “B” IN PASTE

  16. CONSTRAINTS AC + BC <= 14,400,000 AJ + BJ <= 1,000,000 AP + BP <= 2,000,000 AC + AJ + AP <= 600,000 BC + BJ + BP <= 2,400,000 AC - 3BC >= 0 3AJ - BJ >= 0 AC,BC, AJ, BJ, AP, BP >= 0 • DEMAND FOR CWT • DEMAND FOR JUICE • DEMAND FOR PASTE • AVAILABILITY OF “A” • AVAILABILITY OF “B” • QUALITY CWT • QUALITY TJ • NON-NEGATIVITY

  17. OBJECTIVE FUNCTION MAXIMIZE 8.22(AC+BC) + 6.6(AJ+BJ) + 7.4(AP+BP) NOTE: profit coefficients in cents per pound! (For instance: 8.22 = $1.48 / 18 pounds per case)

  18. MAXIMIZE 8.22 (AC+BC) + 6.6(AJ+BJ) + 7.4(AP+BP) • AC + BC <= 14,400,000 • AJ + BJ <= 1,000,000 • AP + BP <= 2,000,000 • AC + AJ + AP <= 600,000 • BC + BJ + BP <= 2,400,000 • AC - 3BC >= 0 • 3AJ - BJ >= 0 • AC,BC,AJ,BJ,AP,BP >= 0

  19. PROFIT: $45,355,55!!!!!

  20. IS IT PROFITABLE TO BUY ADDITIONAL GRADE “A”, OFFERED AT 8.5 CENTS PER POUND FOR ANY AMOUNT UP TO 80,000 POUNDS?

  21. IS IT PROFITABLE TO BUY ADDITIONAL GRADE “A”, OFFERED AT 8.5 CENTS PER POUND FOR ANY AMOUNT UP TO 80,000 POUNDS? • SENSITIVITY ANALYSIS: • SHADOWPRICE OF “A” IS 9.0333 …. • ALLOWABLE MAXIMUM OF RHS IS …. 600,000 pounds • NEW PROFIT = OLD PROFIT + 80,000 * (9.0333-8.5) = 45,355.55 + 402.664 = 45,758.21

  22. IS IT PROFITABLE TO BUY ADDITIONAL GRADE “A”, OFFERED AT 8.5 CENTS PER POUND FOR ANY AMOUNT UP TO 80,000 POUNDS? • MODIFY THE LINEAR PROGRAMMING FORMULATION TO INCLUDE ADDITIONAL PURCHASES …….

  23. - 8.5 EA MAX 8.22 (AC+BC) + 6.6(AJ+BJ) + 7.4(AP+BP) • AC + BC <= 14,400,000 • AJ + BJ <= 1,000,000 • AP + BP <= 2,000,000 • AC + AJ + AP <= 600,000 • BC + BJ + BP <= 2,400,000 • AC - 3BC >= 0 • 3AJ - BJ >= 0 • AC,BC,AJ,BJ,AP,BP >= 0 + EA • EA <= 80,000

  24. SUMMARY • Mr. Coopers (accounting) $ 33,777.60 profit • Mr. Myers (marketing) $ 34,000 • We … (management science) $45,355 and if we buy the extra tomatoes: $45,758

  25. HURREH FOR MANAGEMENT SCIENCE

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