Management Science III

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Management Science III

• RED BRAND CANNERS CASE
• MEDIA SELECTION PROBLEM
• vehicle to introduce integer linear programming
• CASE GLOBAL SOURCING OF MALT

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RED BRAND CANNERS CASE

• DECIDING ON THE PRODUCTION QUANTITIES OF
• WHOLE CANNED TOMATOES
• TOMATO JUICE
• TOMATO PASTE

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• 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)

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QUESTION

• WHAT PRODUCT MIX TO MAKE
• IN ORDER TO MAXIMIZE PROFIT?

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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 gt 8 , QA QB WHICH
IMPLIES THAT QA gt 3QB
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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
• COOPERs NET PROFIT
• 17,777.60 28,000.00 45,777.60
• (44,444 x 0.40 80.000 x 0.35)

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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
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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

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Contd

• 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

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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)
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WHAT ARE BOTH COOPER AND MYERS FUNDAMENTALLY
DOING WRONG?
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• 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 ..

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LINEAR PROGRAMMING!

• BY THE WAY .. ARE ALL NECESSARY
• CONDITIONS FOR THE APPLICATIONS
• OF LP SATISFIED?

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LINEAR PROGRAMMING

• DECISION VARIABLES
• OBJECTIVE FUNCTION
• CONSTRAINTS

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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

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CONSTRAINTS
AC BC lt 14,400,000 AJ BJ lt 1,000,000 AP
BP lt 2,000,000 AC AJ AP lt 600,000 BC
BJ BP lt 2,400,000 AC - 3BC gt 0 3AJ - BJ gt
0 AC,BC, AJ, BJ, AP, BP gt 0

• DEMAND FOR CWT
• DEMAND FOR JUICE
• DEMAND FOR PASTE
• AVAILABILITY OF A
• AVAILABILITY OF B
• QUALITY CWT
• QUALITY TJ
• NON-NEGATIVITY

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OBJECTIVE FUNCTION

• MAXIMIZE
• 8.22(ACBC) 6.6(AJBJ) 7.4(APBP)

NOTE profit coefficients in cents per
pound! (For instance 8.22 1.48 / 18 pounds
per case)
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MAXIMIZE 8.22 (ACBC) 6.6(AJBJ) 7.4(APBP)

• AC BC lt 14,400,000
• AJ BJ lt 1,000,000
• AP BP lt 2,000,000
• AC AJ AP lt 600,000
• BC BJ BP lt 2,400,000
• AC - 3BC gt 0
• 3AJ - BJ gt 0
• AC,BC,AJ,BJ,AP,BP gt 0

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PROFIT 45,355,55!!!!!
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IS IT PROFITABLE TO BUY ADDITIONAL GRADE A,
OFFERED AT 8.5 CENTS PER POUND FOR ANY AMOUNT
UP TO 80,000 POUNDS?
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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

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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 .

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MAX 8.22 (ACBC) 6.6(AJBJ) 7.4(APBP)
- 8.5 EA

• AC BC lt 14,400,000
• AJ BJ lt 1,000,000
• AP BP lt 2,000,000
• AC AJ AP lt 600,000
• BC BJ BP lt 2,400,000
• AC - 3BC gt 0
• 3AJ - BJ gt 0
• AC,BC,AJ,BJ,AP,BP gt 0

EA

• EA lt 80,000

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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

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HURREH FOR MANAGEMENT SCIENCE