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Operations Management. Module B – Linear Programming. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e . © 2006 Prentice Hall, Inc. Lecture Outline. More Examples Practice formulating Practice solving

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Operations management 1337784

Operations Management

Module B – Linear Programming

PowerPoint presentation to accompany

Heizer/Render

Principles of Operations Management, 6e

Operations Management, 8e

© 2006 Prentice Hall, Inc.


Lecture outline
Lecture Outline

  • More Examples

    • Practice formulating

    • Practice solving

    • Practice interpreting the results


Mixed nuts
Mixed Nuts

  • Crazy Joe makes two blends of mixed nuts: party mix and regular mix.

  • Crazy Joe has 10 lbs of cashews and 24 lbs of peanuts

  • Crazy Joe wants to maximize revenue. Please help him.


Ah nuts formulation
Ah, Nuts Formulation

  • Let

    • p = lbs of party mix to make

    • r = lbs of regular mix to make

    • z = total revenue

  • Max z = 6p + 4r

  • Subject to

    • 0.6p + 0.9r< 24 (peanut constraint)

    • 0.4p + 0.1r< 10 (cashew constraint)

    • p, r> 0 (non-negativity constraints)


Solving minimization problems
Solving Minimization Problems

  • Formulated and solved in much the same way as maximization problems

  • In the graphical approach an iso-cost line is used

  • The objective is to move the iso-cost line inwards until it reaches the lowest cost corner point


Minimization example
Minimization Example

Let

X1 = number of tons of black-and-white chemical produced

X2 = number of tons of color picture chemical produced

Minimize total cost = 2,500X1 + 3,000X2

Subject to:

X1 ≥ 30 tons of black-and-white chemical

X2 ≥ 20 tons of color chemical

X1 + X2≥ 60 tons total

X1, X2≥ 0 nonnegativity requirements


Minimization example1

X2

60 –

50 –

40 –

30 –

20 –

10 –

X1 + X2= 60

X1= 30

X2= 20

| | | | | | |

0 10 20 30 40 50 60

X1

Minimization Example

Table B.9

Feasible region

b

a


Minimization example2
Minimization Example

Total cost at a = 2,500X1 + 3,000X2

= 2,500 (40) + 3,000(20)

= $160,000

Total cost at b = 2,500X1 + 3,000X2

= 2,500 (30) + 3,000(30)

= $165,000

Lowest total cost is at point a


Lp applications

Feed

Product Stock X Stock Y Stock Z

A 3 oz 2 oz 4 oz

B 2 oz 3 oz 1 oz

C 1 oz 0 oz 2 oz

D 6 oz 8 oz 4 oz

LP Applications

Diet Problem Example


Lp applications1
LP Applications

X1 = number of pounds of stock X purchased per cow each month

X2 = number of pounds of stock Y purchased per cow each month

X3 = number of pounds of stock Z purchased per cow each month

Minimize cost = .02X1 + .04X2 + .025X3

Ingredient A requirement:3X1 + 2X2 + 4X3 ≥ 64

Ingredient B requirement:2X1 + 3X2 + 1X3 ≥ 80

Ingredient C requirement: 1X1 + 0X2 + 2X3 ≥ 16

Ingredient D requirement: 6X1 + 8X2 + 4X3 ≥ 128

Stock Z limitation:X3 ≤ 80

X1, X2, X3 ≥ 0

Cheapest solution is to purchase 40 pounds of grain X

at a cost of $0.80 per cow


Multiple optimal solutions
Multiple Optimal Solutions

  • Often, real world problems can have more than one optimal solution

  • When would this happen?

  • What does the graph have to look like?

  • Do want to have “ties”?


No solutions
No Solutions

  • Can we ever have a problem without a feasible solution?

  • When would this happen?

  • What would the graph look like?

  • Does this mean we did something wrong?


The simplex method
The Simplex Method

  • Real world problems are too complex to be solved using the graphical method

  • The simplex method is an algorithm for solving more complex problems

  • Developed by George Dantzig in the late 1940s

  • Most computer-based LP packages use the simplex method