Operations Management
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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≥ 30tons of black-and-white chemical

X2≥ 20tons of color chemical

X1 + X2≥ 60tons total

X1, X2≥ 0nonnegativity requirements


Minimization example1

X2

60 –

50 –

40 –

30 –

20 –

10 –

X1 + X2= 60

X1= 30

X2= 20

|||||||

0102030405060

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

ProductStock XStock YStock Z

A3 oz2 oz 4 oz

B2 oz3 oz 1 oz

C1 oz0 oz 2 oz

D6 oz8 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


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