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Chapter 14 Making the most of things: Linear Programming

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Procedure1.    Define carefully the DECISION

VARIABLES.2.    Define the OBJECTIVE FUNCTION3.    Write down in mathematical notation the

CONSTRAINTS of the problem.4.    Solve the problem. Where there are only 2 variables it is possible to

solve the model using a graphical technique.

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ExampleA small company assembles two types of cooker, the Ambassador and the Baron. Each Ambassador which is produced earns £40 contribution, while each Baron cooker earns £50. The company's management need to decide the number of cookers of each type which should be assembled next week in order to maximise contribution.An Ambassador cooker takes 4 hours to assemble and a Baron takes 6 hours. Both cookers need to be assembled by a skilled worker, but currently the company only employ two of these workers, both of whom work a 48 hour week.The Ambassador cooker has two built‑in clocks, while the Baron has only one clock. Because of a strike at the suppliers, only 20 of these clocks can be obtained per week.For marketing reasons, the company has decided that at least 6 Baron cookers should be produced per week.How many cookers of each type should be produced?

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The objective function

Let A = number of Ambassador cookers to make per weekB = number of Baron cookers to make per week

This is a maximization problemThe objective function is:-Max Profit = 40A + 50B

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The constraints

Subject to

4A + 6B 96 Time constraint (2 times 48)

2A + B 20 Clocks constraint

B 6 Marketing constraint

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

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The marketing constraint

The line B = 6 is a horizontal line and the region B 6 is above the line

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

Marketing constraint: B 6

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The clock constraintThe straight line 2A + B = 20 can be plotted by noting that when A = 0, B = 20 And when B = 0, A = 10These two points can now be plotted and a line drawn between them.To find the region that satisfies the constraint test a point on one side of the line (the origin is a good point).Since 0,0 satisfies the constraint the region is below the line

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

Marketing constraint: B 6

Clocks constraint: 2A + B 20

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The time constraint

Again the points where the line crosses the axes can be found, i.e.

A = 0, B = 16

B = 0, A = 24

The region is again below the line

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

B 6

2A + B 20

Time constraint 4A + 6B 48

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The feasible region

• The feasible region is that region that satisfies all constraints.

• It is shown shaded in the last graph• The optimal solution cannot be outside

this region.• How do we find the optimal combination

of the number of Ambassador and Baron cookers to make?

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What combination of A and B will give a profit of say £500

i.e. 40A + 50B = 500

This is another straight line so

A = 0, B = 10

B = 0, A = 12.5

Plot these two points on the graph.

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

B 6

2A + B 20

4A + 6B 48

40A + 50B = 500

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• The line lies partly inside and partly outside the feasible region.

• All points on the line that are within the feasible region will give a profit of £500.

• For example A = 5 and B = 6 is a feasible combination and gives a profit of £500

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Can we get better than £500?

Try £750

40A + 50B = 750

A = 0, B = 15

B = 0, A = 18.75

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

40A + 50B = 750

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Try £1000

40A + 50B = 1000

A = 0, B= 20

B= 0, A =25

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B 40A + 50B = 1000

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The line is completely outside the feasible region so none of the points on the line would be acceptable.

Notice that all the profit lines we have drawn are parallel. They are called isoprofit lines.

If you move the line representing £750 outwards it leaves the feasible region at a corner point. This is the optimal solution.

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

(3, 14)

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The optimal solution

The optimal solution is when

A = 3 and B =14

This gives a profit of

40 3 + 50 14 = £820

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LP Theorems

1. The optimal solution of a linear programming problem will always lie at a corner point of the feasible region

2. The optimal solution can involve fractional values of the decision variables

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Finding the optimal solution

• Either draw in the isoprofit line and move it to the furthest point of the feasible region

• Or evaluate each corner point and chose the largest value

In the next graph the 4 corner points have been labelled p, q, r and s

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0 5 10 15 25200

5

10

15

20

Number of Ambassadors A

Num

ber

of B

aron

s B

B 6

2A + B 20

Time constraint 4A + 6B 48

p

q

rs

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Using corner points

A B Profit p 0 16 £800 q 3 14 £820 r 7 6 £580 s 0 6 £300

Again you can see that point q is the optimum

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Minimisation problems

• In some situations you may want to minimise costs.

• The procedure is exactly the same except the you chose the corner point that is closest to the origin when you plot your isocost line

• Or you choose the corner point that gives you the minimum cost (2nd method)