Linear Programming

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Course Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Why Talk About Linear Programming? ,[object Object],[object Object],[object Object],[object Object],[object Object]

EXAMPLE ,[object Object],[object Object],[object Object]

0 12 20 25 33 40 50 60 80 100 110 100 80 70 60 40 35 25 20 Linear Programming (Graphical method) Constraints Equation Coordinates 1 3x 1 + 10x 2 = 330 (33,110) 2 16x 1 + 4x 2 = 400 (25,100) 3 6x 1 + 6x 2 = 240 (40,40) 4 X 2 = 12 (0,12)

0 12 20 25 33 40 50 60 80 100 110 100 80 70 60 40 35 25 20 Drawing ISO Contribution line and Feasible Region End select two convenient value of Z substitute into The objective function and plot into the resulting equation ; Here 1750 & 3500 appear to be sensible. 1750 = 50X 1 + 70X 2 x 1 = 35; x 2 = 0 3500 = 50X 1 + 70X 2 x 1 = 70; x 2 = 0

0 12 20 25 33 40 50 60 80 100 110 100 80 70 60 40 35 25 20 Feasible Region E D C B A Points X1 x2 Z A 22 12 1940 B 20 20 2400 C 10 30 2600 D 0 33 2310 E 0 12 840

[object Object],[object Object],Shadow Pricing In our example following is the equation of labour. 6x 1 + 6x 2 = 240 If one more labour hour will be available then, 6x 1 + 6x 2 = 241 know the value of Z will be Z=2607.1 The contribution has increased by 7.1 by increase 1hour of scarce resource. This is the shadow price a unit of labour

The Simplex Method ,[object Object],Woodhurst is a furniture company that specialises in high-quality products. The company can manufacture four different types of coffee table ( small, meduim, large and ornate) Each type of table requires time for the cutting of the component parts, for assembly and for finishing. The data in the table below has been collected for the year now bieng planned. Owing to other commitments, no more than a total of 1,800 coffee tables can be made in any given year. Also, market analysis reveals that the annual demand for the company`s small coffee table is at least 800. The Company wishes to determine how many of each type of coffee table it should produce in the coming year to maximise contribution hours required per table Contribution on each table Table Cutting Assembly Finishing small 2 5 1 60 Medium 2 4 4 123 Large 1 3 5 135 Ornate 6 2 3 90 Capacity in hours 3,000 9,000 4,950

Formulating into the equations ,[object Object],[object Object],Subject to : 2x 1 + 2x 2 + 1x 3 + 6x 4 + ≤ 3,000 5x 1 + 4x 2 + 3x 3 + 2x 4 + ≤ 9,000 1x 1 + 4x 2 + 5x 3 + 3x 4 + ≤ 4,950 x 1 + x 2 + x 3 + x 4 + ≤ 1,800 x 1 ≥ 800 x 1 ,x 2 ,x 3 ≥ 0 Objective funtion variable(z) 168,750 Variable Value relative loss X1 950 0 X2 250 0 X3 600 0 X4 0 48 Constraint slack/surplus worth 1 0 9 2 1450 0 3 0 21 4 0 21 5 150 0

Linear Programming

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Linear Programming