The learning outcomes of this topic are: - Recognise the concept of constrained optimisation - Formulate a two variable linear programme (maximisation and minimisation problems) - Find a graphical solution to a two variable LP - Appreciate the process of sensitivity analysis

1. Topic10:
Linear Programming

2. This topic will cover:
◦ Formulating a two variable linear programme
◦ Graphical solution of a linear programme
◦ Introduction to sensitivity analysis

3. By the end of this topic students will be able
to:
◦ Recognise the concept of constrained optimisation
◦ Formulate a two variable linear programme
 Maximisation and minimisation problems
◦ Find a graphical solution to a two variable LP
◦ Appreciate the process of sensitivity analysis

4. ◦ A company produces two specialist materials
 Available demand known for following week:
 Material A 90 square metres
 Material B 150 square metres
 Two internal processes are constrained:
 Process 1 has 140 hours available per
week
 Process 2 has 70 hours available per week
 No input material supply side constraints

5. ◦ Resource use and profit from materials
◦ Profit = 50A + 25B
◦ Process 1: ≤ 140
◦ Process 2: ≤ 70
Material A Material B
Process 1 (time per m2) 24 minutes 42 minutes
Process 2 (time per m2) 12 minutes 24 minutes
Profit per m2 £50 £25
0.4A + 0.7B
0.2A + 0.4B

6. ◦ What to produce this week to maximise profit?
◦ Objective function
 Maximise: profit = 50A + 25B
◦ Constraints
 Process 1: 0.4A + 0.7B ≤ 140
 Process 2: 0.2A + 0.4B ≤ 70
 Demand A: A ≤ 90
 Demand B: B ≤ 150
 A ≥ 0
 B ≥ 0
non-negativity constraints

7. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B
Demand A: A ≤ 90

8. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B
Demand B: B ≤ 150

9. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B
Process 1: 0.4A + 0.7B ≤ 140

10. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B
Process 2: 0.2A + 0.4B ≤ 70

11. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B
The Feasible Area

12. 0 30 60 90
150
120
90
60
30
0
A
B
The Feasible Area • Graph the objective
function
• Profit = 50A + 25B
• Profit =
£3,000
• Profit =
£5,250
• Profit =
£6,750
• Profit =
£7,750

13. ◦ Produce 90 m2 of material A and 130 m2 of B
 Profit = (90 x 50) + (130 x 25) = £7,750
◦ Available process resource constraints
 Process 1: 0.4A + 0.7B ≤ 140, non-binding &
redundant
 Process 2: 0.2A + 0.4B ≤ 70, binding
◦ Available demand constraints
 Demand A: A ≤ 90, binding
 Demand B: B ≤ 150, non-binding

14. 0 30 60 90
150
120
90
60
30
0
A
B • Solutions to Linear
Programs always lie
on a corner of the
feasible area.
• Occasionally also on
the line between two
corners.

15. 0 30 60 90
150
120
90
60
30
0
A
B • Identify variable
mix at each corner
• Evaluate objective
function
− e.g. profit at each
corner
• Assess and decide(90, 0)
(90, 130)
(50, 150)(0, 150)

16. ◦ What to produce this week?
Material A (m2) Material B (m2) Profit (50A +
25B)
0 150 £3,750
50 150 £6,250
90 130 £7,750
90 0 £4,500

17. Minimize:
◦ Objective function: Cost = 42A + 100B
Constraints
◦ A ≥ 25
◦ B ≥ 10
◦ A + B ≥ 50
◦ 3A + 7B ≥ 210

18. 0 10 20 30 40 50 60 70 80
50
40
30
20
10
0
A
B
A = 25
B = 10
A + B = 50
3A + 7B = 210
(25, 25)
(46⅔, 10)
(35, 15)

19. ◦ What to produce this week?
A B Cost (42A +
100B)
25 25 £3,550
35 15 £2,970
46⅔ 10 £2,960

20. 0 30 60 90
150
120
90
60
30
0
A
B • Assume unit profit
of one product
changes, then the
• Gradient of isoprofit
line changes, and
eventually
• Product mix
changes

21. Material A (m2) Material B (m2) Profit (50A + 25B)
0 150 £3,750
50 150 £6,250
90 130 £7,750
90 0 £4,500
Material A (m2) Material B (m2) Profit (25A + 50B)
0 150 £7,500
50 150 £8,750
90 130 £8,750
90 0 £2,250
Material A (m2) Material B (m2) Profit (20A + 60B)
0 150 £9,000
50 150 £10,000
90 130 £9,600
90 0 £1,800
PB:PA = 0.5
PB:PA = 2.0
PB:PA = 3.0

22. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B

23. Material A (m2) Material B (m2) Profit (50A +
25B)
0 150 £3,750
50 150 £6,250
90 130 £7,750
90 0 £4,500
Material A (m2) Material B (m2) Profit (50A +
25B)
0 150 £3,750
50 150 £6,250
100 125 £8,125
100 0 £5,000

24. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B

25. 0 50 100 150 200 250 300 350 400
250
200
150
100
50
0
A
B

26. By the end of this topic students will be able
to:
◦ Recognise the concept of constrained optimization
◦ Formulate a two variable linear programme
 Maximisation and minimisation problems
◦ Find a graphical solution to a two variable LP
◦ Appreciate the process of sensitivity analysis

27. ◦ Hillier, F. and Lieberman. G. Introduction to
Operations Research. McGraw Hill
◦ Keast, S. and Towler M. Rational Decision-Making
for Managers. Wiley.
◦ Wisniewski, M. Quantitative Methods for Decision
Makers. FT Prentice Hall.

28. Any Questions?