Lecture 11 - Linear Programming With Solver Routines

©NCC Education LimitedV1.0

Advanced Mathematics for Business

Topic 11:

Linear Programming with Solver Routines


LP with Solver Routines Topic 11 - 1.2

Scope and Coverage

This topic will cover:• Multiple variable linear programmes• Solving linear programmes with Excel• Interpreting the solutions from linear programmes• Extensions to linear programmes.



Learning Outcomes

By the end of this topic students will be able to:• Formulate an LP for solution with a Solver• Interpret LP output

- Binding & Non-binding constraints

- Allowed increases and decreases

- Shadow price

• Recognise extensions to LP e.g. integer programming



Summary of Management Problem• 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



Graph the Constraints

0 50 100 150 200 250 300 350 400

250

200

150

100

50

0 A

BThe Feasible Area



Corner Solution Method• 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



More than 2 Variables?

n-variables?

Supposing a solution will lie on a corner.

If two corners take the same value then all solutions on the line between those two corners will also be optimal.

2-variables

3-variables



Solving LPs with Solver Routines• We will use Excel

- Others include OpenOffice Calc, LINDO/LINGO etc.

• Formulate clearly- Objective and constraints as rows

- Variable as columns

- Group like constraints

• Input to Excel (or other), solve and interpret



Solving LPs with Excel - 1

Profit: 50 A + 25 B

Process 1: 0.4 A + 0.7 B ≤ 140

Process 2: 0.2 A + 0.4 B ≤ 70

Demand A: A ≤ 90

Demand B: B ≤ 150

Non-negativity: A ≥ 0

Non-negativity: B ≥ 0



Solving LPs with Excel



The Answer Report



Sensitivity Report



Integer Constraints - 1

0 1 2 3

5

4

3

2

1

0

• For some problems at least one management variable is constrained to be an INTEGER

• Algorithms have also been designed to solve such problems

• Simply add in and solve



Integer Constraints - 2



Recap

By the end of this topic students will be able to:• Formulate an LP for solution with a Solver• Interpret LP output

- Binding & Non-binding constraints

- Allowed increases and decreases

- Shadow price

• Recognise extensions to LP e.g. integer programming



Bibliography• 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.



Topic 11 – Linear Programming with Solver Routines

Any Questions?

Lecture 11 - Linear Programming With Solver Routines

Documents

Transcript of Lecture 11 - Linear Programming With Solver Routines