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©NCC Education LimitedV1.0
Advanced Mathematics for Business
Topic 11:
Linear Programming with Solver Routines
©NCC Education LimitedV1.0
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.
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.3
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.4
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.5
Graph the Constraints
0 50 100 150 200 250 300 350 400
250
200
150
100
50
0 A
BThe Feasible Area
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.6
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.7
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.8
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.9
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.22
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.24
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
©NCC Education LimitedV1.0
LP with Solver Routines Topic 11 - 1.25
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.
LP with Solver Routines Topic 11 - 1.26
©NCC Education LimitedV1.0
Topic 11 – Linear Programming with Solver Routines
Any Questions?