Module 2
Dynamic
Programming
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to
• Understand the overall approach
of dynamic programming
• Use dynamic programming to
solve the shortest-route problem.
• Develop dynamic programming
stages.
• Describe important dynamic
programming terminology.
• Describe the use of dynamic
programming in solving knapsack
problems.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Module Outline
M2.1 Introduction
M2.2 Shortest-Route Problem
Solved by Dynamic
Programming
M2.3 Dynamic Programming
Terminology
M2.4 Dynamic Programming
Notation
M2.5 Knapsack Problem
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Four Steps in Dynamic
Programming
• Divide the original problem into
subproblems called stages.
• Solve the last stage of the
problem for all possible
conditions or states.
• Working backward from that
last stage, solve each
intermediate stage.
• Obtain the optimal solution for
the original problem by solving
all stages sequentially.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dynamic Programming
George Yates
Lakecity
4
Rice
1
Athens
10 miles
5
Brown
5 miles
3
7
2
6
10 miles
Hope
Georgetown
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
George Yates
Stages
Lakecity
4
Rice
1
Athens
10 miles
5
Brown
5 miles
7
3
Dixieville
2
6
10 miles
Hope
Georgetown
Stage 3
Stage 2
Stage 1
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
George Yates
Stage 1
14
4
1
5 miles
10 miles
5
3
2
7
10 miles
6
2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
George Yates
Stage 2
14
24
4
10 miles
5
8
1
5 miles
3
2
7
10 miles
12
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
6
2
M2-8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
George Yates
Stage 3
24
4
14
10 miles
5
13
8
1 5 miles
7
3
2
10 miles
2
12
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
6
M2-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dynamic Programming
Terminology
1. Stage - A logical subproblem
2. State Variable - Possible
condition
3. Decision Variable - Alternative
4. Decision Criterion - Problem
objective
5. Optimal Policy - A set of decision
rules
6. Transformation - Relationship
between stages
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-10
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Roller Transport
Problem
Items to be Shipped
Item Weight Profit($) Number
Available
1
1
3
6
2
4
9
1
3
3
8
2
4
2
5
2
The Relationship Between Items
and Stages
Item 1
Stage 4
Item 2
Stage 3
Item 3
Stage 2
Item 4
Stage 1
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-11
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Roller Transport
Problem
Solution
Final Solution
Stage
Optimal Optimal
Decision Return
4
6
18
3
0
0
2
0
0
1
2
10
Total
8
28
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-12
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dynamic Programming
Key Equations
• sn  Input to stage n
• dn  Decision at stage n
• rn  Return at stage n
• sn-1  Input to stage n-1
• tn  Transformation function at
stage n
• sn-1 = tn [sn dn] General
relationship between stages
• fn  Total return at stage n
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
M2-13
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458