1. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Module 2 Dynamic Programming

2. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-2 © 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.

3. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-3 © 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

4. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-4 © 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.

5. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dynamic Programming George Yates Dixieville 1 4 3 2 5 6 7 4 miles 10 miles 14 miles 2 miles 10 miles 6 miles 4 miles 2 miles 12 miles 5 miles Rice LakecityAthens HopeGeorgetown Brown

6. 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 Stages Dixieville 1 4 3 2 5 6 7 4 miles 10 miles 14 miles 2 miles 10 miles 6 miles 4 miles 2 miles 12 miles 5 miles Rice LakecityAthens HopeGeorgetown Brown Stage 1 Stage 2Stage 3

7. 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 1 1 4 3 2 5 6 7 4 miles 10 miles 14 miles 2 miles 10 miles 6 miles 4 miles 2 miles 12 miles 5 miles 14 2

8. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 George Yates Stage 2 1 4 3 2 5 6 7 4 miles 10 miles 14 miles 2 miles 10 miles 6 miles 4 miles 2 miles 12 miles 5 miles 2 14 24 12 8

9. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 1 4 3 2 5 6 7 4 miles 10 miles 14 miles 2 miles 10 miles 6 miles 4 miles 2 miles 12 miles 5 miles George Yates Stage 3 2 14 12 24 8 13

10. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-10 © 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

11. 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

12. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Roller Transport Problem Solution Final Solution StageOptimal Decision Optimal Return 4618 30 0 20 0 1210 Total828

13. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna M2-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dynamic Programming Key Equations s n  Input to stage n d n  Decision at stage n r n  Return at stage n s n-1  Input to stage n-1 t n  Transformation function at stage n s n-1 = t n [s n d n ]  General relationship between stages f n  Total return at stage n