dynamic programing

1. OPERATIONS RESEARCH
Dynamic programming
Topic: Product Allocation Problem
UNIT 5

2. • Dynamic programming is a technique for getting solutions for multistage
decision problems.
• In dynamic programing, the original problem is subdivided into sub problems
and the solution of these sub problems are integrated to attain the solution of the
main problem.
INTRODUCTION:
Applications dynamic programming
• Inventory problems
• Waiting line problems
• Resource allocation problems

3. Characteristics of dynamic programming
Stage: A stage signifies a portion of the total problem for which a decision can be
taken. At each stage there are a number of alternatives, and the best out of those is
called stage decision, which may be optimal for that stage, but contributes to obtain
the optimal decision policy.
State: The condition of the decision process at a stage is called its state. The
variables, which specify the condition of the decision process, i.e. describes the
status of the system at a particular stage are called state variables. The number of
state variables should be as small as possible, since larger the number of the state
variables, more complicated is the decision process.
Policy: A rule, which determines the decision at each stage, is known as Policy.
Bellman’s Principle of optimality states that ‘‘An optimal policy (a sequence of
decisions) has the property that whatever the initial state and decision are, the
remaining decisions must constitute an optimal policy with regard to the state
resulting from the first decision.’’

4. Problem 1. (capital budgeting problem)
A company has 8 salesmen, who have to be allocated to four marketing zones. The return of
profit from each zone depends upon the number of salesmen working that zone. The expected
returns for different number of salesmen in different zones, as estimated from the past records,
are given below. Determine the optimal allocation policy.

5. Solution:
The problem here is how many salesmen are to be allocated to each zone to maximize the total
return. In this problem each zone can be considered as a stage, number of salesmen in each
stage as decision variables. Number of salesmen available for allocation at a stage is the
state variable of the problem.

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9. Assignment: Students may try different combinations
Comb. 1. First combining zone 1 and zone 3 and then zones 2 and 4 and then combining both.
Comb. 2. Add 1 and 2 zones, then add zone 3 and then zone 4 to it.
Prove that above combinations the optimal outcome will be same.
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10. Problem 2.
The owner of a chain of four grocery stores has purchased six crates of fresh strawberries. The
estimated probability distribution of potential sales of the strawberries before spoilage differs
among the four stores. The following table gives the estimated total expected profit at each store,
when it is allocated various numbers of crates:
For administrative reasons, the owner does not wish to split crates between stores. However he is
willing to distribute zero crates to any of his stores.

14. Shortest path problem
Problem: Mr. Banerjee, a sales manager, has decided to travel from city 1 to city 10. He wants
to plan for minimum distance programme and visit maximum number of branch offices as
possible on the route. The route map of the various ways of reaching city 10 from city 1 is
shown below. The numbers on the arrow indicates the distance in km. (× 100). Suggest a
feasible minimum path plan to Mr. Banerjee.

15. Solution
Hence the minimum distance from stations 1 to 10 on the path is 2000 km on routes 1 – 3 –
5 – 8 – 10 and 1 – 2 – 6 – 8 – 10.

16. Problem: The following figure shows the route map of various branch offices of a company. The
marketing executive of the company should like to start from Head office at A and reach the
branch office at B by traveling shortest path and visiting as many as branch offices. Help him to
plan his journey by using dynamic programming technique.

17. The optimal route is 1– 2 – 6 – 8 – 11 and the optimal distance is 21 km.