Theory of Operations research. Decision Theory Dynamic Programming Replacement and Reliability models Operations research for MSc. Mathematics.

1. DECISION THEORY

2. Decision Theory
Decision making is an integral part of management
planning, organizing, controlling and motivation
processes. The decision maker selects one stratergy over
others depending on some criteria like : utility, sales, cost
or rate of return.
Decision theory provides a framework for better
understanding of the decision situation and for evaluating
the alternatives when the criteria are not defined.

3. Types of Decisions
In general, decisions can be classified into three categories:
1. Strategic decision : These decisions are concerned with external environment of organisation.
2. Administrative decision : This is concerned with structuring and acquisition of the
organization’s resources so as to optimize the performance of the organization.
3. Operating decision : This is primarily concerned with day to day operations.
Decision making under certainty :
This is an easiest form of decision making. The outcome resulting from the selection of
particular course of action given in certainty. There is just one state of nature for each course
of action and has probability 1. We are given complete and accurate knowledge of the
consequences of each choice. Since the decision maker has perfect knowledge of future and
outcome , he simply selects that course of action for which the pay off is optimum.
Decision making under risk ;
Under the condition of risk, the various states of nature can be enumerated and the long run
relative frequency of their occurrence are assumed to be given.

4. Definitions
 Expected Monetary Value (EMV) : The EMV for the specified course of action
is the weighted average payoff , i.e., the sum of product of the payoff for the
several combination of courses of action and states of nature multiplied by the
probability of occurrence of each outcome.
 Expected Profit with Perfect Information (EPPI) : The EPPI is the maximum
attainable EMV based on perfect information about the state of nature that will
occur.
 Expected Value of Perfect Information (EVPI) : The EVPI may now be defined
as the maximum amount one would be willing to pay to obtain perfect
information about the state of nature that would occur. EVPI = EPPI – EMV* .
 Condition Opportunity Loss (COL) : The opportunity loss for each course of
action.
 Expected Opportunity Loss (EOL) : The expected difference between the pay
off of right decision and the payoff of actual decision.

5. DYNAMIC PROGRAMMING

6. Dynamic Programming
 Dynamic Programming is a mathematical technique dealing with the optimization of
multistage decision problem.
 DP is a technique for solving problem in which decision have to be made at each successive
stages.
 It provides a systematic procedure for determining the combination of decisions that
maximize over all effectiveness.
 In DP, the problem is divide into number of subproblems are then combined to get over all
solutions.
 The DPP is broken into sub problems and each sub problem is called stage.
 The variables that links up two stage is called state variable.
 At any stage, the status of the problem can be described by the values the state variable can
take. These values are referred to as status.
 A multistage decision system, in which each decision and state variable can be taken only
finite no. of values, can be represented graphically by a decision tree.
 Bellman’s Principle of optimality :
“An optimal policy (set of decisions) has the property that whatever the initial stateand
decisions are , the remaining decisions must constitute an optimal policy with regard to the
state resulting from the first decision.”

7. REPLACEMENT AND RELIABILITY
MODELS

8. Replacement Problems
 The replacement problems are concerned with the situations the arises when some items such
as men, machines, electric bulbs, etc., need replacement due to their decreased efficiency or
failure or break down.
 The replacement problem arises because of the following factors :
1) The old item has become in worse condition and work badly or require expensive
maintenance.
2) The old item has failed has failed due to accident or otherwise and does not work at all, or the
old item is expected to fail shortly.
3) A better or more efficient design of machine or equipment has become available in the
market.
 The replacement situations may be placed into four categories :
1) Replacement of capital equipment that becomes worse with time.
2) Group replacement of items that fail completely.
3) Problems of mortality and staffing.
4) Miscellaneous problems.

9. Failure Mechanism of items
 Gradual failure :
The mechanism under this category is progressive . That is, as the life of
an item increases, its efficiency deteriorates causing : (i) increased
expenditure for operating costs, (ii) decreased productivity of equipments,
(iii) decrease in the value of equipment, i.e., the resale of saving value
decreases.
 Sudden failure :
This type of failure is applicable to those items that do not deteriorate
markedly with service but which ultimately fail after some period of using.
The period between installation and failure is not constant for any
particular type of equipment but will follow some frequency distribution
which may be :
a) Progressive
b) Retrogressive
c) Random

10. Replacement of items that deteriorate
Replacement policy of items whose maintenance cost increases with time and money value is
constant :
The cost of maintenance of a machine is given as a function increasing with time and scrap
value is constant.
a) If time is measured continuously and its average annual cost will be minimized by replacing
the machine when the average cost to date becomes equal to the current maintenance cost.
b) If time is measured in discrete units then the average annual cost will be minimized by
replacing the machine when the next period’s maintenance cost becomes greater than the
current average cost.
Replacement policy of items whose maintenance cost increases with time and money value
changes with constant rate :
The maintenance cost increases with time and the money value decreases with constant rate,
i.e., depreciation value is given. Then replacement policy will be :
a) Replace if the next period’s cost is greater than the weighted average of previous costs.
b) Do not replace if the bext period’s cost is less than the weighted average of previous costs.

11. Replacement of Items that fail completly
Individual replacement policy :
A large population is subject to a given mortality aw for a very long time.
All deaths are immediately replaced by births and there are no other entries
or exists. The age distribution ultimately becomes stable and that the
number of deaths per unit time becomes constant.
Group replacement policy :
a) One should group replace at the end of tth period if the cost of individual
replacements for the tth period is greater than the average cost per period
through the end of tth period.
b) One should not replace at the end of tth period if the cost of individual
replacement at the end of (t - 1)th period is less than the average cost per
period through the end of tth period.