Operation research complete note

Operations Research Techniques

Introduction• Operations Research is an Art and Science

• It had its early roots in World War II and is flourishing in business and industry with the aid of computer

• Primary applications areas of Operations Research include forecasting, production scheduling, inventory control, capital budgeting, and transportation.

Operations Research

• “Operational Research is the scientific study of

operations for the purpose of making better

decisions.”

Terminology• The British/Europeans refer to “Operational Research",

the Americans to “Operations Research" - but both are often shortened to just "OR".

• Another term used for this field is “Management Science" ("MS"). In U.S. OR and MS are combined together to form "OR/MS" or "ORMS".

• Yet other terms sometimes used are “Industrial Engineering" ("IE") and “Decision Science" ("DS").

History of OR Operational Research has been existed as a science since

1930‘s.

But as a formal discipline Operational Research originated by the efforts of military planner during World War II .

In the decade after World War-II the techniques began to be applied more widely in problems of business, industries and societies.

Objectives

Decision making and improve its quality.• Identify optimum solution.• Integrating the systems.• Improve the objectivity of analysis.• Minimize the cost and maximize the profit.• Improve the productivity.• Success in competition and market leadership.

Applications of OR

1. Finance and Accounting:

a. Dividend policies, investment and portfolio

management, auditing, balance sheet

b. Claim and Complaint procedure and public accounting

c. Break-even analysis, capital budgeting, cost allocation

and control, and financial planning

2. Marketing

• Selection of product mix• Marketing and export planning • Sales force allocations • Assignment allocations • Media planning • Advertising

3. Purchasing & Procurement

• Optimal buying • Transportation planning • Replacement policies • Bidding Policies • Vendor analysis

4.Production management

• Facility location • Logistics arrangement • Layout design • Engineering design • Aggregate production planning • Transportation • Planning and scheduling

5. Personnel Management

• Manpower planning • Wage/ salary administration • Negotiation in bargaining situation• Skills and Wages Balancing • Schedule of training programmes • Skill development and retention

6. General Management

• Decision Support system • Forecasting • Making quality control more effective • Project Management and strategic planning

methods

Most projects of Operational Research apply one of three broad groups of methods :-

1.Simulation methods.

2.Optimization methods.

3.Data-analysis methods.

1.Simulation methods

It gives ability to conduct sensitive study to - (a). search for improvements and (b). test the improvement ideas that are being made.2.Optimization methods. Here goal is to enable the decision makers to identify and locate the

very best choice, where innumerable feasible choices are available and comparing them is difficult.

3.Data-analysis methods The goal is to aid the decision-maker in detecting actual patterns and

inter-connections in the data set and Use of this analysis for making solutions.

This method is very useful in Public Health.

Deterministic vs. Stochastic Models

Deterministic models (Non-Probabilistic) assume all data are known with certainty

Stochastic models (Probabilistic Models)

explicitly represent uncertain data via random variables or stochastic processes.

Deterministic models involve optimization

Stochastic models characterize / estimate system performance.

Operations Research Models

Deterministic Models Stochastic Models

• Linear Programming • Discrete-Time Markov Chains

• Network Optimization • Continuous-Time Markov Chains

• Integer Programming • Queuing Theory (waiting lines)

• Nonlinear Programming • Decision Analysis

Process1. Identification of program problem. Most critical step in the process. Unless problem is clearly defined it is impossible

to develop good solutions.

2: Identification of possible reasons and solutions . Once the problem has been identified , it is the

job of the program implementer and researcher to determine the reasons for the problem and generate possible solutions.

3.Testing of potential solution A good solution must be measurable, easy to implement and sustainable. To determine effectiveness of proposed solution two designs are used- (a) quasi-experimental design.

- comparison of situations before and after the solution. (b) true experiment. - comparison of outcome between experimental and control groups.

4.Result utilization It is necessary to decide how its results are meant to be used. This determine to some extent that what information should be collected.

5.Results dissemination

Results dissemination are done in the form of seminars or by meeting with decision makers.

Examples of OR Applications

• Rescheduling aircraft in response to groundings and delays

• Planning production for printed circuit board assembly

• Scheduling equipment operators in mail processing & distribution centers

• Developing routes for propane delivery

• Adjusting nurse schedules in light of daily fluctuations in demand

Introduction to Decision Analysis

Decision Making Overview

Decision Making

Certainty Nonprobabilistic

Uncertainty Probabilistic

Decision Environment Decision Criteria

The Decision Environment

Certainty

Uncertainty

Decision Environment

Certainty: The results of decision alternatives are known

Decision Making Under Certainty When it is known for certain which is of the possible future conditions will happen, just choose the alternative that has the best payoff under the state of nature

Example:

Must print 10,000 color brochures

Offset press A: 2,000 fixed cost + 0.24 per page

Offset press B: 3,000 fixed cost + 0.12 per page

*

The Decision Environment

Uncertainty

Certainty

Decision EnvironmentUncertainty: The outcome that will occur after a choice is unknown

Example:

You must decide to buy an item now or wait. If you buy now the price is 2,000. If you wait the price may drop to 1,500 or rise to 2,200. There also may be a new model available later with better features.

*

Decision Criteria

Nonprobabilistic

Probabilistic

Decision CriteriaNonprobabilistic Decision Criteria: Decision rules that can be applied if the probabilities of uncertain events are not known.

* maximax criterion

maximin criterion

minimax regret criterion

Decision Criteria

Nonprobabilistic

Probabilistic

Decision Criteria

*

Probabilistic Decision Criteria: Consider the probabilities of uncertain events and select an alternative to maximize the expected payoff of minimize the expected loss

maximize expected value

minimize expected opportunity loss

PAYOFF

• The payoffs of a decision analysis problem are the benefits or rewards that result from selecting a particular decision alternative

A Payoff TableA payoff table shows alternatives, states of nature,

and payoffs

Investment Choice

(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

5012030

-120-30 20

Maximax Solution

Investment Choice

(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

1.

Maximum Profit

200120 40

The maximax criterion (an optimistic approach):

1. For each option, find the maximum payoff

Maximax Solution

Investment Choice

(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

1.

Maximum Profit

200120 40

The maximax criterion (an optimistic approach): 1.For each option, find the maximum payoff

2.Choose the option with the greatest maximum payoff

2.

Greatest maximum

is to choose Large

factory

(continued)

Maximin Solution

Investment Choice

(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

1.

Minimum Profit

-120 -30 20

The maximin criterion (a pessimistic approach):

1. For each option, find the minimum payoff

Maximin Solution

Investment Choice

(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

1.

Minimum Profit

-120 -30 20

The maximin criterion (a pessimistic approach):

1. For each option, find the minimum payoff

2. Choose the option with the greatest minimum payoff

2.

Greatest minimum

is to choose Small

factory

Opportunity Loss

Investment Choice(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

The choice “Average factory” has payoff 90 for “Strong Economy”. Given “Strong Economy”, the choice of “Large factory” would have given a payoff of 200, or 110 higher. Opportunity loss = 110 for this cell.

Opportunity loss is the difference between an actual payoff for a decision and the optimal payoff for that state of nature

Payoff Table

Opportunity Loss

Investment Choice(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

Investment Choice(Alternatives)

Opportunity Loss in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

0110160

70 0

90

140500

Payoff Table

Opportunity Loss Table

Minimax Regret Solution

Investment Choice(Alternatives)

Opportunity Loss in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

0110160

70 0

90

140500

Opportunity Loss Table

The minimax regret criterion:1. For each alternative, find the maximum opportunity loss (or “regret”)

1.

Maximum Op. Loss

140110160

Minimax Regret Solution

Investment Choice(Alternatives)

Opportunity Loss in $1,000’s (States of Nature)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

0110160

70 0

90

140500

Opportunity Loss Table

The minimax regret criterion:1. For each alternative, find the maximum opportunity loss (or “regret”)

2. Choose the option with the smallest maximum loss

1.

Maximum Op. Loss

140110160

2.

Smallest maximum loss is to choose

Average factory

Expected Value Solution• The expected value is the weighted average payoff,

given specified probabilities for each state of nature

Investment Choice(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

Suppose these probabilities have been assessed for these states of nature

Expected Value Solution

Investment Choice(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

Expected Values

618131

Maximize expected value by choosing Average factory

Example: EV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81Example: EV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81

Expected Opportunity Loss Solution

Investment Choice(Alternatives)

Opportunity Loss in $1,000’s (States of Nature)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

Large factoryAverage factorySmall factory

0110160

70 0

90

140500

Expected Op. Loss

(EOL)634393

Minimize expected

op. loss by choosing Average factory

Opportunity Loss Table

Example: EOL (Large factory) = 0(.3) + 70(.5) + (140)(.2) = 63

Cost of Uncertainty• Cost of Uncertainty (also called Expected Value of Perfect

Information, or EVPI)

• Cost of Uncertainty = Expected Value Under Certainty (EVUC)

– Expected Value without information (EV)

so: EVPI = EVUC – EV

Expected Value Under CertaintyExpected Value Under Certainty (EVUC):EVUC = expected value of the best decision, given perfect information

Investment Choice(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

Example: Best decision given “Strong Economy” is “Large factory”

200 120 20

Expected Value Under Certainty– Now weight these outcomes with their probabilities to find

EVUC:

Investment Choice(Alternatives)

Profit in $1,000’s (States of Nature)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20

200 120 20

EVUC = 200(.3)+120(.5)+20(.2) = 124

Cost of Uncertainty Solution• Cost of Uncertainty (EVPI)

= Expected Value Under Certainty (EVUC)

– Expected Value without information (EV)

so: EVPI = EVUC – EV = 124 – 81 = 43

Recall: EVUC = 124

EV is maximized by choosing “Average factory”, where EV = 81

Decision Tree Analysis• A Decision tree shows a decision problem, beginning with

the initial decision and ending will all possible outcomes and payoffs.

Use a square to denote decision nodes

Use a circle to denote uncertain events

Sample Decision Tree

Large factory

Small factory

Average factory

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

Add Probabilities and Payoffs

Large factory

Small factory

Decision

Average factory

Uncertain Events(States of Nature)

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

PayoffsProbabilities

200

50

-120

40

30

20

90

120

-30

(.3)

(.5)

(.2)

(.3)

(.5)

(.2)

(.3)

(.5)

(.2)

Fold Back the Tree

Large factory

Small factory

Average factory

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

200

50

-120

40

30

20

90

120

-30

(.3)

(.5)

(.2)

(.3)

(.5)

(.2)

(.3)

(.5)

(.2)

EV=200(.3)+50(.5)+(-120)(.2)=61

EV=90(.3)+120(.5)+(-30)(.2)=81

EV=40(.3)+30(.5)+20(.2)=31

Make the Decision

Large factory

Small factory

Average factory

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

Strong Economy

Stable Economy

Weak Economy

200

50

-120

40

30

20

90

120

-30

(.3)

(.5)

(.2)

(.3)

(.5)

(.2)

(.3)

(.5)

(.2)

EV=61

EV=81

EV=31

Maximum

EV=81

•Unit2

DEFINITION OF LPP

• Mathematical programming is used to find the best or optimal solution to a problem that requires a decision or set of decisions about how best to use a set of limited resources to achieve a state goal of objectives.

• It is a mathematical model or technique for efficient and effective utilization of limited recourses to achieve organization objectives (Maximize profits or Minimize cost).

Assumption of LP

• Proportionality - The rate of change (slope) of the objective function and constraint equations is constant.

• Additivity - Terms in the objective function and constraint equations must be additive.

• Divisibility -Decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature.

• Certainty - Values of all the model parameters are assumed to be known with certainty (non-probabilistic)

REQUIREMENTS

• There must be well defined objective function.

• There must be a constraint on the amount.• There must be alternative course of action.• The decision variables should be interrelated

and non negative.• The resource must be limited in supply.

AREAS OF APPLICATION OF LINEAR PROGRAMMING

• Business• Industrial• Military• Economic• Marketing• Distribution

ADVANTAGES and DISADVANTAGE OF L.P

ADVANTAGES OF L.P.• It helps in attaining optimum use of productive factors. It improves the quality of the decisions. It provides better tools for meeting the changing conditions. It highlights the bottleneck in the production process.DISADVANTAGE OF L.P•For large problems the computational difficulties are enormous. It may yield fractional value answers to decision variables. It is applicable to only static situation. LP deals with the problems with single objective.

. IMPORTANT DEFINITIONS IN L.P.

Solution: A set of variables [X1,X2,...,Xn+m] is called a solution to L.P. Problem if it satisfies its constraints. Feasible Solution: A set of variables [X1,X2,...,Xn+m] is called a feasible solution to L.P. Problem if it satisfies its constraints as well as non-negativity restrictionsOptimal Feasible Solution: The basic feasible solution that optimizes the objective function. Unbounded Solution: If the value of the objective function can be increased or decreased indefinitely, the solution is called an unbounded solution

Unit 3

Transportation problem

• A TRANSPORTATION PROBLEM (TP) CONSISTS OF DETERMINING HOW TO ROUTE PRODUCTS IN A SITUATION WHERE THERE ARE SEVERAL SUPPLY LOCATIONS AND ALSO SEVERAL DESTINATIONS IN ORDER THAT THE TOTAL COST OF TRANSPORTATION

IS MINIMISED

Application of Transportation Problem

• Minimize shipping costs• Determine low cost location• Find minimum cost production schedule• Military distribution system• Etc……

Two Types of Transportation Problem•

1. Balanced Transportation Problem where the total supply equals total demand

2. • Unbalanced Transportation Problem where

the total supply is not equal to the total demand

Phases of Solution of Transportation Problem Phase I- obtains the initial basic feasible solution

Phase II-obtains the optimal basic solution

1: Initial Basic Feasible Solution methodNorth West Corner Rule (NWCR)Row Minima MethodColumn Minima MethodLeast Cost MethodVogle Approximation Method (VAM)

2:Optimum Basic SolutionStepping Stone MethodModified Distribution Method a.k.a. MODI Method

NORTH WEST CORNER METHOD

• . North- West Corner Method (NWCM)The simplest of the procedures, used to generate an initial feasible solution is, NWCM. It is so called because we begin with the North West or upper left corner cell of our transportation table.

STEPS of NORTH WEST CORNER METHOD

• Step1: construct an empty m*n matrix, completed with

rows & columns.• Step2: indicate the rows and column totals at the end.• Step3: starting with (1,1)cell at the north west corner of the

matrix, allocate maximum possible quantity keeping in view that allocation can neither be more than the quantity required by the respective warehouses nor more than quantity available at the each supply center.

• Step 4: adjust the supply and demand nos. in the rows and columns allocations.

• Step5: if the supply for the first row is exhausted then move down to the first cell in the second row and first column and go to the step 4.

STEPS of NORTH WEST CORNER METHOD

• Step 6: if the demand for the first column is satisfied, then move to the next cell in the second column and first row and go to step 4.

• Step 7: if for any cell, supply equals demand then the next allocation can be made in cell either in the next row or column.

• Step 8: continue the procedure until the total available quantity is fully allocated to the cells as required.

LEAST COST ENTRY METHOD

• This method takes into consideration the lowest cost and therefore takes less time to solve the problem

• Least-Cost Method consist in allocating as much as possible in the lowest cost cell and then further allocation is done in the cell with second lowest cost cell and so on.

Steps in LEAST COST ENTRY METHOD

Step1: select the cell with the lowest transportation cost among all the rows and columns of the transportation table. If the minimum cost is not unique then select arbitrarily any cell with the lowest cost.Step2: allocate as many units as possible to the cell determined in step 1 and eliminate that row in which either capacity or requirement is exhausted.• Step3:adjust the capacity and the requirement for the next allocations.• Step4: repeat the steps1to3 for the reduced table until the entire capacities are exhausted to fill the requirements at the different destinations.

Vogel’s Approximation Method

• this method, each allocation is made on the basis of the opportunity (or penalty or extra) cost that would have been incurred if allocations in certain cells with minimum unit transportation cost were missed. In this method allocations are made so that the penalty cost is minimized.

Assignment problems

• Assignment The Name “assignment problem” originates from the classical problem where the objective is to assign a number of origins (Job) to the equal number of destinations (Persons) at a minimum cast (or maximum profit).

Application areas of Assignment Problem

• In assignment machines to factory orders.• In assigning sales/marketing people to sales

territories.• In assignment contracts to bidders by systematic

bid evaluation.• In assignment teachers to classes.• In assigning accountants to account of the

clients.• In assignment police vehicles to patrolling areas.

Game theoryGame theory

Game theory

• Game theory is the science of strategy. It attempts to determine mathematically and logically the actions that “players” should take to secure the best outcomes for themselves in a wide array of “games.

What is a dominated strategy?

One strategy (strategy A) is said to "dominate" another strategy (strategy B) if a player is always better off choosing A instead of choosing B, regardless of the strategies chosen by other players.•A strategy is called a dominant strategy if it dominates all other strategies.

Replacement TheoryReplacement Theory

Replacement Theory

• It is used in the decision making process of replacing a used equipment with a substitute; mostly a new equipment of better usage.

• The replacement might be necessary due to the deteriorating property or failure or breakdown of particular equipment.

• The ‘Replacement Theory’ is used in the cases like; existing items have out-lived, or it may not be economical anymore to continue with them, or the items might have been destroyed either by accident or otherwise

Types of failureThere are two types of failure

• Gradual failure

• Sudden failure

Gradual failure cases Gradual failure cases

• Increased running costs ( maintenance +

operating cost) •Decrease in productivity • Decrease in the resale or salvage value

Sudden failure casesSudden failure cases

• This type of failure occurs in items after some period of giving

desired service rather than deteriorating while in service

Example:• A transport company purchased a motor vehicle for

rupees 80000/-. The resale value of the vehicle keeps on decreasing from Rs 70000/- in the first year to Rs 5000/- in the eighth year while, the running cost in maintaining the vehicle keeps on increasing with Rs. 3000/- in the first year till it goes to Rs. 20000/- in the eighth year as shown in the below table. Determine the optimum replacement policy?

•

• Solution: The vehicle needs to be replaced after four years of its purchase wherein the cost of maintaining that vehicle would be lowest at an average of Rs 11850/- per year.

Notations Used:

• C – (Capital) Cost of Equipment• S – Scrap (or Resale) Value• Rn – Running (or Maintenance) Cost• E Rn – Cumulative Running Cost• (C-S) – Depreciation• TC – Total Cost• ATC – Average Total Cost