Lecture Introduction to Management Science with Spreadsheets: Chapter 12 - Stevenson, Ozgur

Chapter 12 Decision theory, after completing this chapter, you should be able to: Outline the characteristics of a decision theory approach to decision making; describe and give examples of decisions under certainty, risk, and complete uncertainty; make decisions using maximin, maximax, minimax regret, Hurwicz, equally likely, and expected value criteria and use Excel to solve problems involving these techniques;...

Stevenson and Ozgur

First Edition

Introduction to Management Science

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Learning Objectives

1 Give examples of systems that may lend

themselves to be analyzed by a Markov model

2 Explain the meaning of transition probabilities

3 Describe the kinds of system behaviors that Markov

analysis pertains to

4 Use a tree diagram to analyze system behavior

5 Use matrix multiplication to analyze system

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Learning Objectives (cont’d)

7 Analyze absorbing states, namely accounts

receivable, using a Markov model

8 List the assumptions of a Markov model

9 Use Excel to solve various problems pertaining to a

Markov model

After completing this chapter, you should be able to:

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Characteristics of a Markov System Characteristics of a Markov System

1 It will operate or exist for a number of periods

2 In each period, the system can assume one of a

number of states or conditions

3 The states are both mutually exclusive and collectively exhaustive

4 System changes between states from period to period can be described by transition probabilities, which

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Markov Analysis: Assumptions

Markov Analysis: Assumptions

• Markov Analysis Assumptions

–The probability that an item in the system either will change from one state (e.g., Airport A) to another or remain in its current state is a function of the transition probabilities only

–The transition probabilities remain constant

–The system is a closed one; there will be no arrivals to the system or exits from the system

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by the system’s transition probabilities.

• Short-term behavior is solely dependent on the

system’s state in the current period and the

transition probabilities.

• The long-run proportions are referred to as the

steady-state proportions, or probabilities, of the

system.

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Methods of System Behavior Analysis

Methods of System Behavior Analysis

• Tree Diagram

– A visual portrayal of a system’s transitions composed of a series

of branches, which represent the possible choices at each stage (period) and the conditional probabilities of each choice being selected.

• Matrix Multiplication

– Assumes that “current” state proportions are equal to the product

of the proportions in the preceding period multiplied by the matrix

of transition probabilities.

– Involves the multiplication of the “current” proportions, which is referred to as a probability vector, by the transition matrix.

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Methods of System Behavior Analysis

(cont’d)

Methods of System Behavior Analysis

(cont’d)

• Algebraic Solution

–The basis for an algebraic solution is a set of

equations developed from the transition matrix

–Because the states are mutually exclusive and

collectively exhaustive, the sum of the state

probabilities must be 1.00, and another equation can

be developedf rom this requirement

–The result is a set of equations that can be used to solve for the steady-state probabilities

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Steady-State Proportions Based on Matrix Multiplications

Steady-State Proportions Based on Matrix Multiplications

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Maintenance Problem

Maintenance Problem

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Problem: Initial State = Operation

Problem: Initial State = Operation

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Problem Initial State = Broken

Problem Initial State = Broken

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Maintenance Problem

Maintenance Problem

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(Initial State = X)

(Initial State = X)

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(Initial State =Y)

(Initial State =Y)

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Problem

Problem

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University Problem

University Problem

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Probabilities of the Acorn University Problem

Probabilities of the Acorn University Problem

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Problem

Problem

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Cyclical, Transient, and Absorbing Systems

Cyclical, Transient, and Absorbing Systems

• Cyclical system

–A system that has a tendency to move from state to state in a definite pattern or cycle

• Transient system

–A system in which there is at least one state—the

transient state—where once a system leaves it, the

system will never return to it

• Absorbing system

–A system that gravitates to one or more states—once

a member of a system enters an absorbing state, it

becomes trapped and can never exit that state

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Given in Tables 12-6, 12-7, and 12-8

Given in Tables 12-6, 12-7, and 12-8

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Absorbing State Problem

Absorbing State Problem

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Calculations for Solved Problem 4

Calculations for Solved Problem 4

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Problem 5

Problem 5

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Calculations for Solved Problem 5

Calculations for Solved Problem 5

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—Absorbing State Problem

—Absorbing State Problem