SOLUTIONS MANUAL AND WORKBOOK
to accompany
Quantitative Methods for
Management
WILLIAM E. PINNEY
Alcorn State University
DONALD B. McWILLIAMS
Texas Wesleyan University
JOSEPH G. ORMSBY
Stephen F. Austin State University
MARK ATCHISON
Electronic Data Systems
January, 2004
International Thomson Publications, Inc.
Acknowledgements
The authors would like to express their gratitude and appreciation to Dianne Cole, Noushin
Ashrafi, Roy Tan, Hang Vu, Thanh Nguyen, Thuy Nguyen, Yen Le, and Mai Nguyen for their
assistance in the completion of this manual. Special thanks go to Andrea Webb and Janice Camp
for their support in the editing of the revised edition
.
Solutions Manual and Workbook
Table of Contents
CHAPTER 1 PREFACE...........................................................................................................................................4
CHAPTER 2 LINEAR PROGRAMMING: PROBLEM FORMULATION.......................................................8
CHAPTER 3 LINEAR PROGRAMMING: GRAPHICAL ANALYSIS ...........................................................14
CHAPTER 4 LINEAR PROGRAMMING: THE SIMPLEX ALGORITHM...................................................33
CHAPTER 5 LINEAR PROGRAMMING: COMPUTER APPLICATIONS...................................................58
CHAPTER 6 GOAL PROGRAMMING...............................................................................................................68
CHAPTER 7 INTEGER LINEAR PROGRAMMING........................................................................................73
CHAPTER 8 THE TRANSPORTATION MODEL.............................................................................................77
CHAPTER 9 THE ASSIGNMENT MODEL........................................................................................................94
CHAPTER 10 NETWORK MODELS .................................................................................................................105
CHAPTER 11 FORECASTING............................................................................................................................117
CHAPTER 12 INVENTORY AND PRODUCTION MODELS ........................................................................136
CHAPTER 13 LEARNING CURVES..................................................................................................................144
CHAPTER 14 DYNAMIC PROGRAMMING....................................................................................................147
CHAPTER 15 DECISION ANALYSIS................................................................................................................171
CHAPTER 16 WAITING LINES (QUEUES) .....................................................................................................189
CHAPTER 17 SIMULATION...............................................................................................................................197
CHAPTER 18 MARKOV ANALYSIS .................................................................................................................218
APPENDIX A: REVIEW OF MATHEMATICAL TOOLS ..............................................................................228
APPENDIX B: PRACTICE/HOMEWORK QUIZZES .....................................................................................237
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Chapter 1 Preface
Chapter 1 Preface
A NOTE TO THE STUDENT:
The secret to success in a course in Quantitative Methods lies in getting plenty of practice
in solving problems. If used properly, this computer disk can be a very useful tool. If used
improperly, it can actually reduce your effectiveness in learning by giving you a false sense of
mastery of the models and their application in solving novel problem situations.
The recommended approach is:
a. Work the assigned problems from the text.
b. Check your answer against the one on the CD.
c. If your answer is incorrect, attempt to rework the problem to obtain the correct answer.
d. Review the solution procedure on the CD to be certain that you really understand the method
involved in solving the problem.
e. After you have satisfactorily completed the (assigned) endofchapter problems, work the
PRACTICE QUIZ on the CD for that chapter. Your instructor may or may not assign
these as homework, but they are an excellent tuneup for your exams, and we strongly
urge you to use them. Your instructor has the solutions to each of them in his/her
INSTRUCTOR'S CD.
The incorrect way to use this CD is to copy the solutions instead of working assigned
homework problems, or to "browse" over the worked out solutions before attempting to work
the problems yourself. The solutions always look easy when someone else has worked them out.
If you have not had practice at facing a problem "cold" and unraveling the mystery of what is to
be optimized, subject to what constraints; determining what variables should be included and
what values are fixed, or constant; and selecting the practice or model which is most appropriate
for the problem, you are likely to face a very uncomfortable experience when your first quiz
arrives.
Our advice, then, is to use this CD as it was intended to be usedas a study aid, not as a
substitute for your own learning by doing.
TOPICS COVERED IN THE TEXT
Introduction
(Chapter 1) presents a brief review of the history of Management Science
models, details the steps in the systems approach to problem solving, discusses the roles of
models and optimization in the decisionmaking process, and suggests areas of application for
Management Science tools in business decisionmaking settings.
Linear Programming: Problem Formulation
(Chapter 2) sets forth the steps in
formulating a linear programming problem: (1) identify the variables, (2) define the objective
function, (3) formulate the constraints, and (4) solve the problem. Examples from production
management, marketing, and finance are presented.
Linear Programming: Graphical Analysis
(Chapter 3) presents the basic model of the
linear programming problem, and explains graphical solution procedures. Topics introduced
include constraints (GTE, LTE, and EQ), the feasible region, objective functions, maximization
Solutions Manual and Workbook
5
and minimization, bounded and unbounded solutions, binding and nonbinding constraints,
degenerate and multiple solutions, and sensitivity analysis.
Linear Programming: The Simplex Algorithm
(Chapter 4) formulates the LP problem in a
standard simplex format and explains the simplex procedure in detail. Strong use is made of
graphical concepts to give the students a visual picture of exactly what the mathematical model
is accomplishing at each step of the analysis. Special problems (infeasible, unbounded,
degenerate, and multiple solutions) are explained, the dual problem is formulated and solved, and
a detailed sensitivity analysis is presented. A new addition is the introduction of the dual simplex
model as an attractive alternative to the BIG M method.
Linear Programming: Computer Applications
(Chapter 5) describes, through the use of
several examples, the process of formulating, inputting, and interpreting the output of a
computerized LP program. The aim of this chapter is to make the power of linear programming
available to the student who is neither a management scientist nor a computer programmer. It is
written to be used either in addition to or instead of the simplex chapter.
Goal Programming
(Chapter 6) covers problems "not solvable" by regular Linear
Programming, including the infeasible problem and problems with multiple objectives. A
notation which is simpler than the one usually presented is introduced.
Integer Linear Programming
(Chapter 7) includes pure integer, mixed integer, and zero
one programming. Stick trees and the branch and bound method are treated extensively.
The Transportation Model
(Chapter 8) is a special class of LP problems and specialized
procedures for their solution. Initial solution procedures include the Northwest Corner Model,
the Best Cell Method, and Vogel's Approximation Method (VAM). The Modified Distribution
(MODI) Method for testing solutions for optimality is presented, and the Stepping Stone method
for improving a suboptimal solution is covered. Unbalanced problems, degeneracy, multiple
solutions, and the Transhipment problem are treated.
The Assignment Model
(Chapter 9) is a special case of the Transportation Model, with all
sources and destinations having exactly one unit. The Hungarian Method of solution is
presented, and unbalanced problems and multiple solutions are covered.
Network Models
(Chapter 10) presents the critical path method (CPM), program
evaluation and review technique (PERT), and CPM/COST models for the analysis of networks
of activities. Methods are developed for determining which activities are critical to the timely
completion of a project, the amount of slack available in the noncritical activities and the least
expensive way to reduce the expected length of the project. Also presented are models for
finding the SHORTEST PATH from one point in a network to another, for reducing a network to
its MINIMAL SPANNING TREE, and for determining the MAXIMUM FLOW through a
network. The TRAVELING SALES REP model is covered, and software solution for all of the
models are discussed.
Forecasting
(Chapter 11) includes coverage of subjective forecasting methods, simple
and weighted moving averages, exponential smoothing, and linear regression and correlation.
Trend, seasonal, cyclical, and random variations in time series data are treated.
Inventory and Production Models
(Chapter 12) provide an application area for classical
optimization techniques. The approach used is to develop a total cost function and minimize the
sum of competing costs. Optimum values are found for order size, number of orders per year,
length of the inventory cycle, and reorder level. The treatment includes consideration of
shortages (stock outs), buffer stock (safety stock), variable lead time, variable demand, and the
development of the noninstantaneous inventory replenishment (production) model.
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Chapter 1 Preface
Learning Curves
(Chapter 13) discuss a mathematical method for explaining the
improvement in productivity as a result of worker learning. Briefly stated, every doubling of
work repetitions results in a constant percentage decrease in the time per repetition.
Dynamic Programming
(Chapter 14) addresses the Sales Distribution, Production
Planning, Investment, Knapsack, and Stagecoach problems utilizing the Transition Function and
employing Bellman's Principle of Optimality.
Decision Analysis
(Chapter 15) presents a thorough discussion of decisionmaking under
conditions of certainty, uncertainty, and risk. Strategies for decisions under uncertainty include
the Maximax, Maximin, Hurwicz, Savage, and LaPlace criteria. Under conditions of risk, the
strategies are optimizing expected payoff value and minimizing the expected value of regret
(opportunity loss). Probability trees, decision trees, the expected value of perfect information
(EVPI) and sample information (EVSI), the expected net gain from sampling (ENGS), and
determination of optimal sample size are covered, and a simplified approach to computation of
revised (Bayesian) probabilities using a contingency table format is presented. Utility functions
and risk preference are also treated.
Waiting Lines (Queues)
(Chapter 16) treats the class of problems involving bottlenecks in
the smooth flow of a system. Two analytical approaches are presented: (1) a graphical model for
analysis of simple, deterministic systems; and (2) a mathematical model for analysis of the
steady state, or average, conditions of systems which meet certain requirements. The solutions
focus on the average length of the queue, the average time spent waiting for service, and the
determination of the optimal number of servers. Both single channel and multichannel queuing
systems are analyzed.
Simulation
(Chapter 17) presents an approach for solving complex problems, usually
those of a stochastic nature. The technique does not attempt to solve model equations
analytically, but by following the changes over time of a dynamic model of the system. In this
chapter the major emphasis is placed on Monte Carlo simulation of stochastic systems. Random
number tables are employed to demonstrate the approximation of probability distributions and
the general approach to simulating systems is discussed and illustrated at length.
Markov Analysis
(Chapter 18) provides an interesting application of the matrix
multiplication and the simultaneous linear equations reviewed in Appendix A. First order
Markov chains of market shares of competing firms are analyzed, and both nth period and
equilibrium solutions are presented. The assumptions of the model, absorbing and disappearing
states, higher order chains, and generalizations of the model to other applications are discussed.
Review of Mathematical Tools
(Appendix A) is presented to reacquaint the student with
the basic building blocks that form the basis for the models to be developed in the remainder of
the text. Functional relationships, graphical representation, and elements of scalar algebra are
covered briefly. Addition, subtraction, multiplication, and division (through use of the inverse)
of matrices are covered. Methods for evaluating determinants are presented, and Cramer's rule
for solving sets of linear equations is explained. The Gaussian reduction method of finding the
inverse, that forms the basis for the simplex method of linear programming, is presented in
detail.
SOLUTIONS TO ENDOFCHAPTER PROBLEMS
This CD contains the complete solution to each of the endorchapter problems in the text.
Solutions Manual and Workbook
7
PRACTICE QUIZZES
For Chapters 2 through 18 there are practice quizzes over materials covered in the chapter.
These may be used for review, as preparation for inclass quizzes, or as homework assignments.
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