Quantitative business analysis by ron davis

Th is problem defi nition is usually not written by the model builders, but is rather provided by various professionals in the organization who know what the “real world” problem is.. Th

Quantitative Business Analysis

MANAGEMENT SCIENCE READER for Quantitative Business Analysis

Ron Davis

San Jose State University

Copyright © 2010 by Ron Davis All rights reserved No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any elec- tronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfi lming, and recording, or in any information retrieval system without the written permission of University Readers, Inc.

First published in the United States of America in 2010 by Cognella, a division of University Readers, Inc.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifi cation and explanation without intent to infringe.

Previously published by Mathematical Programming Services

3790 El Camino Real #219 Palo Alto, California 94306 http://www.mathproservices.com

14 13 12 11 10 1 2 3 4 5

Printed in the United States of America

ISBN: 978-1-935551-32-4

0 OR/MS Methodology/Terminology

0.1 Model Building Processes 1

0.2 A Taxonomy of OR/MS Model Types 4

0.3 OR/MS Glossary 5

0.4 OR/MS Links 6

1 Network Models 1.0 Defi nition of Terms 7

1.1 Minimal Spanning Tree Problem 8

1.2 Shortest Route Problem 12

1.3 The Critical Path Method (CPM) 14

1.3.1 AON Network Representation 15

1.3.2 Time Concepts 16

1.3.3 A Numerical Example 17

1.4 CPM in the Spreadsheet 20

1.5 Practice Problems 22

2 Transportation Models

2.1 Problem Statement 25

2.2 Linear Programming Formulation 26

2.3 Transportation Solution Algorithm 27

2.4 Transportation Solution Example 29

2 5 Transportation Practice Problems 33

3 Linear Programming 3.1 Mathematical Form 35

3.2 Graphical Interpretation and Solution of LP Problems 37

3.3 LP Problem Formulation Procedure 41

3.4 Spreadsheet LP Modeling for the EXCEL SOLVER add-in .48

3.5 Spreadsheet LP Output Interpretation 54

3.6 Linear Programming Formulation Problems 56

3.7 LP Modeling Problems for Solution Using the SOLVER 58

3.8 Graphical LP Problems 60

4 Project Crashing a CPM Model 4.1 Project Crashing for Small Models 61

4.2 Project Crashing via Linear Programming 68

4.3 Project Crashing Problems 69

5 PERT: Program Evaluation & Review Techique 5.0 Beta Distributions 74

5.1 PERT Approximation Formulas 75

5.2 PERT Approximation Procedure 75

5.3 Example PERT Computation 76

5.4 PERT Practice Problems 78

6 PDFs & CDFs for Continuous Probability Distributions 6.1 PDFs 81

6.2 CDFs 81

6.3 Uniform Distribution 83

6.4 Histogram Distribution 83

6.4.1 Example Histogram Computation 84

6.4.2 Solution of Histogram Example 84

6.5 Beta Distribution for PERT Simulation Analysis 87

6.6 Problems on PDFs, CDFs, and Histogram Distributions 89

7 Monte Carlo Simulation of a PERT-beta Model 7.1 Monte Carlo Simulation Procedure 92

7.2 Automatic CPM: Forward Pass 92

7.3 Automatic CPM: Backward Pass 93

7.4 The Simulation Advantage 93

7.5 The Evergreen Foothills Winery Case 93

7.6 PERT Analysis vs PERT-beta Simulation Analysis 95

7.7 EXCEL TIPS – PERT-beta Simulation Analysis 95

7.8 Simulation Output Reports 100

8 Risk Neutral Decision Analysis 8.0 Decision Making Processes 103

8.1 Finite Discrete Distributions 104

8.2 Payoff Table Analysis 110

8.4 Evaluation of Sample Information – EVSI 116

8.4.1 Bayesian Probability Inversion 117

8.4.2 Structuring the Decision Tree 119

8.4.3 Be-Sure Survey Company Example – EVSI 120

8.5 Payoff Distribution for the Optimal Policy 123

8.6 Practice Problems 125

Appendix: Problem Solutions 129

In past decades, the term Operations Research (OR) connoted a more

mathematical or algorithmic emphasis than did the term Management Science (MS) which was used in more practical or applied contexts But

in recent years, the two terms have become blurred, and overlap so much that in fact the former Operations Research Society of America and the former Institute of Management Sciences merged and became the unifi ed INFORMS Th is acronym stands for Institute For Operations Research and the Management Sciences Two INFORMS conferences are held each year, one in the spring and one in the fall, where thousands of presentations are made by academics, practitioners, government employees, consulting fi rms, and the military Th e best way to get a feeling for the breadth and depth of the papers presented is to go to the INFORMS web site at www.informs.org and take a look at their listings for recent and upcoming conferences on the web

Th ere is also the Decision Sciences Institute, another professional society in this area You can visit their web site at http://www.decisionsciences.org

0.1 Model Building Processes

What unifi es the MS/OR community is the commitment to a particular methodology for problem solving, one which is analogous to the scientifi c method Th ere is a fundamental diff erence, however, in that management sci-ence models incorporate one or more quantitative objective functions used as performance measures for the evaluation of system performance resulting from selected decisions or controls Th ere is a universal desire to improve system performance through the selection of those decision variable values or control settings that give rise to the “best” performance with respect to the performance indices which have been selected for the problem If there is only one perfor-mance index, we seek optimal solutions that either minimize or maximize the performance index If there are two or more performance indices, then we seek the Pareto-optimal set of non-dominated or maximal-value solutions Th is concept is defi ned more precisely later in the chapter

OR/MS Methodology/

Terminology

0

2 Quantitative Business Analysis

Besides the presence of one or more objective functions for evaluating system performance, another ubiquitous commitment which MS/OR practitioners share is that mathematical and computer models are central to the analysis and computation of improved solutions MS/OR practitioners are uniformly model builders, and these models invariably have a mathematical aspect and a computational aspect It would be a mistake to conclude that model building is the province of mathematician and computer scientists only In order that the model built has suffi cient “reality” built into it, inputs from other dis-ciplines are required It frequently requires the combined eff orts of a team of specialists with knowledge

of the engineering, production, logistics, marketing and fi nancial aspects, all providing critical inputs to the model building process Managing the model development process becomes a job in itself, and the process can be described in greater detail by reference to Figure 0-1 shown on the next page

Th e process begins in the upper left corner, with a “real-world problem.” Since model building is not free, the process must begin with the realization that there is “room for improvement” in some aspect of a business’s operation, and a consequent commitment to expend the resources necessary to carry out a model-building eff ort In short, the “higher-ups” in the organization must be convinced that the prospects for a positive return on investment are good Two aspects of obtaining support and approval from the “higher-ups” are a clear demonstration that current practice is not nearly as eff ective

or effi cient as it might be And secondly, they must be convinced that the present “state-of-the-art” tools are suffi ciently powerful to handle the dimensionality and complexity of the requisite models for the decision problem at hand

Once a decision problem has been identifi ed for which a model-building eff ort is desired, the fi rst step is to prepare a written problem statement Th is gives a detailed account of the alternatives to be considered, the system structure which relates actions taken and performance indices used for defi ning optimal or Pareto-optimal solutions, and the data to be taken into account by the model Th is problem defi nition is usually not written by the model builders, but is rather provided by various professionals in the organization who know what the “real world” problem is

Figure 0-1: OR/MS MODELING METHODOLOGY

Th e next step is to translate the verbal statement of the problem into a mathematical formulation of the problem Th e defi nition of the mathematical formulation involves some mathematical notation and the model builders rather than the model sponsors usually provide the mathematical thinking behind the model Th is step is referred to as problem formulation since the mathematical model usually includes a number of quantitative formulas useful in stating the objectives and constraints in the model.

Once formulated, the knowledgeable model builder must then classify the model and come to one

of two conclusions Either (1) this is a model type which is known and for which a solution algorithm already exists; or (2) this model is of a type which has not yet been analyzed, or for which solution procedures have not yet been developed In the former case, it is then a matter of applying the known modeling and solution algorithm to the data associated with the problem at hand In the latter case, the services of a mathematician and a computer scientist must be secured to develop new analysis and new solution algorithms for the problem at hand

Once the mathematical model has been specifi ed and a solution method either selected or developed, then a computer model must be developed which embodies both the mathematical formulation of the problem and the “real-world” data associated with the model Th is enables the solution algorithm to be run on the “real” data respecting the relationships and objectives in the mathematical model Th e outputs from these runs then constitute the computer solution to the problem at hand

Since there are many points at which errors can creep into the process, it is necessary to maintain

a healthy skepticism about the computer results until a thorough testing process has been complete

Th e model usually goes through an evolutionary process in which errors and glitches of all types are gradually eliminated from the model Errors can occur at the formulation stage Data entry or data alignment or scaling errors can easily creep in Th ere can be errors in the solution algorithms or glitches

in the computer programs that implement those algorithms Th ere can even be errors in the report generators that create false output reports based on correct internal solution values Hence, for a model

of any meaningful size or complexity, it is extremely unlikely that the model will function correctly on the fi rst run It is much more common that a period of “debugging” must be endured until the errors are systematically removed from the model, until it functions correctly Model building sponsors and managers alike must be prepared to carry out this “debugging” eff ort, lest the project fail before all errors are removed

To facilitate this debugging process, two types of formal testing are generally carried out to “fl ush out” the glitches which need to be handled Th e fi rst round of testing is referred to as “verifi cation” testing

Th is entails a comparison of model outputs with model inputs to see if the results are mathematically correct Th is is a test which can be carried out by the model builders, since there will be technical speci-

fi cations for how the model is supposed to work In some cases, alternate software systems may be run in parallel on the same data to see if they produce equivalent results During this phase of the development, the goal is to eliminate all modeling, data, algorithmic and programming errors which may be present after the fi rst pass Reworking of all these aspects may be necessary to pass the verifi cation tests

At the successful conclusion of the verifi cation testing there follows another round of testing known

as “validation” testing Th e model building team, since they may have made a number of hidden tions that are not correct, must NOT conduct these tests Rather, the decision support system must be

assump-“turned-over” to the model sponsors for independent testing by users not involved in the development

4 Quantitative Business Analysis

of the system Th ese “real-world” testers may provide data sets other than those used by the developers, and may turn up some malfunctions that were not seen during the verifi cation tests Problems identifi ed during validation may also entail reworking of many of the same areas as verifi cation testing required Untrained or non-technical users may try features or exercise options that had not been tested as part of verifi cation testing

Once the decision model sponsors are satisfi ed with the performance of the decision support system, then it becomes a matter of accounting to see if use of the model leads to superior results compared to current practice While success is not guaranteed, the long list of success stories is now, after 50 years,

so long that the building of models is now a standard business practice Th is is true also for government agencies, in the military, and even among non-profi ts As they say on TV, “try it, you’ll like it.”

0.2 A Taxonomy of OR/MS Model Types

Th ere have now been over 50 years since George Dantzig discovered the Simplex Algorithm for solving linear programming problems Th e publication of this algorithm sparked a tremendous explosion in the development of various model types and various solution algorithms It was found that readily apparent extensions of the Simplex Method enabled one to solve problems with non-linear objective functions and also non-linear constraints Also a number of branch-and-bound algorithms were developed for solving problems in which some or all of the variables are either integer or binary instead of continuous

Th e parallel development of mathematical model types and mathematical solution algorithms has led to

an ever wider class of problem types for which solution algorithms and hence also computer programs have been developed Today, most comprehensive Management Science textbooks have over 15 chapters that discuss no less than a dozen diff erent model types

Figure 0-2 on this page gives a model classifi cation that helps identify the scope of existing knowledge

as well as the scope of the present Course Reader Areas that are covered in this one-semester tory survey are marked in red, while those areas we have to omit for lack of time are shown in black In order to understand the structure of this diagram it is necessary to defi ne a few terms that are commonly used in this discipline but are frequently not known or understood by the layman Th e fi rst dichotomy indicated by the chart is the distinction between a “deterministic” and a “stochastic” model Th is relates

introduc-to the fact that the data or parameters for a model are of two types Parameters that are known with a fair degree of accuracy are represented as constants, i.e by single values that may be particular specifi ed numbers in a spreadsheet model On the other hand, parameters about which there is considerable uncertainty, such as demand forecasts for example must instead be represented by probability distribu-tions Th ese uncertain quantities are said to be random variables, and are generally described by means of standard density functions, or by means of a histogram distribution If any of the parameters in a model are represented as random variables, then the model is said to be a STOCHASTIC MODEL Whereas, if all parameters are represented by constants, then the model is said to be a DETERMINISTIC MODEL

Figure 0-2: A TAXONOMY OF OR/MS MODELS

Another dichotomy indicated by the diagram is between linear and nonlinear models A linear expression

is one which can be stated as a summation of a set of “coeffi cient * variable” expressions Th e coeffi cients are constants, and the variables are included at the fi rst power - no higher order polynomials, no trig functions, no logs or exponential functions, etc If there are no nonlinearities then the model is said to

be a LINEAR MODEL

Nonlinearities, if they exist at all, can occur in three distinct ways Th ere may be a need to include nonlinear terms in one or more of the objective functions Th e standard Markowitz model for portfolio optimization, for example, has a quadratic form in the objective function representing the variance of the portfolio return Nonlinearities can also occur in the constraints of the model, as can be the case, for example, if both Cartesian and polar coordinates must be included in the same formulation And nonlinearity can arise if some of the variables in the model must be integers or binary variables Th is causes the model to become an integer or mixed integer model We will not treat any of these extensions

in this course, but it is important to note their existence since many current applications utilize nonlinear formulations

pos-6 Quantitative Business Analysis

An OPTIMIZATION ALGORITHM is a detailed step-by-step procedure for maximizing or ing the value of an objective function, possibly subject to a set of constraints on the decision variables in the problem formulation

minimiz-An OPTIMAL SOLUTION is the set of decision variable values that yield the minimum or maximum value of the objective function (possibly subject to constraints)

Th e OPTIMAL SOLUTION VALUE is the value of the objective function at an optimal solution point

A HEURISTIC ALGORITHM is an approximation procedure which is designed to give a good tion with a relatively small computational eff ort

solu-Heuristics are often used to initialize an optimization algorithm Th ey are also used when the tional burden for obtaining a truly optimal solution is too great

computa-0.4 OR/MS Links

Since new sites are going up all the time, you need to do your own searches to fi nd the latest ments Th ere are a few “old standbys” that you should be aware of, however, which will get you started in your search for interesting applications of the OR/MS methodology Th ese are presented below

Many practical applications of management science involve the use

of network models Th ese arise in diff erent ways, so the tation of the network elements may vary from one application to the next For a transportation network, the nodes represent locations and the arcs represent routes between the locations Whereas, for project analyses, the nodes represent project activities and the arcs represent precedence relation-ships between the activities Th ere are certain fundamental concepts about networks that are common to all such applications, and for our purposes,

interpre-a key underlying concept is the notion of interpre-a spinterpre-anning tree Th is chapter is devoted primarily to explaining what a spanning tree is, and how various algorithms discover spanning trees that are associated with solutions to opti-mization problems posed for network models

1.0 Defi nition of Terms

Network Diagram - set of nodes and arcs, G = {V,E} where V is the set of nodes (vertices) and E is the set of arcs (edges) in the network

Nodes (vertices) - junction points; at which a fl ow originates, terminates or is relayed

Arcs (branches or edges) - connect pairs of nodes Represent fl ow or precedence Undirected Arcs - No arrow

Directed Arcs - Arrow

Nodes i=1,2,… , 6Undirected arc (i,j) - between i & j (3,4) (3,5) (4,5)Directed arc (i->j) - from i to j (1->2) (1->3) (1->4) (2->5) (4->6) (5->6)

Network Models

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8 Quantitative Business Analysis

Chain - sequence of arcs and nodes connecting any two nodes (i&j), disregarding direction on directed arcs

Eg) 6 (5->6) 5 (3,5) 3 (1->3) 1 is a chain from node 6 to node 1

Path - a chain which respects specifi ed direction along directed arcs in the chain

Cycle - a chain connecting a node to itself, in which each arc and each node (other than the starting node) occurs only once

Eg) (3,4) (4,5) (5,3) is a cycle

Source Node or Origin Node - an origin of fl ow in a network

Sink Node or Destination Node - a termination of fl ow in a network

Other nodes in transportation networks are sometimes called Transshipment Nodes

Two nodes are connected if there is a chain joining them Two nodes are adjacent if there is an arc necting them A network is connected if there is a chain joining any pair of distinct nodes in the network

con-A Sub-Network of a reference network G = {V, E} is a network G’ = {V’,E’} where G contains G’,

V contains V’, and E contains E’, such that the nodes connected by each arc in E’ are contained in V’.Tree - a connected sub-network containing no cycles

Spanning Tree - A Tree which contains every node in the network

Note: A spanning tree on N nodes contains N-1 arcs Th is is proved by induction on N, noting that

a tree on 2 nodes contains 1 arc, and each additional node requires the addition of one additional arc

1.1 Minimal Spanning Tree Problem

Given: A connected network with branch lengths (distance, time or cost)

Find: A spanning tree with minimum total length, time or cost

Th e solution algorithm for this problem, perhaps the simplest you will encounter, proceeds by “growing

a tree” according to what is sometimes called the “Greedy Algorithm.” Th is name refers to the fact that

at each step, the cost of the next branch to be included is minimized It is a “myopic” algorithm in that it only looks one step ahead at each iteration, but has the pleasing property that for this problem type, the optimal solution is obtained in spite of its near sightedness

Th e description of the algorithm makes use of two terms which are defi ned for networks a little diff erently than they are in common parlance Two nodes are said to be adjacent to each other if they are connected by a single arc, no matter how far away they may be from each other spatially

And by the connected nodes we mean those which have been included in the growing tree which has been built so far, up to the present iteration Th e algorithm description which follows has two parts, the initialization step, and the iterative steps, which are repeated until the stopping condition has been met