Operations research methodologies a ravi ravindran, CRC press, 2009 scan

The most effective method for solving systems of linear equations, called the elimination method, was discovered by the Chinese and the Indians over 2500 years ago and this method is stil

OPERATIONS RESEARCH METHODOLOGIES

The Operations Research Series

Series Editor: A Ravi Ravindran

Dept of Industrial & Manufacturing Engineering The Pennsylvania State University, USA

Integer Programming: Theory and Practice

John K Karlof

Operations Research Applications

A Ravi Ravindran

Operations Research: A Practical Approach

Michael W Carter and Camille C Price

Operations Research Calculations Handbook

Probability Models in Operations Research

Richard C Cassady and Joel A Nachlas

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Boca Raton London New York

OPERATIONS

RESEARCH

METHODOLOGIES

CRC Press

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© 2009 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

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Preface ix

Acknowledgments xi

Editor xiii

Contributors xv

History of Operations Research xvii

1 Linear Programming

Katta G Murty 1-1

1.1 Brief History of Algorithms for Solving Linear Equations, Linear

Inequalities, and LPs 1-1

1.2 Applicability of the LP Model: Classical Examples of

Direct Applications 1-4

1.3 LP Models Involving Transformations of Variables 1-12

1.4 Intelligent Modeling Essential to Get Good Results, an Example from

Container Shipping 1-18

1.5 Planning Uses of LP Models 1-22

1.6 Brief Introduction to Algorithms for Solving LP Models 1-26

1.7 Software Systems Available for Solving LP Models 1-31

1.8 Multiobjective LP Models 1-31

2 Nonlinear Programming

Theodore B Trafalis and Robin C Gilbert 2-1

2.1 Introduction 2-1

2.2 Unconstrained Optimization 2-3

2.3 Constrained Optimization 2-15

2.4 Conclusion 2-19

3 Integer Programming

Michael Weng 3-1

3.1 Introduction 3-1

3.2 Formulation of IP Models 3-3

3.3 Branch and Bound Method 3-7

3.4 Cutting Plane Method 3-12

3.5 Other Solution Methods and Computer Solution 3-15

4 Network Optimization

Mehmet Bayram Yildirim 4-1

4.1 Introduction 4-1

4.2 Notation 4-2

4.3 Minimum Cost Flow Problem 4-3

v

4.4 Shortest Path Problem 4-4

4.5 Maximum Flow Problem 4-8

4.6 Assignment Problem 4-13

4.7 Minimum Spanning Tree Problem 4-14

4.8 Minimum Cost Multicommodity Flow Problem 4-18

4.9 Conclusions 4-19

5 Multiple Criteria Decision Making

Abu S M Masud and A Ravi Ravindran 5-1

5.1 Some Definitions 5-3

5.2 The Concept of “Best Solution” 5-4

5.3 Criteria Normalization 5-5

5.4 Computing Criteria Weights 5-6

5.5 Multiple Criteria Methods for Finite

6.2 Terminology for Decision Analysis 6-2

6.3 Decision Making under Risk 6-3

6.4 Decision Making under Uncertainty 6-17

6.5 Practical Decision Analysis 6-21

7.2 Deterministic Dynamic Programming Models 7-3

7.3 Stochastic Dynamic Programming Models 7-19

8.3 Discrete-Time Markov Chains 8-14

8.4 Continuous-Time Markov Chains 8-27

8.5 Renewal Theory 8-39

8.6 Software Products Available for Solving

Stochastic Models 8-46

9 Queueing Theory

Natarajan Gautam 9-1

9.1 Introduction 9-1

9.2 Queueing Theory Basics 9-2

9.3 Single-Station and Single-Class Queues 9-8

9.4 Single-Station and Multiclass Queues 9-21

9.5 Multistation and Single-Class Queues 9-28

9.6 Multistation and Multiclass Queues 9-34

9.7 Concluding Remarks 9-37

10 Inventory Control

Farhad Azadivar and Atul Rangarajan 10-1

10.1 Introduction 10-1

10.2 Design of Inventory Systems 10-4

10.3 Deterministic Inventory Systems 10-9

10.4 Stochastic Inventory Systems 10-20

10.5 Inventory Control at Multiple Locations 10-27

10.6 Inventory Management in Practice 10-34

10.7 Conclusions 10-35

10.8 Current and Future Research 10-36

11 Complexity and Large-Scale Networks

Hari P Thadakamalla, Soundar R.T Kumara, and R´ eka Albert 11-1

11.1 Introduction 11-1

11.2 Statistical Properties of Complex Networks 11-6

11.3 Modeling of Complex Networks 11-11

11.4 Why “Complex” Networks 11-16

11.5 Optimization in Complex Networks 11-18

12.3 Simulation Languages and Software 12-15

12.4 Simulation Projects—The Bigger Picture 12-20

12.5 Summary 12-22

13 Metaheuristics for Discrete Optimization Problems

Rex K Kincaid 13-1

13.1 Mathematical Framework for Single Solution Metaheuristics 13-3

13.2 Network Location Problems 13-3

13.3 Multistart Local Search 13-5

13.4 Simulated Annealing 13-6

13.5 Plain Vanilla Tabu Search 13-8

13.6 Active Structural Acoustic Control (ASAC) 13-10

13.7 Nature Reserve Site Selection 13-13

13.8 Damper Placement in Flexible Truss Structures 13-21

13.9 Reactive Tabu Search 13-29 13.10 Discussion 13-35

Operations research (OR), which began as an interdisciplinary activity to solve complexproblems in the military during World War II, has grown in the past 50 years to a full-fledged academic discipline Now OR is viewed as a body of established mathematical modelsand methods to solve complex management problems OR provides a quantitative analy-sis of the problem from which the management can make an objective decision OR hasdrawn upon skills from mathematics, engineering, business, computer science, economics,and statistics to contribute to a wide variety of applications in business, industry, govern-ment, and military OR methodologies and their applications continue to grow and flourish

in a number of decision-making fields

The objective of this book is to provide a comprehensive overview of OR models andmethods in a single volume This book is not an OR textbook or a research monograph.The intent is that the book becomes the first resource a practitioner would reach for whenfaced with an OR problem or question The key features of this book are as follows:

• Single source guide to OR techniques

• Comprehensive resource, but concise

• Coverage of emerging OR methodologies

• Quick reference guide to students, researchers, and practitioners

• Bridges theory and practice

• References to computer software availability

• Designed and edited with nonexperts in mind

• Unified and up-to-date coverage ideal for ready reference

This book contains 14 chapters that cover not only the fundamental OR models andmethods such as linear, nonlinear, integer and dynamic programming, networks, simula-tion, queueing, inventory, stochastic processes, and decision analysis, but also emerging ORtechniques such as multiple criteria optimization, metaheuristics, robust optimization, andcomplexity and large-scale networks Each chapter gives an overview of a particular ORmethodology, illustrates successful applications, and provides references to computer soft-ware availability Each chapter in this book is written by leading authorities in the field and

is devoted to a topic listed as follows:

book titled Operations Research Applications, which contains both functional and

industry-specific applications of the OR methodologies discussed here

A Ravi Ravindran

University Park, Pennsylvania

First and foremost I would like to thank the authors, who have worked diligently in writingthe various handbook chapters that are comprehensive, concise, and easy to read, bridgingthe gap between theory and practice The development and evolution of this handbookhave also benefited substantially from the advice and counsel of my colleagues and friends

in academia and industry, who are too numerous to acknowledge individually They helped

me identify the key topics to be included in the handbook, suggested chapter authors, andserved as reviewers of the manuscripts

I express my sincere appreciation to Atul Rangarajan, an industrial engineering toral student at Penn State University, for serving as my editorial assistant and for hiscareful review of the page proofs returned by the authors Several other graduate stu-dents also helped me with the handbook work, in particular, Ufuk Bilsel, Ajay Natarajan,Richard Titus, Vijay Wadhwa, and Tao Yang Special thanks go to Professor PrabhaSharma at the Indian Institute of Technology, Kanpur, for her careful review of severalchapter manuscripts I also acknowledge the pleasant personality and excellent typing skills

doc-of Sharon Frazier during the entire book project

I thank Cindy Carelli, Senior Acquisitions Editor, and Jessica Vakili, project coordinator

at CRC Press, for their help from inception to publication of the handbook Finally, I wish

to thank my dear wife, Bhuvana, for her patience, understanding, and support when I wasfocused completely on the handbook work

A Ravi Ravindran

xi

A Ravi Ravindran, Ph.D., is a professor and the past department head of Industrial

and Manufacturing Engineering at the Pennsylvania State University Formerly, he was afaculty member at the School of Industrial Engineering at Purdue University for 13 yearsand at the University of Oklahoma for 15 years At Oklahoma, he served as the director ofthe School of Industrial Engineering for 8 years and as the associate provost of the universityfor 7 years, with responsibility for budget, personnel, and space for the academic area Heholds a B.S in electrical engineering with honors from the Birla Institute of Technology andScience, Pilani, India His graduate degrees are from the University of California, Berkeley,where he received an M.S and a Ph.D in industrial engineering and operations research

Dr Ravindran’s area of specialization is operations research with research interests inmultiple criteria decision-making, financial engineering, health planning, and supply chain

optimization He has published two major textbooks (Operations Research: Principles and Practice and Engineering Optimization: Methods and Applications) and more than 100 jour-

nal articles on operations research He is a fellow of the Institute of Industrial Engineers In

2001, he was recognized by the Institute of Industrial Engineers with the Albert G HolzmanDistinguished Educator Award for significant contributions to the industrial engineeringprofession by an educator He has won several Best Teacher awards from IE students Hehas been a consultant to AT&T, General Motors, General Electric, IBM, Kimberly Clark,Cellular Telecommunication Industry Association, and the U.S Air Force He currentlyserves as the Operations Research Series editor for Taylor & Francis/CRC Press

xiii

R´ eka Albert

Pennsylvania State University

University Park, Pennsylvania

Farhad Azadivar

University of Massachusetts–Dartmouth

North Dartmouth, Massachusetts

Natarajan Gautam

Texas A&M University

College Station, Texas

Pennsylvania State University

University Park, Pennsylvania

Pennsylvania State University

University Park, Pennsylvania

Mehmet Bayram Yildirim

Wichita State University Wichita, Kansas

xv

History of Operations

Research

A Ravi Ravindran

Pennsylvania State University

Origin of Operations Research

To understand what operations research (OR) is today, one must know something of itshistory and evolution Although particular models and techniques of OR can be tracedback to much earlier origins, it is generally agreed that the discipline began during WorldWar II Many strategic and tactical problems associated with the Allied military effort weresimply too complicated to expect adequate solutions from any one individual, or even asingle discipline In response to these complex problems, groups of scientists with diverseeducational backgrounds were assembled as special units within the armed forces Theseteams of scientists started working together, applying their interdisciplinary knowledge andtraining to solve such problems as deployment of radars, anti-aircraft fire control, deploy-ment of ships to minimize losses from enemy submarines, and strategies for air defense.Each of the three wings of Britain’s armed forces had such interdisciplinary research teamsworking on military management problems As these teams were generally assigned to the

commanders in charge of military operations, they were called operational research (OR) teams The nature of their research came to be known as operational research or operations research.

The work of these OR teams was very successful and their solutions were effective inmilitary management This led to the use of such scientific teams in other Allied nations,

in particular the United States, France, and Canada At the end of the war, many of thescientists who worked in the military operational research units returned to civilian life inuniversities and industries They started applying the OR methodology to solve complexmanagement problems in industries Petroleum companies were the first to make use of

OR models for solving large-scale production and distribution problems In the universitiesadvancements in OR techniques were made that led to the further development and appli-cations of OR Much of the postwar development of OR took place in the United States

An important factor in the rapid growth of operations research was the introduction ofelectronic computers in the early 1950s The computer became an invaluable tool to the

operations researchers, enabling them to solve large problems in the business world.

The Operations Research Society of America (ORSA) was formed in 1952 to serve theprofessional needs of these operations research scientists Due to the application of OR inindustries, a new term called management science (MS) came into being In 1953, a nationalsociety called The Institute of Management Sciences (TIMS) was formed in the UnitedStates to promote scientific knowledge in the understanding and practice of management

The journals of these two societies, Operations Research and Management Science, as well

as the joint conferences of their members, helped to draw together the many diverse resultsinto some semblance of a coherent body of knowledge In 1995, the two societies, ORSAand TIMS, merged to form the Institute of Operations Research and Management Sciences(INFORMS)

xvii

Another factor that accelerated the growth of operations research was the introduction

of OR/MS courses in the curricula of many universities and colleges in the United States.Graduate programs leading to advanced degrees at the master’s and doctorate levels wereintroduced in major American universities By the mid-1960s many theoretical advances

in OR techniques had been made, which included linear programming, network analysis,integer programming, nonlinear programming, dynamic programming, inventory theory,queueing theory, and simulation Simultaneously, new applications of OR emerged in serviceorganizations such as banks, health care, communications, libraries, and transportation Inaddition, OR came to be used in local, state, and federal governments in their planning andpolicy-making activities

It is interesting to note that the modern perception of OR as a body of established modelsand techniques—that is, a discipline in itself—is quite different from the original concept of

OR as an activity, which was preformed by interdisciplinary teams An evolution of this kind

is to be expected in any emerging field of scientific inquiry In the initial formative years,there are no experts, no traditions, no literature As problems are successfully solved, thebody of specific knowledge grows to a point where it begins to require specialization even

to know what has been previously accomplished The pioneering efforts of one generationbecome the standard practice of the next Still, it ought to be remembered that at least aportion of the record of success of OR can be attributed to its ecumenical nature

Meaning of Operations Research

From the historical and philosophical summary just presented, it should be apparent that theterm “operations research” has a number of quite distinct variations of meaning To some,

OR is that certain body of problems, techniques, and solutions that has been accumulatedunder the name of OR over the past 50 years and we apply OR when we recognize a prob-lem of that certain genre To others, it is an activity or process, which by its very nature isapplied It would also be counterproductive to attempt to make distinctions between “oper-ations research” and the “systems approach.” For all practical purposes, they are the same.How then can we define operations research? The Operational Research Society of GreatBritain has adopted the following definition:

Operational research is the application of the methods of science to plex problems arising in the direction and management of large systems of men,machines, materials and money in industry, business, government, and defense.The distinctive approach is to develop a scientific model of the system, incor-porating measurement of factors such as chance and risk, with which to predictand compare the outcomes of alternative decisions, strategies or controls Thepurpose is to help management determine its policy and actions scientifically.The Operations Research Society of America has offered a shorter, but similar,description:

com-Operations research is concerned with scientifically deciding how to best designand operate man–machine systems, usually under conditions requiring the allo-cation of scarce resources

In general, most of the definitions of OR emphasize its methodology, namely its uniqueapproach to problem solving, which may be due to the use of interdisciplinary teams ordue to the application of scientific and mathematical models In other words, each prob-lem may be analyzed differently, though the same basic approach of operations research isemployed As more research went into the development of OR, the researchers were able to

classify to some extent many of the important management problems that arise in practice.Examples of such problems are those relating to allocation, inventory, network, queuing,replacement, scheduling, and so on The theoretical research in OR concentrated on devel-oping appropriate mathematical models and techniques for analyzing these problems underdifferent conditions Thus, whenever a management problem is identified as belonging to aparticular class, all the models and techniques available for that class can be used to studythat problem In this context, one could view OR as a collection of mathematical mod-els and techniques to solve complex management problems Hence, it is very common tofind OR courses in universities emphasizing different mathematical techniques of operationsresearch such as mathematical programming, queueing theory, network analysis, dynamicprogramming, inventory models, simulation, and so on.

For more on the early activities in operations research, see Refs 1–5 Readers interested

in the timeline of major contributions in the history of OR/MS are referred to the excellentreview article by Gass [6]

References

1 Haley, K.B., War and peace: the first 25 years of OR in Great Britain, Operations Research,

50, Jan.–Feb 2002

2 Miser, H.J., The easy chair: what OR/MS workers should know about the early formative

years of their profession, Interfaces, 30, March–April 2000.

3 Trefethen, F.N., A history of operations research, in Operations Research for Management,

J.F McCloskey and F.N Trefethen, Eds., Johns Hopkins Press, Baltimore, MD, 1954

4 Horner, P., History in the making, ORMS Today, 29, 30–39, 2002.

5 Ravindran, A., Phillips, D.T., and Solberg, J.J., Operations Research: Principles and tice, Second Edition, John Wiley & Sons, New York, 1987 (Chapter 1).

Prac-6 Gass, S.I., Great moments in histORy, ORMS Today, 29, 31–37, 2002.

1.3 LP Models Involving Transformations ofVariables . 1-12

Min–Max, Max–Min Problems•Minimizing Positive Linear Combinations of Absolute Values

of Affine Functions1.4 Intelligent Modeling Essential to Get GoodResults, an Example from Container Shipping . 1-18

1.5 Planning Uses of LP Models . 1-22

Finding the Optimum Solutions•Infeasibility Analysis

•Values of Slack Variables at an Optimum Solution•

Marginal Values, Dual Variables, and the Dual Problem, and Their Planning Uses•Evaluating the Profitability of New Products

1.6 Brief Introduction to Algorithms for Solving

LP Models . 1-26

The Simplex Method•Interior Point Methods for LP1.7 Software Systems Available for Solving LPModels . 1-31

1.8 Multiobjective LP Models . 1-31

References . 1-34

Equations, Linear Inequalities, and LPs

The study of mathematics originated with the construction of linear equation models for realworld problems several thousand years ago As an example we discuss an application thatleads to a model involving a system of simultaneous linear equations from Murty (2004)

1-1

Example 1.1: Scrap Metal Blending Problem

A steel company has four different types of scrap metal (SM-1 to SM-4) with the followingcompositions (Table 1.1)

The company needs to blend these four scrap metals into a mixture for which the position by weight is: Al—4.43%, Si—3.22%, C—3.89%, and Fe—88.46% How should theyprepare this mixture? To answer this question, we need to determine the proportions of thefour scrap metals SM-1 to SM-4 in the blend to be prepared The most fundamental idea in mathematics that was discovered more than 5000 years ago

com-by the Chinese, Indians, Iranians, Bacom-bylonians, and Greeks is to represent the quantitiesthat we wish to determine by symbols, usually letters of the alphabet likex, y, z, and then

express the relationships between the quantities represented by these symbols in the form

of equations, and finally use these equations as tools to find out the true values represented

by the symbols The symbols representing the unknown quantities to be determined are

nowadays called unknowns or variables or decision variables The process of representing

the relationships between the variables through equations or other functional relationships

is called modeling or mathematical modeling.

This process gradually evolved into algebra, one of the chief branches of mathematics.

Even though the subject originated more than 5000 years ago, the name algebra itself

came much later; it is derived from the title of an Arabic book Al-Maqala fi Hisab al-jabr w’almuqabalah written by Al-Khawarizmi around 825 AD The term “al-jabr” in Arabic

means “restoring” in the sense of solving an equation In Latin translation the title of this

book became Ludus Algebrae, the second word in this title surviving as the modern word

“algebra” for the subject, and Al-Khawarizmi is regarded as the father of algebra Theearliest algebraic systems constructed are systems of linear equations

In the scrap metal blending problem, the decision variables are:x j= proportion of SM-j

by weight in the mixture, forj = 1–4 Then the percentage by weight of the element Al in

the mixture will be 5x1+ 7x2+ 2x3+x4, which is required to be 4.43 Arguing the sameway for the elements Si, C, and Fe, we find that the decision variablesx1tox4 must satisfy

each equation in the following system of linear equations to lead to the desired mixture:

5x1+ 7x2+ 2x3+x4= 4.43

3x1+ 6x2+x3+ 2x4= 3.22

4x1+ 5x2+ 3x3+x4= 3.8988x1+ 82x2+ 94x3+ 96x4= 88.46

x1+x2+x3+x4= 1The last equation in the system stems from the fact that the sum of the proportions ofvarious ingredients in a blend must always be equal to 1 This system of equations is themathematical model for our scrap metal blending problem; it consists of five equations

TABLE 1.1 Scrap Metal Composition Data

% of Element by Weight, in Type

in four variables It is clear that a solution to this system of equations makes sense forthe blending application only if all the variables in the system have nonnegative values

in it The nonnegativity restrictions on the variables are linear inequality constraints They

cannot be expressed in the form of linear equations, and as nobody knew how to handlelinear inequalities at that time, they ignored them

Linear algebra dealing with methods for solving systems of linear equations is the classical

subject that initiated the study of mathematics a long time ago The most effective method

for solving systems of linear equations, called the elimination method, was discovered by the

Chinese and the Indians over 2500 years ago and this method is still the leading method inuse today This elimination method was unknown in Europe until the nineteenth centurywhen the German mathematician Gauss rediscovered it while calculating the orbit of theasteroid Ceres based on recorded observations in tracking it The asteroid was lost fromview when the Sicilian astronomer Piazzi tracking it fell ill Gauss used the method of leastsquares to estimate the values of the parameters in the formula for the orbit It led to asystem of 17 linear equations in 17 unknowns that he had to solve, which is quite a largesystem for mental computation Gauss’s accurate computations helped in relocating theasteroid in the skies in a few months’ time, and his reputation as a mathematician soared.Another German, Wilhelm Jordan, popularized the algorithm in a late nineteenth-centurybook that he wrote From that time, the method has been popularly known as the Gauss–Jordan elimination method Another version of this method, called the Gaussian eliminationmethod, is the most popular method for solving systems of linear equations today

Even though linear equations were resolved thousands of years ago, systems of linearinequalities remained unsolved until the middle of the twentieth century The followingtheorem (Murty, 2006) relates systems of linear inequalities to systems of linear equations

THEOREM 1.1 Consider the system of linear inequalities in variables x

where A i. is the coefficient vector for the i-th constraint If this system has a feasible solution,

then there exists a subset P = {p1, , p s } ⊂ {1, , m} such that every solution of the system

of equations: A i. x = b i , i ∈ P is also a feasible solution of the original system of linear

inequalities (Equation 1.1).

This theorem can be used to generate a finite enumerative algorithm to find a feasiblesolution to a system of linear constraints containing inequalities, based on solving subsys-tems in each of which a subset of the inequalities are converted into equations and the otherinequality constraints are eliminated However, if the original system hasm inequality con-

straints, in the worst case this enumerative algorithm may have to solve 2msystems of linearequations before it either finds a feasible solution of the original system, or concludes that

it is infeasible The effort required grows exponentially with the number of inequalities inthe system in the worst case

In the nineteenth century, Fourier generalized the classical elimination method for solvinglinear equations into an elimination method for solving systems of linear inequalities The

method called Fourier elimination, or the Fourier–Motzkin elimination method, is very

elegant theoretically However, the elimination of each variable adds new inequalities to theremaining system, and the number of these new inequalities grows exponentially as moreand more variables are eliminated So this method is also not practically viable for largeproblems

The simplex method for linear programming developed by Dantzig (1914–2005) in themid-twentieth century (Dantzig, 1963) is the first practically and computationally viable

method for solving systems of linear inequalities This has led to the development of linear programming (LP), a branch of mathematics developed in the twentieth century as an extension of linear algebra to solve systems of linear inequalities The development of LP is

a landmark event in the history of mathematics and its applications that brought our ability

to solve general systems of linear constraints (including linear equations, inequalities) to astate of completion

A general system of linear constraints in decision variablesx = (x1, , x n)T is of the form:

Ax ≥ b, Dx = d, where the coefficient matrices A, D are given matrices of orders m × n,

p × n, respectively The inequality constraints in this system may include sign restrictions

or bounds on individual variables

A general LP is the problem of finding an optimum solution for the problem of minimizing(or maximizing) a given linear objective function z = cx say, subject to a system of linear

constraints

Suppose there is no objective function to optimize, and only a feasible solution of a system

of linear constraints is to be found When there are inequality constraints in the system, theonly practical method to even finding a feasible solution is to solve a linear programmingformulation of it as a Phase I linear programming problem Dantzig developed this Phase Iformulation as part of the simplex method for LPs that he developed in the mid-twentiethcentury

Examples of Direct Applications

LP has now become a dominant subject in the development of efficient computationalalgorithms, the study of convex polyhedra, and in algorithms for decision making Butfor a short time in the beginning, its potential was not well recognized Dantzig tells thestory of how when he gave his first talk on LP and his simplex method for solving it at aprofessional conference, Hotelling (a burly person who liked to swim in the sea; the popularstory about him was that when he does, the level of the ocean rises perceptibly) dismissed

it as unimportant since everything in the world is nonlinear But Von Neumann came tothe defense of Dantzig saying that the subject will become very important (Dantzig andThapa, 1997, vol 1, p xxvii) The preface in this book contains an excellent account ofthe early history of LP from the inventor of the most successful method in OR and in themathematical theory of polyhedra

Von Neumann’s early assessment of the importance of LP turned out to be astonishinglycorrect Today, the applications of LP in almost all areas of science are numerous The LPmodel is suitable for modeling a real world decision-making problem if

• All the decision variables are continuous variables

• There is a single objective function that is required to be optimized

• The objective function and all the constraint functions defining the constraints

in the problem are linear functions of the decision variables (i.e., they satisfy the

usual proportionality and additivity assumptions)

There are many applications in which the reasonableness of the linearity assumptions can

be verified and an LP model for the problem constructed by direct arguments We presentsome classical applications like this in this section; this material is from Murty (1995, 2005b)

In all these applications you can judge intuitively that the assumptions needed to handlethem using an LP model are satisfied to a reasonable degree of approximation.

Of course LP can be applied to a much larger class of problems Many important tions involve optimization models in which a nonlinear objective function that is piecewiselinear and convex is to be minimized subject to linear constraints These problems can

applica-be transformed into LPs by introducing additional variables These transformations arediscussed in the next section

This is an extremely important class of problems that manufacturing companies face.Normally the company can make a variety of products using the raw materials, machinery,labor force, and other resources available to them The problem is to decide how much

of each product to manufacture in a period, to maximize the total profit subject to theavailability of needed resources

To model this, we need data on the units of each resource necessary to manufacture oneunit of each product, any bounds (lower, upper, or both) on the amount of each productmanufactured per period, any bounds on the amount of each resource available per period,the expected demand for each product, and the cost or net profit per unit of each productmanufactured

Assembling this type of reliable data is one of the most difficult jobs in constructing aproduct mix model for a company, but it is very worthwhile The process of assembling

all the needed data is sometimes called the input–output analysis of the company The

coefficients, which are the resources necessary to make a unit of each product, are called

input–output (I/O) coefficients, or technology coefficients.

Example 1.2: The Fertilizer Product Mix Problem

As an example, consider a fertilizer company that makes two kinds of fertilizers calledHi-phosphate (Hi-ph) and Lo-phosphate (Lo-ph) The manufacture of these fertilizersrequires three raw materials called RM 1, RM 2, and RM 3 At present their supply ofthese raw materials comes from the company’s own quarry that is only able to supply max-imum amounts of 1500, 1200, 500 tons/day, respectively, of RM 1, RM 2, and RM 3 Eventhough there are other vendors who can supply these raw materials if necessary, at themoment they are not using these outside suppliers

They sell their output of Hi-ph and Lo-ph fertilizers to a wholesaler who is willing to buyany amount that they can produce, so there are no upper bounds on the amounts of Hi-phand Lo-ph manufactured daily

At the present rates of operation their Cost Accounting Department estimates that it iscosting the quarry $50, $40, and $60/ton respectively to produce and deliver RM 1, RM 2,and RM 3 at the fertilizer plant Also, at the present rates of operation, all other productioncosts (for labor, power, water, maintenance, depreciation of plant and equipment, floorspace, insurance, shipping to the wholesaler, etc.) come to $7/ton to manufacture Hi-ph orLo-ph and deliver to the wholesaler

The sale price of the manufactured fertilizers to the wholesaler fluctuates daily, but theiraverages over the last one month have been $222 and $107 per ton, respectively, for Hi-Phand Lo-ph fertilizers We will use these prices to construct the mathematical model The Hi-ph manufacturing process needs as inputs 2 tons RM 1 and 1 ton each of RM 2and RM 3 for each ton of Hi-ph manufactured Similarly, the Lo-ph manufacturing processneeds as inputs 1 ton RM 1 and 1 ton of RM 2 for each ton of Lo-ph manufactured

So, the net profit/ton of fertilizer manufactured is $(222− 2 × 50 − 1 × 40 − 1 × 60 − 7) = 15

and (107− 1 × 50 − 1 × 40 − 7) = 10, respectively, for Hi-ph and Lo-ph.

There are clearly two decision variables in this problem; these are:x1= the tons of Hi-phproduced per day, x2 the tons of Lo-ph produced per day Associated with each variable

in the problem is an activity that the decision maker can perform The activities in this

example are: Activity 1: to make 1 ton of Hi-ph, Activity 2: to make 1 ton of Lo-ph The

variables in the problem just define the levels at which these activities are carried out.

As all the data are given on a per ton basis, they provide an indication that the linearityassumptions are quite reasonable in this problem Also, the amount of each fertilizer man-ufactured can vary continuously within its present range So, LP is an appropriate modelfor this problem

Each raw material leads to a constraint in the model The amount of RM 1 used is2x1+x2 tons, and it cannot exceed 1500, leading to the constraint 2x1+x2≤ 1500 As

this inequality compares the amount of RM 1 used to the amount available, it is called a

material balance inequality All goods that lead to constraints in the model for the problem are called items The material balance equations or inequalities corresponding to the various

items are the constraints in the problem When the objective function and all the constraintsare obtained, the formulation of the problem as an LP is complete The LP formulation ofthe fertilizer product mix problem is given below

Maximizep(x) = 15x1+ 10x2 Itemsubject to 2x1+ x2≤ 1500 RM 1

This is another large class of problems in which LP is applied heavily Blending is

con-cerned with mixing different materials called the constituents of the mixture (these may

be chemicals, gasolines, fuels, solids, colors, foods, etc.) so that the mixture conforms tospecifications on several properties or characteristics

To model a blending problem as an LP, the linear blending assumption must hold for each

property or characteristic This implies that the value for a characteristic of a mixture isthe weighted average of the values of that characteristic for the constituents in the mixture,the weights being the proportions of the constituents As an example, consider a mixtureconsisting of four barrels of fuel 1 and six barrels of fuel 2, and suppose the characteristic

of interest is the octane rating (Oc.R) If linear blending assumption holds, the Oc.R of themixture will be equal to (4 times the Oc.R of fuel 1 + 6 times the Oc.R of fuel 2)/(4 + 6).The linear blending assumption holds to a reasonable degree of precision for many impor-tant characteristics of blends of gasolines, crude oils, paints, foods, and so on This makes itpossible for LP to be used extensively in optimizing gasoline blending, in the manufacture

of paints, cattle feed, beverages, and so on

The decision variables in a blending problem are usually either the quantities or theproportions of the constituents in the blend If a specified quantity of the blend needs to

be made, then it is convenient to take the decision variables to be the quantities of the

various constituents blended; in this case one must include the constraint that the sum ofthe quantities of the constituents is equal to the quantity of the blend desired.

If there is no restriction on the amount of blend made, but the aim is to find an optimumcomposition for the mixture, it is convenient to take the decision variables to be the propor-tions of the various constituents in the blend; in this case one must include the constraintthat the sum of all these proportions is 1

We provide a gasoline blending example There are more than 300 refineries in the UnitedStates processing a total of more than 20 million barrels of crude oil daily Crude oil is acomplex mixture of chemical components The refining process separates crude oil into itscomponents that are blended into gasoline, fuel oil, asphalt, jet fuel, lubricating oil, andmany other petroleum products Refineries and blenders strive to operate at peak economicefficiencies, taking into account the demand for various products To keep the example sim-ple, we consider only one characteristic of the mixture, the Oc.R of the blended fuels in thisexample In actual application there are many other characteristics to be considered also

A refinery takes four raw gasolines and blends them to produce three types of fuel Thecompany sells raw gasoline not used in making fuels at $38.95/barrel if its Oc.R is >90,

and at $36.85/barrel if its Oc.R is≤90 The cost of handling raw gasolines purchased and

blending them into fuels or selling them as is is estimated to be $2 per barrel by the CostAccounting Department Other data are given in Table 1.2

The problem is to determine how much raw gasoline of each type to purchase, the blend

to use for the three fuels, and the quantities of these fuels to make to maximize total dailynet profit

We will use the quantities of the various raw gasolines in the blend for each fuel as thedecision variables, and we assume that the linear blending assumption holds for the Oc.R.Let

RG i= raw gasoline typei to purchase/day, i = 1–4

x ij=



barrels of raw gasoline typei used in making fuel

typej per day, i = 1 to 4, j = 1, 2, 3

y i= barrels of raw gasoline typei sold as is/day

F j= barrels of fuel typej made/day, j = 1, 2, 3

So, the total amount of fuel type 1 made daily isF1=x11+x21+x31+x41 If this is>0, by

the linear blending assumption its Oc.R will be (68x11+ 86x21+ 91x31+ 99x41)/F1 This

is required to be ≥95 So, the Oc.R constraint on fuel type 1 can be represented by the

linear constraint: 68x11+ 86x21+ 91x31+ 99x41− 95F1≥ 0 Proceeding in a similar manner,

we obtain the following LP formulation for this problem

TABLE 1.2 Data for the Fuel Blending Problem

A diet has to satisfy many constraints; the most important is that it should be palatable (i.e.,

be tasty) to the one eating it This is a very difficult constraint to model mathematically,particularly if the diet is for a human individual So, early publications on the diet problemhave ignored this constraint and concentrated on meeting the minimum daily requirement(MDR) of each nutrient identified as being important for the individual’s well-being Also,these days most of the applications of the diet problem are in the farming sector, and farmanimals and birds are usually not very fussy about what they eat

The diet problem is one among the earliest problems formulated as an LP The first paper

on it was by Stigler (1945) Those were the war years, food was expensive, and the problem

of finding a minimum cost diet was of more than academic interest Nutrition science was

in its infancy in those days, and after extensive discussions with nutrition scientists, Stigleridentified nine essential nutrient groups for his model His search of the grocery shelvesyielded a list of 77 different available foods With these, he formulated a diet problem thatwas an LP involving 77 nonnegative decision variables subject to 9 inequality constraints.Stigler did not know of any method for solving his LP model at that time, but he obtained

an approximate solution using a trial and error search procedure that led to a diet meetingthe MDR of the nine nutrients considered in the model at an annual cost of $39.93 at

1939 prices! After Dantzig developed the simplex method for solving LPs in 1947, Stigler’sdiet problem was one of the first nontrivial LPs to be solved by the simplex method on acomputer, and it gave the true optimum diet with an annual cost of $39.67 at 1939 prices

So, the trial and error solution of Stigler was very close to the optimum

The Nobel prize committee awarded the 1982 Nobel prize in economics to Stigler for hiswork on the diet problem and later work on the functioning of markets and the causes andeffects of public regulation

TABLE 1.3 Data on the Nutrient Content of Grains

The data in the diet problem consist of a list of nutrients with the MDR for each; a list

of available foods with the price and composition (i.e., information on the number of units

of each nutrient in each unit of food) of every one of them; and the data defining any otherconstraints the user wants to place on the diet As an example we consider a very simplediet problem in which the nutrients are starch, protein, and vitamins as a group; the foodsare two types of grains with data given in Table 1.3

The activities and their levels in this model are: activityj: to include 1 kg of grain type j in

the diet, associated level =x j, forj = 1, 2 The items in this model are the various nutrients,

each of which leads to a constraint For example, the amount of starch contained in the diet

x is 5x1+ 7x2, which must be≥8 for feasibility This leads to the formulation given below.

Minimize z(x) = 0.60x1+0.35x2 Itemsubject to 5x1+ 7x2≥ 8 Starch

He always has his laptop with LP-based diet models for the various cattle and chicken feedformulations inside it He told me that before accepting an offer from a farm on raw materialsfor the feed, he always uses his computer to check whether accepting this offer would reducehis overall feed costs or not, using a sensitivity analysis feature in the LP software in hiscomputer He told me that this procedure has helped him save his costs substantially

An essential component of our modern life is the shipping of goods from where they areproduced to markets worldwide Nationally, within the United States alone transportation

of goods is estimated to cost over 1 trillion/year The aim of this problem is to find a way

of carrying out this transfer of goods at minimum cost Historically, it was among the firstLPs to be modeled and studied The Russian economist L V Kantorovitch studied thisproblem in the 1930s and developed the dual simplex method for solving it, and published

a book on it, Mathematical Methods in the Organization and Planning of Production, in

TABLE 1.4 Data for the Transportation Problem

Russian in 1939 In the United States, (Hitchcock, 1941) developed an algorithm similar tothe primal simplex algorithm for finding an optimum solution to the transportation prob-lem And (Koopmans, 1949) developed an optimality criterion for a basic solution to thetransportation problem in terms of the dual basic solution (discussed later on) The earlywork of L V Kantorovitch and T C Koopmans in these publications was part of theireffort for which they received the 1975 Nobel prize for economics

The classical single commodity transportation problem is concerned with a set of nodes

or places called sources that have a commodity available for shipment, and another set

of places called sinks or demand centers or markets that require this commodity The data consists of the availability at each source (the amount available there to be shipped out), the requirement at each market, and the cost of transporting the commodity per unit

from each source to each market The problem is to determine the quantity to be ported from each source to each market so as to meet the requirements at minimum totalshipping cost

trans-As an example, we consider a small problem where the commodity is iron ore, the sourcesare mines 1 and 2 that produce the ore, and the markets are three steel plants that requirethe ore Letc ij= cost (cents per ton) to ship ore from minei to steel plant j, i = 1, 2, j = 1,

2, 3 The data are given in Table 1.4 To distinguish between different data elements, weshow the cost data in normal size letters, and the supply and requirement data in bold faceletters

The decision variables in this model are: x ij= ore (in tons) shipped from mine i to

plantj The items in this model are the ore at various locations We have the following LP

formulation for this problem

LetG denote the directed network with the sources and sinks as nodes, and the various

routes from each source to each sink as the arcs Then this problem is a single ity minimum cost flow problem in G So, the transportation problem is a special case of

commod-single commodity minimum cost flow problems in directed networks Multicommodity flowproblems are generalizations of these problems involving two or more commodities.The model that we presented for the transportation context is of course too simple Realworld transportation problems have numerous complicating factors, both in the constraints

to be satisfied and the objective functions to optimize, that need to be addressed Starting

with this simple model as a foundation, realistic models for these problems are built bymodifying it, and augmenting as necessary.

Problems

The LP model finds many applications for making production allocation, planning, storage,and distribution decisions in companies Companies usually like to plan ahead; when theyare planning for one period, they usually like to consider also a few periods into the future.This leads to multiperiod planning problems

To construct a mathematical model for a multiperiod horizon, we need reliable data onthe expected production costs, input material availabilities, production capacities, demandfor the output, selling prices, and the like in each period With economic conditions changingrapidly and unexpectedly these days, it is very difficult to assemble reliable data on suchquantities beyond a few periods from the present That is why multiperiod models used inpractice usually cover the current period and a few periods following it, for which data can

be estimated with reasonable precision

For example, consider the problem of planning the production, storage, and marketing

of a product whose demand and selling price vary seasonally An important feature in thissituation is the profit that can be realized by manufacturing the product in seasons duringwhich the production costs are low, storing it, and putting it in the market when the sellingprice is high Many products exhibit such seasonal behavior, and companies and businessestake advantage of this feature to augment their profits A linear programming formulation

of this problem has the aim of finding the best production-storage-marketing plan overthe planning horizon, to maximize the overall profit For constructing a model for thisproblem we need reasonably good estimates of the demand and the expected selling price ofthe product in each period of the planning horizon; availability and cost of raw materials,labor, machine times, etc necessary to manufacture the product in each period; and theavailability, and cost of storage space

As an example, we consider the simple problem of a company making a product subject

to such seasonal behavior The company needs to make a production plan for the comingyear, divided into six periods of 2 months each, to maximize net profit (= sales revenue –production and storage costs) Relevant data are in Table 1.5 The production cost thereincludes the cost of raw material, labor, machine time, and the like, all of which fluctuatefrom period to period And the production capacity arises due to limits on the availability

of raw material and hourly labor

Product manufactured during a period can be sold in the same period, or stored and soldlater on Storage costs are $2/ton of product from one period to the next Operations begin

in period 1 with an initial stock of 500 tons of the product in storage, and the companywould like to end up with the same amount of the product in storage at the end of period 6

TABLE 1.5 Data for the 6-Period Production Planning Problem

Period Cost ($/Ton) Capacity (Tons) (Tons) Price ($/Ton)

The decision variables in this problem are, for periodj = 1–6

x j= product made (tons) during periodj

y j= product left in storage (tons) at the end of periodj

z j= product sold (tons) during periodj

In modeling this problem the important thing to remember is that inventory equations(or material balance equations) must hold for the product for each period For periodj this

equation expresses the following fact

Amount of product in storage

at the beginning of periodj +

the amount manufactured

storage at the end of periodj

The LP model for this problem is given below:

Maximize 180(z1+z2) + 250z3+ 270z4+ 300z5+ 320z6

− 20x1− 25x2− 30x3− 40x4− 50x5− 60x6

− 2(y1+y2+y3+y4+y5+y6)subject to x j , y j , z j ≥ 0 for all j = 1 to 6

to the material balance constraints of the type discussed in the example above for eachproduct and facility

In this section, we will extend the range of application of LP to include problems that can

be modeled as those of optimizing a convex piecewise linear objective function subject tolinear constraints These problems can be transformed easily into LPs in terms of additionalvariables This material is from Murty (under preparation)

Letθ(λ) be a real valued function of a single variable λ ∈ R1.θ(λ) is said to be a piecewise linear (PL) function if it is continuous and if there exists a partition of R1 into intervals

TABLE 1.6 The PL Functionθ(λ)

Interval forλ Slope Value ofθ(λ)

λ1≤ λ ≤ λ2 c2 θ(λ1 ) +c2 λ − λ1

λ2≤ λ ≤ λ3 c3 θ(λ2 ) +c3 λ − λ2

.

.

.

λ ≥ λ r c r+1 θ(λ r) +c r+1(λ − λ r

of the form [−∞, λ1] ={λ ≤ λ1}, [λ1, λ2], , [λr−1 , λ r], [λr , ∞] (where λ1< λ2< · · · < λ r

are the breakpoints in this partition) such that inside each interval the slope of θ(λ) is

a constant If these slopes in the various intervals are c1, c2, , c r+1, the values of thisfunction at various values ofλ are tabulated in Table 1.6.

This PL function is said to be convex if its slope is monotonic increasing with λ, that is, if

c1< c2· · · < c r+1 If this condition is not satisfied it is nonconvex Here are some numericalexamples of PL functions of the single variableλ (Tables 1.7 and 1.8).

Example 1.3: PL Functionθ(λ)

TABLE 1.7 PL Convex Functionθ(λ)

10≤ λ ≤ 25 5 30 + 5(λ − 10)

λ ≥ 25 9 105 + 9(λ − 25)

Example 1.4: PL Functiong(λ)

TABLE 1.8 Nonconvex PL Functionθ(λ)

Interval forλ Slope Value ofθ(λ)

100≤ λ ≤ 300 5 1000 + 5(λ − 100)

300≤ λ ≤ 1000 11 2000 + 11(λ − 300)

λ ≥ 1000 20 9700 + 20(λ − 1000)

Both functions θ(λ), g(λ) are continuous functions and PL functions θ(λ) is convex

because its slope is monotonic increasing, butg(λ) is not convex as its slope is not monotonic

increasing withλ.

A PL function h(λ) of the single variable λ ∈ R1 is said to be a PL concave function

iff −h(λ) is a PL convex function; that is, iff the slope of h(λ) is monotonic decreasing as

λ increases.

PL Functions of Many Variables

Letf(x) be a real valued function of x = (x1, , x n)T.f(x) is said to be a PL (piecewise

lin-ear) function ofx if there exists a partition of R ninto convex polyhedral regionsK1, , K r

such that f(x) is linear within each K t, for t = 1 to r; and a PL convex function if it is

also convex It can be proved mathematically that f(x) is a PL convex function iff there

exists a finite number,r say, of linear (more precisely affine) functions c t+c t x, (where c t,

andc t ∈ R nare the given coefficient vectors for thet-th linear function) t = 1 to r such that

0+c t x, t = 1 to r PL convex functions of many variables

that do not satisfy the additivity hypothesis always appear in this form (Equation 1.3) inreal world applications

Similarly, a PL function h(x) of x = (x1, , x n)T is said to be a PL concave function if

there exist a finite numbers of affine functions d t

0+d t x, t = 1 to s, such that h(x) is their pointwise infimum, that is, for each x ∈ R n

h(x) = Minimum{d t

0+d t x : t = 1, , s}

Now we show how to transform various types of problems of minimizing a PL convexfunction subject to linear constraints into LPs, and applications of these transformations

Minimizing a Separable PL Convex Function Subject to Linear Constraints

A real valued functionz(x) of variables x = (x1, , x n)T is said to be separable if it satisfies

the additivity hypothesis, that is, if it can be written as the sum of n functions, each

one involving only one variable as in: z(x) = z1(x1) +z2(x2) +· · · + z n(xn) Consider thefollowing general problem of this type:

As the objective function to be minimized does not satisfy the proportionality assumption,this is not an LP However, the convexity property can be used to transform this into an LP

by introducing additional variables This transformation expresses each variablex j as a sum

ofr j+ 1 variables, one associated with each interval in which its slope is constant

Denot-ing these variables byx j1 , x j2 , , x j,r j+1, the variablex j becomes =x j1+· · · + x j,r j+1 and

z j(xj) becomes the linear functionc j1 x j1+· · · + c j,r j+1x j,r j+1in terms of the new variables.The reason for this is that as the slopes are monotonic (i.e., c j1 < c j2 < · · · < c j,r j+1), forany value of ¯x j ≥ 0, if (¯x j1 , ¯x j2 , , ¯x j,r j+1) is an optimum

Minimize c j1 x j1+· · · + c j,r j+1x j,r j+1

Subject to x j1+· · · + x j,r j+1= ¯x j

0≤ x jt <  jt , t = 1, , r j+ 1

TABLE 1.9 The PL FunctionZ j(x j)

Interval Slope in Interval Value ofz j(x j) Length of Interval

k j1 ≤ x j ≤ k j2 c j2 z j(k j1) +c j2(x j − k j1)  j2=k j2 − k j1

.

.

.

.

x j,t+1 will not be positive unless ¯x jk= jk for k = 1 to t, for each t = 1 to r j Hence theoptimum objective value in this problem will be equal toz j(xj) This shows that our originalproblem (Equation 1.4) is equivalent to the following transformed problem which is an LP

Minimize

j=n j=1

Each input is available from one or more sources The company has its own quarry for

LI, which can supply up to 250 units/day at a cost of $20/unit Beyond that, LI can bepurchased in any amounts from an outside supplier at $50/unit EP is only available fromthe local utility Their charges for EP are: $30/unit for the first 1000 units/day, $45/unit for

up to an additional 500 units/day beyond the initial 1000 units/day, $75/unit for amountsbeyond 1500 units/day Up to 800 units/day of water is available from the local utility at

$6/unit; beyond that they charge $7/unit of water/day There is a single supplier for F whocan supply at most 3000 units/day at $40/unit; beyond that there is currently no supplierfor F From their regular workforce they have up to 640 man hours of labor/day at $10/manhour; beyond that they can get up to 160 man hours/day at $17/man hour from a pool ofworkers

They can sell up to 50 units ofP1 at $3000/unit/day in an upscale market; beyond thatthey can sell up to 50 more units/day ofP1 to a wholesaler at $250/unit They can sell

up to 100 units/day ofP2 at $3500/unit They can sell any quantity ofP3 produced at aconstant rate of $4500/unit

Data on the inputs needed to make the various products are given in Table 1.10 Formulatethe product mix problem to maximize the net profit/day at this company

Maximizing the net profit is the same as minimizing its negative, which is = (the costs

of all the inputs used/day)− (sales revenue/day) We verify that each term in this sum

TABLE 1.10 I/O Data

Input Units/Unit Made

is a PL convex function So, we can model this problem as an LP in terms of variablescorresponding to each interval of constant slope of each of the input and output quantities.LetLI, EP , W , F , L denote the quantities of the respective inputs used/day; and P1,

P2,P3denote the quantities of the respective products made and sold/day LetLI1andLI2

denote the units of limestone used daily from own quarry and outside supplier Let EP1,

EP2, andEP3denote the units of electricity used/day at $30, 45, 75/unit, respectively Let

W1 andW2 denote the units of water used/day at rates of $6, 7/unit, respectively LetL1

andL2 denote the man hours of labor used/day from regular workforce, pool, respectively.LetP11 and P12 denote the units ofP1 sold at the upscale market and to the wholesaler,respectively

Then the LP model for the problem is:

Minimize z = 20LI1+ 50LI2+ 30EP1+ 45EP2+ 75EP3+ 6W1+ 7W2+ 40F + 10L1

As discussed above, a PL convex function in variables x = (x1, , x n)T can be expressed

as the pointwise maximum of a finite set of linear functions Minimizing a function likethat is appropriately known as a min–max problem Similarly, a PL concave function inx

can be expressed as the pointwise minimum of a finite set of linear functions Maximizing

a function like that is appropriately known as a max–min problem Both min–max andmax–min problems can be expressed as LPs in terms of just one additional variable

If the PL convex function f(x) = min{c t

0+c t x : t = 1, , r}, then −f(x) = max{−c t

0−

c t x : t = 1, , r} is PL concave and conversely Using this, any min–max problem can be

posed as a max–min problem and vice versa So, it is sufficient to discuss max–min problems.Consider the max–min problem

linear constraints is:

The fact that x n+1 is being maximized and the additional constraints together imply

that if (¯x, ¯x n+1) is an optimum solution of this LP model, then ¯x n+1= min{c1+c1x, , c¯ r

0+

c r x} = z(¯x), and that ¯x¯ n+1is the maximum value ofz(x) in the original max–min problem.

Example 1.6: Application in Worst Case Analysis

Consider the fertilizer maker’s product mix problem with decision variablesx1andx2

(Hi-ph, Lo-ph fertilizers to be made daily in the next period) discussed in Example 1.2, tion 1.2 There we discussed the case where the net profit coefficients c1 and c2 of thesevariables are estimated to be $15 and $10, respectively In reality, the prices of fertilizersare random variables that fluctuate daily Because of unstable conditions, and new agri-cultural research announcements, suppose market analysts have only been able to estimatethat the expected net profit coefficient vector (c1, c2) is likely to be one of {(15, 10),

Sec-(10, 15), (12, 12)} without giving a single point estimate So, here we have three possible

scenarios In scenario 1, (c1, c2) = (15, 10), expected net profit = 15x1+ 10x2; in scenario

2, (c1, c2) = (10, 15), expected net profit = 10x1+ 15x2; in scenario 3 (c1, c2) = (12, 12),expected net profit = 12x1+ 12x2 Suppose the raw material availability data in the prob-lem is expected to remain unchanged The important question is: which objective function

to optimize for determining the production plan for the next period Irrespective of which of the three possible scenarios materializes, at the worst the minimumexpected net profit of the company will be p(x) = min {15x1+ 10x2, 10x1+ 15x2, 12x1+

12x2} under the production plan x = (x1, x2)T Worst case analysis is an approach that

advocates determining the production plan to optimize this worst case net profitp(x) in

this situation This leads to the max–min model: maximizep(x) = min {15x1+ 10x2, 10x1+15x2, 12x1+ 12x2} subject to the constraints in Equation 1.2 The equivalent LP model cor-

responding to this is:

where the weights w1, , w r are all strictly positive In this problem the objective tion to be minimized, z(x), is a PL convex function; hence this problem can be trans-

func-formed into an LP To transform, define for eacht = 1 to r two new nonnegative variables

t) Using this, we can

trans-form the above problem into the following LP:

Results, an Example from Container Shipping

To get good results from a linear programming application, it is very important to develop

a good model for the problem being solved There may be several ways of modeling theproblem, and it is very important to select the one most appropriate to model it intelligently

to get good results Skill in modeling comes from experience; unfortunately there is no theory

to teach how to model intelligently We will now discuss a case study of an applicationcarried out for routing trucks inside a container terminal to minimize congestion Threedifferent ways of modeling the problem have been tried The first two approaches lead to(1) an integer programming model and (2) a large-scale multicommodity flow LP model,respectively Both these models gave very poor results The third and final model developeduses a substitute objective function technique; that is, it optimizes another simpler objectivefunction that is highly correlated to the original, because that other objective function ismuch easier to control This approach led to a small LP model, and gives good results.Today most of the nonbulk cargo is packed into steel boxes called containers (typically

of size 40× 8 × 9 in feet) and transported in oceangoing vessels A container terminal in a

port is the place where these vessels dock at berths for unloading of inbound containers andloading of outbound containers The terminals have storage yards for the temporary storage

of these containers The terminal’s internal trucks (TIT) transport containers between theberth and the storage yard (SY) The SY is divided into rectangular areas called blocks, eachserved by one or more cranes (rubber tired gantry cranes, or RTGC) to unload/load con-tainers from/to trucks Customers bring outbound containers into the terminal in their owntrucks (called external trucks, or XT), and pick up from the SY and take away their inboundcontainers on these XT Each truck (TIT or XT) can carry only one container at a time.The example (from Murty et al., 2005a,b) deals with the mathematical modeling of theproblem of routing the trucks inside the terminal to minimize congestion We represent

the terminal road system by a directed network G = (N , A) where N is the set of nodes

(each block, berth unloading/loading position, road intersection, terminal gate is a node),

A is the set of arcs (each lane of a road segment joining a pair of nodes is an arc) Each

(berth unloading/loading position, block), (block, berth unloading/loading position), (gate,block), (block, gate) is an origin–destination pair for trucks that have to go from the origin

to the destination; they constitute a separate commodity that flows inG Let T denote the

number of these commodities Many terminals use a 4-hour planning period for their truckrouting decisions

Let f = (f r

ij) denote the flow vector of various commodities on G in the planning

period, where f r

ij= expected number of trucks of commodity r passing through arc (i, j)

in the planning period for r = 1 to T , and (i, j) ∈ A Let θ = max T

r=1 f r

ij: (i, j) ∈ A ,

μ = min T

r=1 f r

ij: (i, j) ∈ A Then eitherθ or θ − μ can be used as measures of

conges-tion onG during the planning period, to optimize.

As storage space allocation to arriving containers directly determines how many truckstravel between each origin–destination pair, the strategy used for this allocation plays acritical role in controlling congestion This example deals with mathematical modeling ofthe problem of storage space allocation to arriving containers to minimize congestion.Typically, a block has space for storing 600 containers, and a terminal may have 100(some even more) blocks At the beginning of the planning period, some spaces in the

SY would be occupied by containers already in storage, and the set of occupied storagepositions changes every minute; it is very difficult to control this change Allocating a specificopen storage position to each container expected to arrive in the planning period has beenmodeled as a huge integer program, which takes a long time to solve In fact, even beforethis integer programming model is entered into the computer, the data change So thesetraditional integer programming models are not only impractical but also inappropriate forthe problem

So a more practical way is to break up the storage space allocation decision into two stages:

Stage 1 determines only the container quota x i, for each block i, which is the number of

newly arriving containers that will be dispatched to blocki for storage during the planning

period Stage 1 will not determine which of the specific arriving containers will be stored

in any block; that decision is left to Stage 2, which is a dispatching policy that allocateseach arriving container to a specific block for storage at the time of its arrival, based onconditions prevailing at that time So, Stage 2 makes sure that by the end of the planningperiod the number of new containers sent for storage to each block is its quota numberdetermined in Stage 1, while minimizing congestion at the blocks and on the roads.Our example deals with the Stage 1 problem The commonly used approach is based on abatch-processing strategy Each batch consists of all the containers expected to arrive/leave

at each node during the planning period At the gate, this is the number of outboundcontainers expected to arrive for storage At a block it is the number of stored containersexpected to be retrieved and sent to each berth or the gate At each berth it is the number

of inbound containers expected to be unloaded to be sent for storage to SY With thisdata, the problem can be modeled as a multicommodity network flow problem It is a large-scale LP with many variables and thousands of constraints However, currently available

LP software systems are fast; this model can be solved using them in a few minutes ofcomputer time

But the output from this model turned out to be poor, as the model is based solely onthe total estimated workload during the planning period Such a model gives good resultsfor the real problem only if the workload in the terminal (measured in number of containers