Optimization Techniques

1 1 WEB CHAPTERWEB CHAPTER PREVIEWN ormativeeconomic decision analysis involves determiningthe action that best achieves a desired goal or objec-tive. This means finding the action that optimizes(that is, maximizes or minimizes) the value of anobjective function. For example, in a price-outputdecision-making problem, we may be interested indetermining the output level that maximizes a production problem, the goal may be to find thecombination of inputs (resources) that minimizesthe cost of producing a desired level of output. In acapital budgeting problem, the objective may be toselect those projects that maximize the net presentvalue of the investments chosen. There are manytechniques for solving Optimization problems suchas these. This chapter (and appendix) focuses on theuse of differential calculus to solve certain types ofoptimization problems. In Web Chapter B, linear-programming Techniques , used in solving con-strained Optimization problems, are Techniques are a powerful set of toolsthat are important in efficiently managing an enter-prise s resources and thereby maximizing share-holder TechniquesPHOTO ON PAGE 1 AND 2: DANIEL CHAPTER AOptimization TechniquesTYPES OFOPTIMIZATIONTECHNIQUESIn Chapter 1 we defined the general form of a problem that managerial economics at-tempts to analyze.

2 The basic form of the problem is to identify the alternative means ofachieving a given objective and then to select the alternative that accomplishes the ob-jective in the most efficient manner, subject to constraints on the means. In program-ming terminology, the problem is optimizing the value of some objective function, sub-ject to any resource and/or other constraints such as legal, input, environmental, andbehavioral 1990 the Air Force publicly unveiled itsnewest long-range strategic bomber the B-2 or Stealth bomber. This plane is characterized by aunique flying wing design engineered to evadedetection by enemy radar. Theplane has been controversialbecause of its high cost. However,a lesser-known controversyrelates to its fundamental plane s flying wingdesign originated from a secretstudy of promising military tech-nologies that was undertaken atthe end of World War II. Thegroup of prominent scientistswho undertook the study con-cluded that a plane can achievemaximum range if it has a designin which virtually all the volumeof the plane is contained in thewing.

3 A complex mathematicalappendix was attached to thestudy that purported to showthat range could be maximizedwith the flying wing , a closer examination of the technicalappendix by Joseph Foa, now an emeritus professorof engineering at George Washington University,discovered that a fundamental error had been madein the initial report. It turned out that the originalresearchers had taken the first derivative of a com-plex equation for the range of a plane and foundthat it had two solutions. The original researchersmistakenly concluded that theall-wing design was the one thatmaximized range, when, in fact,it this chapter we introducesome of the same optimizationtechniques applied to an analy-sis of the Stealth bomber develop tools designed tomaximize profits or minimizecosts. Fortunately, the mathe- matical functions we deal within this chapter and throughoutthe book are much simpler thanthose that confronted the origi-nal flying wing engineers.

4 Weintroduce Techniques that can beused to check whether a func-tion, such as profits or costs, isbeing minimized or maximizedat a particular level of Managerial Challenge is based primarily on W. Biddle, Skeleton Alleged in the Stealth Bomber s Closet, Science,12 May 1989, pp. 650 CHALLENGEA Skeleton in the Stealth Bomber s Closet1 WEB CHAPTER AOptimization Techniques3 Mathematically, we can represent the problem asOptimize y f(x1, x2, .. , xn)[ ]subject to gj(x1, x2, .. , xn) bjj 1, 2, .. , m[ ]where Equation is the objective function and Equation constitutes the set of con-straints imposed on the solution. The xivariables, x1,x2, .., xn, represent the set of de-cision variables, and y f(x1, x2, .., xn) is the objective function expressed in terms ofthese decision variables. Depending on the nature of the problem, the term optimizemeans either maximizeor minimizethe value of the objective function. As indicated inEquation , each constraint can take the form of an equality ( ) or an inequality ( or ) Factors in OptimizationSeveral factors can make Optimization problems fairly complex and difficult to such complicating factor is the existence of multiple decision variablesin a simple procedures exist for determining the profit-maximizing output levelfor the single-product firm.

5 However, the typical medium- or large-size firm often pro-duces a large number of different products, and as a result, the profit-maximizationproblem for such a firm requires a series of output decisions one for each product. An-other factor that may add to the difficulty of solving a problem is the complex nature ofthe relationships between the decision variables and the associated example,in public policy decisions on government spending for such items as education, it is ex-tremely difficult to determine the relationship between a given expenditure and the ben-efits of increased income, employment, and productivity it provides. No simple rela-tionship exists among the variables. Many of the Optimization Techniques discussed hereare only applicable to situations in which a relatively simple function or relationship canbe postulated between the decision variables and the outcome variable. A third compli-cating factor is the possible existence of one or more complex constraints on the example, virtually every organization has constraints imposed on its deci-sion variables by the limited resources such as capital, personnel, and facilities overwhich it has control.

6 These constraints must be incorporated into the decision , the Optimization Techniques that are applied to the problem may yield a so-lution that is unacceptable from a practical standpoint. Another complicating factor isthe presence of uncertaintyor this chapter, we limit the analysis to decision mak-ing under certainty,that is, problems in which each action is known to lead to a specificoutcome. Chapter 2 examines methods for analyzing decisions involving risk and un-certainty. These factors illustrate the difficulties that may be encountered and may ren-der a problem unsolvable by formal Optimization versus Unconstrained OptimizationThe mathematical Techniques used to solve an Optimization problem represented byEquations and depend on the form of the criterion and constraint functions. Thesimplest situation to be considered is the unconstrainedoptimization problem. In such aproblem no constraints are imposed on the decision variables, and differential calculuscanbe used to analyze them.

7 Another relatively simple form of the general Optimization prob-lem is the case in which all the constraints of the problem can be expressed as equality 4 WEB CHAPTER AOptimization Techniques ( ) relationships. The technique of Lagrangian multiplierscan be used to find the opti-mal solution to many of these , however, the constraints in an economic decision-making problem take theform of inequalityrelationships ( or ) rather than equalities. For example, limitationson the resources such as personnel and capital of an organization place an upperboundor budget ceiling on the quantity of these resources that can be employed in max-imizing (or minimizing) the objective function. With this type of constraint, all of agiven resource need not be used in an optimal solution to the problem. An example of alower boundwould be a loan agreement that requires a firm to maintain a current ratio(that is, ratio of current assets to current liabilities) of at least Any combination ofcurrent assets and current liabilities having a ratio greater than or equal to wouldmeet the provisions of the loan agreement.

8 Such Optimization procedures as the La-grangian multiplier method are not suited to solving problems of this type efficiently;however, modern mathematical programming Techniques have been developed that canefficiently solve several classes of problems with these inequality constitute the most important class for which efficientsolution Techniques have been developed. In a linear-programming problem, both the ob-jective and the constraint relationships are expressed as linear functions of decision classes of problems include integer-programmingproblems, in which some(or all) of the decision variables are required to take on integer values, and quadratic-programmingproblems, in which the objective relationship is a quadratic function of thedecision computing algorithms exist for solving optimizationproblems that meet these remainder of this chapter deals with the classical Optimization procedures of dif-ferential calculus. Lagrangian multiplier Techniques are covered in the Appendix.

9 Linearprogramming is encountered in Web Chapter : OPTIMIZINGFLIGHTCREWSCHEDULES ATAMERICANAIRLINES4 One problem that faces major airlines, such as American Airlines, is the development ofa schedule for airline crews (pilots and flight attendants) that results in a high level ofcrew utilization. This scheduling problem is rather complex because of Federal AviationAdministration (FAA) rules designed to ensure that a crew can perform its duties with-out risk from fatigue. Union agreements require that flight crews receive pay for a con-tractually set number of hours of each day or trip. The goal for airline planners is to con-struct a crew schedule that meets or exceeds crew pay guarantees and does not violateFAA rules. With the salary of airline captains at $140,000 per year or more, it is impor-tant that an airline make the maximum possible usage of its crew personnel. Crew costsare the second largest direct operating cost of an nature of the constrained Optimization problem facing an airline planner is tominimize the cost of flying the published schedule,subject to the following flight is assigned only one linear relationship of the variables x1, x2.

10 , xnis a function of the form:a1x1 a2x2 .. anxnwhere all the xvariables have exponents of quadratic function contains either squared terms (xi2) or cross-product terms (xixj).4 Ira Gershkoff, Optimizing Flight Crew Schedules, Interfaces,July/August 1989, pp. 29 ://Decision science modeling at American Airlines isdescribed CHAPTER AOptimization pairing of a crew and a flight must begin and end at a home crew base, such as Chicago or Dallas for pairing must be consistent with union work rules and FAA number of jobs at each crew base must be within targeted minimum andmaximum limits as specified in American s personnel was able to develop a sophisticated constrained Optimization model that saved$18 million per year compared with previous crew allocation models Chapter 2, marginal analysis was introduced as one of the fundamental concepts ofeconomic decision making. In the marginal analysis framework, resource-allocation de-cisions are made by comparing the marginal benefits of a change in the level of an ac-tivity with the marginal costs of the change.