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SciAm_excerpt - ALL RIGHTS RESERVED Roam Glam COPYRIGHT ©...

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Unformatted text preview: . ALL RIGHTS RESERVED Roam Glam; COPYRIGHT © 1981 BY excefifl’s Ptorw an Article frOm JUNE. I981 VOL. 244. SCIENTIFIC AMERICAN NO. 6 The Allocation of Resources by Linear Programming Abstract, crystal-like Structures in many geometrical dimensions can help to solve problems in planning and management. A new algorithm has set upper limits on the complexity ofsucli problems ( :onsider the situation of a small brewery whOse ale and beer are always in demand but whose pro- duction is limited by certain raw materi- als that are in short supply. Suppose the scarce ingredients are com, hops and barley malt. The recipe for a barrel of ale calls for the ingredients in propor— tions different from those in the recipe for a barrel of beer. For instance, ale requires more malt per barrel than beer does. Furthermore, the brewer sells ale at a profit of $ 13 per barrel and beer at a profit of $23 per barrel. Subject to these conditions, how can the brewer maxi— mize his profit? It may seem that the brewer‘s best plan would be to devote all his resources to the production of beer, his more prof- itable product. This choice may not be well advised, however, because making beer may consume some of the avail~ able resources much faster than making ale does. If five pounds of corn are re- quired for brewing a barrel of ale and 15 pounds are needed for brewing a barrel of beer, it may be possible to make three times as much ale as beer. Moreover, in brewing only beer the brewer may find that all his corn is used up long before his supplies of hops and malt are ex- hausted. It may turn out that by produc- ing some beer and some ale he can take better advantage of his resources and thereby increase his profit. Determining such an optimum production program is not a trivial problem. It is the kind of problem that can be solved by the tech- nique of linear programming. Linear programming is a mathemati~ cal field of study concerned with the ex- plicit formulation and analysis of such questions. It is a part of the broader field of inquiry called operations research, in which various methods of mathematical modeling and quantitative analysis are applied to large organizations and un- dertakings. Linear programming was developed shortly after World War H in response to logistical problems that 126 by Robert G. Bland arose during the war and immediately after it. One of the earliest publications on the uses of linear programming dis- cussed a model of the 1948 Berlin airlift. Although the computer is an indis~ pensable tool for solving problems in linear programming, the term "pro- gramming” is employed in the sense of planning, not computer programming. “Linear” refers to a mathematical prop- erty of certain problems that simplifies their analysis. in the brewery problem the amount of any one resource needed to make either ale or beer is assumed to be proportional to the amount of the beverage produced. Doubling the amount of beer doubles the amount of each ingredient required for the brewing of beer, and it also doubles the prof- it attributable to the sale of beer. If the amount of corn consumed in mak» ing beer is plotted as a function of the amount of beer produced, the graph is a straight line. In order to apply the tech- niques of linear programming one must also assume that products and resources are divisible, or at least approximately so. For example, half a barrel of beer can be produced, and it has half the val- ue of a full barrel. Problems in linear programming are generally concerned with the al- location ol scarce resources among a number of products or activities, under the proportionality and divisibility con— ditions I have described. The scarce re- sources may be raw materials, partly finished products, labor, investment capital or processing time on large, ex- pensive machines. An optimum alloca— tion may be one that maximizes some measure of benefit or utility, such as profit, or minimizes some measure of cost. In this era of declining productivi- ty and dwindling resources a technique that can aid in allocating resources with the greatest possible efficiency may be well worth examining. The properties of linear-programming problems derive from elementary prin- ciples of algebra and geometry. Solving such problems efliciently depends on algorithms. or step-by-step procedures, that cleverly exploit these algebraic and geometrical principles. The algorithms are also simple in conception, although the details of their Operation can be quite intricate. It is the efficiency and versatility of a single algorithm called the simplex method that is largely re- sponsible for the economic importance of linear programming. The simplex method was introduced in 194'}r by George B. Dantzig, who is now at Staaford University. It has sub- stantial value because it is fast, it is rich in applications and it can answer impor- tant questions about the sensitivity of solutions to variations in the input data. Such questions might take the follow- ing form: How should the production plan of the brewer change if the avail- ability of hops or the profitability of beer is altered? How much should he be willing to pay for additional supplies of the scarce resources? What price should he ask from another entrepreneur who wants to buy some of his supplies? The simplex method can help in determin- ing whether to buy a machine or sell one, whether to borrow money or lend it and whether to pay overtime wages or discontinue overtime labor. With the simplex method one can also im- pose additional constraints and solve the problem again to examine their effect. For example, the method can quickly tell an entrepreneur the cost of provid- ing an unprofitable service in order to maintain the goodwill of a customer. The simplex method has proved ex~ tremer efiicient in solving complex lin— ear-programming problems with thou- sands of constraints. For theoreticians, however, its Speed in solving such prob- lems has been somewhat puzzling. There is no definitive explanation of why the method does so well. Indeed. there are problems devised by mathe‘ maticians for which the simplex method is intolerably slow For reasons that are not entirely clear. such problems do not seem to arise in practice. Mathematicians in the U.S.S.R. have recently developed a new algorithm for linear programming that in a certain sense avoids some of the theoretical dif- ficulties that have been attributed to the simplex method. The development was reported in front-page articles in news- papers throughout the world, which sug— gests the economic significance of linear programming today. Unfortunately the new algorithm. which is called the ellip- soid method. has so far shown little prospect of outperforming the simplex method in practice. For now there is a curious divergence of practical and theoretical measures of computational performance. Even when an efficient algorithm is employed. the setup costs of solving a large linear—programming problem can be considerable. Expressing a practical set of circumstances in terms of linear programming is not a trivial enterprise. and neither is the collection and organi- zation of the data describing the circum- stances. Moreover, solving the problem is feasible only with the aid of a high- speed computer. Nevertheless. the economic benefits SIMPLEX METHOD of solving problems in linear programming finds an optimum allocation of resources by moving from one vertex to another along the edges of a polytope, a three-dimensional solid whose faces are polygons. Each point in the region vrbere the polytope is constructed corresponds to a particular program, or plan, for allo- cating labor, capital or other resources. Associated with each such program is a net benefit or a net cost. The object of linear program- ming is to find the program with the maximum benefit or the mini- mum coat. The polytope define a region of feasibility: all programs of allocation repreSented by a point within the polytope or on its sur- face are feasible, whereas those represented by a point outside the polytope are infeasible because of a scarcity of resources. “hen the relation between resources and benefits or costs is linear, the maxi- mum and minimum values must lie at one of the vertexes of the poly- tope. The simplex algorithm examines the vertexw selectively, trac- ing a path (ABC. . . M) along the edges of the polytope. At each step along the path the measure of benefits or costs is improved until at the point M the maximum or minimum is reached. Often there are many paths from A, the starting vertex, to M. The polytope need not be three-dimensional and it commonly has thousands of dimensions. 12'.lr LABORATORY 6 7 B \I \ ““““‘ BUILDING A BUILDING A BUILDING B BUILDING C BUILDING A BUILDING B BUILDING C ASSIGNIHEZVT PROBLEM seeks to minimize the cost of matching buildings available for renovation with functions that must each be confined to one building. With three buildings and three functions there are 32, or nine. renovation costs that must be considered. The costs [which are given here in millions of dollars) are conveniently represented in a matrix of numbers: a feasible assignment picim one 123 LIBRARY TENNIS COURTS 1 6 2 2 5 4 BUILDING C 2+5+6=14 number from each row and from each column of the matrix. There are 3 X 2 X I. or six, ways of accomplishing this. In general, for as- signments of size n-by-n. the number of assignments is n factorial (written III}. which is equal to n multiplied by all positive integers less than it. Because the value of it! increases rapidly. determining the min- imum cost by enumeration at all possible assignments is impractical. of linear programming are often sub— stantial. In the mid-l950‘s. when linear- programming methods were developed to guide the blending of gasoline, the Exxon Corporation began saving from 2 to 3 percent of the cost of its blend— ing operations. The application soon spread within the petroleum industry to the control of additional refinery opera— tions, including catalytic cracking, dis— tillation and polymerization. At about the same time other industries, notably paper products, food distribution, agri- culture, steel and metalworking, began to adopt linear programming. Charles Boudrye of Linear Programming, Inc, of Silver Spring, Md, estimates that a paper manufacturer increased its profits by $15 million in a single year by em- ploying linear programming to deter- mine its assortment of products. Today “packages” of computer pro- grams based on the simplex algorithm are offered commercially by some 10 companies. Roughly 1,000 customers make use of the packages under licenses from the developers. Each customer pays a sizable monthly fee, and so it is likely that each makes use of the meth- od regularly. Many more organizations have access to the packages through consultants. In addition special-purpose programs for solving problems of flows in networks have been developed, and they may be in even wider service than the general-purpose linear-program- ming algorithms. The broad applicability of linear pro- gramming can be illustrated even with- in a single organization. Exxon current- ly applies linear programming to the scheduling of drilling operations, to the allocation of crude oil among refineries, to the setting of refinery operating con- ditions, to the distribution of products and to the planning of business strategy. David Smith of the Cemmunication and Computer Sciences Department of Ex- xon reports that linear programming and its extensions account for from 5 to 10 percent of the company's total com- puting load. This share has kept pace for the past 20 years with rapidly expand- ing general applications of information processing. Athough the simplex method is a pow- erful tool, it is founded on elemen- tary ideas. in order to understand some of these ideas it is useful to examine a specially structured allocation problem called the assignment problem. Consid- er the situation of a university planning committee with three buildings avail- able for renovation and three functions the buildings are to serve. Suppose the functions are those of a laboratory, a. library and indoor tennis courts, so that there can be only one function to a building. The tables in the illustration on the opposite page indicate the cost of renovation for each of the nine possi- ble matches of a building with a func~ tion. How can the cummittee minimize the renovation cost? The problem can be solved by mark- ing three of the squares in the 3-by-3 table. Precisely one Square must be marked in each row and one must be marked in each column, so that each building has a function and all the func- tions are accommodated. The solution being sought is the one in which the sum of the costs in the marked squares is as SUB‘i'HACT 4 SUBTHACT 6 small as possible. Finding an optimum solution is not dilficult because there are only a few possible ways of marking the squares. Once one of the three squares in the first row is marked, only two squares in the second row remain available for marking. In the third row the choice is forced, because only one square appears in a heretofore unmarked column. Hence there are 3 X 2 X l, or six, ways of making the assignment. It is an easy matter to enumerate all six possibilities, SUBTFIACT 7 SUBTFlACT 1 SUBTRACT 6 0’ SUBTFtACTS 4:- SUBTHACTZ SUBTFIACT 2 SUBTRACT 5 ‘0 SUBTRACT 5 SU BTFIACT 6 SUBYHACT 4 SUBTRACT 8 SUBTFIACT 5 SUBTRACT 3 SUBTHACT 1 SUBTRACT 0 SUBTRACT 7 CONVINCING A SKEP‘I'IC that an assignment is optimum (colored arms in upper illustra- tion) does not entail evaluating the cost of all the feasible assignments. Here the cost matrix is n lO-by-lfl one, and so there are 10!, or 3,628,800, possible assignments. If a number is subtract- ed from each entry In a row or column, the number is also subtracted from the total cost of any possible assignment null the relative costs remain unchanged. The reason is that every assign- ment picks exactly one number from the transformed row or column of the matrix. The set ot numbers to be subtracted can be chosen so that the original matrix is transformed into n matrix that has no negative entries Ind has at least one entry with a cost of zero in each row and col- umn (loner illustration). Becattse no assignment can have a total cost less than zero the opti- mum nssignment for the transformed matrix must be one that has 1 total cost of zero. It lol- lows that the entries in the corresponding positions in the original matrix are also optimum. Efficient algorithms have been devised for generating the set of numbers to be subtracted. I29 CORN 480 POUNDS HOPS 160 OUNCES BARLEY MALT 1,190 POUNDS 5 POUNDS CORN 15 POUNDS CORN 4 OUNCES HOPS 35 POUNDS MALT 4 OUNCES HOPS 20 POUNDS MALT $13 PROFIT $23 PROFIT MOST PROF ITABLE PRODUCT MIX BREWER’S DILEMMA illustrates an application of linear programming to problems of op- timizing the apportionment of resources among various products. The brewer’s production of beer and ale is limited by the scarcity of three essential ingredients: corn, hops and malt. Feasi- ble production levels are determined not only by the total amount of each ingredient on hand but also by the proportions of the ingredients required to make the two products. The objective function, or the quantity to be optimized, is the brewer’s profit. In linear programming all the resources, all the products and all the benefits are assumed to be divisible quantities: the brewer can use half a pound of corn, sell a fourth of a barrel of ale and realize proportional profits. \ 130 evaluate the total cost of each one and select the least expensive assignment. This enumerative approach solves the problem of a 3-by-3 matrix, but it be- comes impractical for larger problems. Suppose there are four buildings and four functions; the number of possible assignments is then 4 X 3 X 2 X 1, or 24. In the general statement of the prob- lem there are 71 buildings and n func- tions, and the number of assignments is n factorial (written n!), which signifies n .multiplied by all the integers from 1 to n — 1. For n = 10 there are 10!, or more than 3.6 million, distinct assignments. The rapid growth of n! dispels any en- thusiasm one might have for the enu- merative method. Suppose one had to solve a 35-by-35 assignment problem by enumeration and one had a computer that could sort through the possible as- signments, evaluate the cost of each one and compare it with the lowest-cost as- signment encountered up to that point at a rate of a billion assignments per sec- ond. (A computer capable of this speed would be much faster than any availa- ble now.) Even if the task of enumerat- ing the 3 5! assignments were to be shared by a billion such computers, only an in- significant fraction of the required com- putations would be completed after a billion years. The.3S-by~35 assignment problem is not large. If the problem were one of assigning personnel to jobs in order to minimize the total cost of job training, It might well be equal to 35. There are nu- merous other assignment problems in which n is equal to 1,000 or more. Clear- ly such problems require a procedure cleverer than enumeration. he burden of enumeration might be greatly reduced if one could avoid examining assignments that are costlier than the ones already checked. This ef- fect would be achieved if there were a stopping rule or optimality criterion that would allow an optimum assign- ment to be recognized quickly once it was encountered. Any algorithm that in- corporates such a criterion offers impor- tant collateral benefits. The benefits are summarized by what Jack Edmonds of the University of Waterloo in Ontario calls “the principle of the absolute su- pervisor,” or what might also be called “the problem of the skeptical boss.” Suppose after tedious enumeration you have solved the 10-by-10 assign- ment problem indicated in the upper il- lustration on the preceding page by ex- amining all 3,628,800 possible assign- ments. The optimum assignment, you maintain, is the one shaded in color in the illustration. You present the solu- tion to your boss. who looks you in the eye, pufls on hafcigar and demands, “How do I know there is no less costly solution?” You might swallow hard at this question, because it seems the only way to demonstrate the merits of your soiution is to repeat the examination of all 3,628,800 possibilities under the scrutiny of your boss. A stepping rule ofi’ers a concise proof of optimality. Suppose you go to your boss not only with the optimum assign- ment but also with a set of numbers to be subtracted from the entries in each row and column. Setting forth how these numbers are obtained would re- quire a detailed discussion of the assign- ment problem; it will suffice to point out that the set of numbers can be specified by an efficient computer algorithm. The utility of the numbers, once they have been found, is readily appreciated. Note that subtracting the same number from every entry in a given row or column is equivalent to subtracting this amount from the total cost of every pessible as- signment. The reason is that every feasi— ble assignment must choose one and only one entry from each row and each column. For example, if 5 is subtracted from every entry in the sixth row, every possible assignment will include exact- ty one entry that is 5 less than the cor— responding assignment made with the original array of costs. The relative costs of all the assignments will therefore re- main unchanged. Such subtraction can be done repeatedly, prov'tded it is always applied uniformly to every entry in a given row or column. By means of repeated subtraction you can transform the original cost matrix into the matrix shown in the lower illus- tration on page 129. The latter array of costs has a remarkable property. You can now point out to your boss that the costs corresponding to the squares se- lected by your assignment are all zero...
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