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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 and that no entries in the matrix are less than zero. Since the sum of the costs FEASIBILITY REGION in the brewer‘s problem is made up of the intersection of five half planes. Any point (A,B) tn the plane corresponds to a production program that calls {or mak- lllg A barrels of ale and B barrels of beer. The first three half planes graphically represent all the production programs that are achievable, given the available quantities of each ingredient. For example, the amount of corn required is SA + 153 (the weight in pounds of the corn need— ed to make a bane! of ale times A plus the Weight o! the corn needed to make a barrel of beer times H). Thls quantity must not exceed the 480 pounds of corn available to the brewer. Hence any point in the half plane to the lower left of the line 5A + 158 = 480 represents a feasible pro- duction program that requires no more than the 480 pounds of com available. The half plan associated with hops and malt can be constructed in a slmllar way. The remaining two half planes express the fact that only those programs having nonnegatlve production are feasible. 13?. marked by your assignment is zero and there are no negative costs, no other pos- sible assignment can have a lower cost. In short, you have shown your boss, with a few hundred calculations rather than tens of millions, that no assignment can be less expensive than the one you have chosen. lthough l have not demonstrated how to solve an assignment problem or how to find the row and column num‘ bers to subtract, the assignment prob- lem does illustrate the necessity of avoiding enumeration and the possibili- ty of doing so by means of a stopping rule that recognizes an optimum solu- tion. These are design characteristics of algorithms that can be applied not mere— ly to the assignment problem but to lin— ear-programming problems in general. Consider again the situation of a brewer whOSe two products, ale and beer, are made from difi'ercnt propor- tions of corn, hops and malt. Suppose 480 pounds of corn. 160 ounces of hOps and 1,190 pounds of malt are immedi- ately available and the output is limited by the scarcity of these raw materials. Other resources, such as water, yeast. labor and energy, may be consumed in the manufacturing process, but they are considered to be plentiful. Although they may influence the brewer‘s willing- ness to produce beer and ale because of their casts, they do not directly limit the ability to produce. ASsume that the brewing of each barrel of ale consumes five pounds of corn. four ounces of hops and 35 pounds of malt, whereas each barrel of beer requires 15 pounds of corn, four ounces of hops and 20 pounds of malt. Assume further that all the ale and beer that can be produced can be sold at current prices, which yield a profit of $13 per barrel of ale and $23 per barrel of beer. The scarcity of corn, hops and malt limits the feasible levels of production. For example, although there are enough hops and malt to brew more than 32 barrels of beer, production of this much beer would exhaust the supply of corn. allowing no greater output of beer and no output of ate at all. Another feasible production program calls for no beer and 34 barrels of ale, depleting all 1,190 pounds of malt. The first alternative seems preferable to the second. The profit realized by the first program is 32 X $23, or $736, whereas the second program yields only 34 X $13, or $442. There are other production programs that are better than either of these. Six barrels of ale and 30 barrels of beer use all 480 pounds of corn, 154 of the 160 ounces of hops and 810 of the 1,190 pounds of malt, yielding a profit of (6 X $13} + (30 X $23), or $768. Many programs earn even greater profit. In this case it is not merely impractical to I; VALUE OF THE OBJECTIVE FUNCTION for any point P in the feasible region can be graphed as the distance ZU’) above or below P measured on the z axis. (Distances along the z axis have not been drawn to the same scale as distances In the plane of the feasible region.) One can think of the distance Z(P) as a point in three-dimensional space. If the function is linear, the graph of the values the function takes along any straight line in the feasible region is also a straight line [upper graph). For any point P in the interior of the region, there is some line through P that intersects the boundary of the region at two points, say the points R and S (loner graph). If the line segment in space connecting 2”?) and 2(3) is not parallel to the plane of the feasible region, the objective function must assume its maximum value along the line segment at one of its endpoints, say 2(5), which corresponds to the point 5 on the edge of lthe feasible region. The graph of the objective function along an edge is also a straight line and it too as- sumes its maximum at one of its endpoints, say 2(0), which corresponds to a vertex of the tea- sible region. Thus there is always an edge point that dominates a given interior point, and there is always a vertex that dominates any point along an edge. To find the maximum value Z(M) of the objective function one need examine only the vertexes. When the feasible region is two- dimensioual, the objective function forms a plane whose maximum height is attained at a vertex. 134 enumerate all the possibilities. as it was in the assignment problem; here it is not even possible. There are infinitely many production programs that meet the con- ditions of the brewer‘s problem. Each such program is called a feasible solu- tion. Fortunately there is a small set of feasible solutions called extremal solu- tions to which attention can be confined. he importance of the extremal solu- tions becomes apparent when the set of all feasible solutions is represented graphically as a set of points in a plane; the set of points constitutes the feasi- ble region. Let A designatethc number of barrels of ale brewed according to any possible production strategy and lcl B designate the number of barrels of beer. A and B are known in linear pro- gramming as decision variables. They can be associated with the coordinate axes of the plane. Any pointon the plane can be specified by a pair of coordinates (AB), which also correspond to a partic— ular set of production levels. Because negative levels of production are not possible, the feasible region can immediately be confined to the upper righthand quadrant of the plane. where A and Bare both nonnegaiive. How does the scarcity of malt affect production? Since 35 pounds of malt are needed for each barrel of ale and 20 pounds are needed for each barrel of beer. the to- tal amount of malt needed to make A barrels of ale and B barrels of beer is 35/! + 203. If all 1,190 pounds of malt are used. 35A + 203 = 1,190. The setol' points (AB) that satisfy this equation form a straight line. All points (AB) corresponding to production plans that call for more than 1,190 pounds of malt lie on one side of the line and the points that require lees malt lie on the other side. Only the latter set of points and the points actually on the line are feasible because of the limited supply of malt. In a similar way the scarcity of hops confines the feasible region to one side of the line 4.4 + 43 = 160 and the scar— city of corn confines the region to one side of the line 5A + 153 = 480. The points that satisfy all these requirements make up the feasible region [388 illustra- lion on page 132]. Note that the feasible region is convex: any line segment that connects two points in the region (in~ cluding the points on the perimeter lines) lies entirely within the region. Since the brewer’s profit is $13 per barrel of ale and $23 per barrel of beer. his problem is to maximize his total profit: 13A + 233. in order to do so he must find a point (AB) in the convex feasible region where 13.4 + 233 has its maximum value. In linear programming such a measure of benefits to be maxi- mized (or sometimes of cests to be mini- mized} is called an objective function. The objective function can be incor~ porated into the graph of the feasible region by adding a third dimension. For any point (A,B) representing a produc~ tion plan the expected profit is given by the height of the function 13A + 233 above the plane at that point. Clearly the task confronting the brewer is to find a point in the feasible region where the objective function is at its greatest height. If the function had to be evaluat- ed at~all the points, the task would be impossible, but two distinctive proper- ties of the problem act to narrow the search. The properties are the convexity of the feasible region and the linearity of the objective function. Since the region is convex, any point in the interior of the region can be in- cluded in a line segment whose end— points lie on the boundary of the region. (Indeed, an infinite number of such line segments can be drawn through any giv- en point; which line segment is chosen is immaterial.) In the space above each such line segment it is possible to con- struct a graph of the objective func— tion giving the profit for each point on the line segment. Since the objective function is linear, the graph is a straight line [see illustration on page 134]. The straight-line graph may be parallel to the plane, in which case all the brewing strategies along the line segment have the same profit. If the graph of the objec- tive function over the line segment is not parallel to the plane, it must assume its ‘ maximum value at one of the two end- points, which lie on the boundary of the feasible region. Hence the maximum of the objective function over such a line segment must always be attained at one of the points where the line segment in- tersects the boundary. Because the same analysis can be applied to any line seg- ment in the feasible region, it follows that the overall maximum of the objec- tive function is invariably found some- where on the boundary of the region. The brewer in search of maximum profit can ignore the entire interior of the fea- sible region and consider only those brewing strategies that correspond to points on the boundary. The analysis can be taken a step fur- ther by the same argument. If the fea- sible region is a polygon, every point on the boundary lies on a line segment whose endpoints are two‘of the vertexes of the polygon. A graph of the objective function for such a boundary segment can be constructed in the same way it is for a line segment that crosses the interi- or. Again the maximum must be found at one of the endpoints (or at both end- points if the objective function is con— stant and parallel to the plane). Thus a maximum value of the objective func- tion throughout the feasible region can be found among the vertexes. The brew- er needs only to check, at most, the val- ue of the function at all the vertexes of the feasible region and select the one yielding the best profit. He can then be certain that no other brewing strategy would bring a higher profit. In the exam- ple considered here there are five vertex- es. The one at point (12,28), which rep- resents the production of 12 barrels of ale and 28 barrels of beer, yields a prof- it of (12 X $13) + (28 X $23), or $800. That is the maximum profit the brewer can realize. he inclusion of an additional con- straint (such as a shortage of yeast) would not significantly alter the geomet- rical interpretation of the brewer’s prob- lem. The polygon might then have six sides instead of five. The introduction of a third product, on the other hand, would have a more profound effect: the geometrical model would then be three- dimensional. Inequalities in three varia- bles correspond to half spaces defined by planes in three-dimensional space in- stead of half planes defined by lines in two—dimensional space. The feasible re- gion is no longer a polygon but might instead look like a cut gemstone, a three— dimensional polytope whose faces are all polygons. As the number n of de- cision variables increases further, the geometrical interpretation remains valid but it becomes more difficult to visualize the n-dimensional polytope formed by the intersections of (n — l)-dimensional hyperplanes. Vertexes retain their spe- cial status, however, and their positions can be determined by algebraic meth- ods that replace geometrical intuition. It may appear that by confining the evaluation of the objective function to the vertexes of the feasible region, the solution of linear-programming prob- lems by enumeration becomes practical. As in the assignment problem,_however, the number of possibilities to be enu- merated grows explosively. A problem with 35 decision variables and 35 con- straints would be impossible. Dantzig’s simplex method examines vertexes, but it does so selectively. In the brewery example the method might be- gin with the vertex at the origin, (0,0). Here nothing is produced and the prof- it is zero. Following either of the inci- dent edges away from the origin leads to points with larger objective-function values. The simplex method selects such an edge, say the B axis, and follows it to its other end, the vertex (0,32). The pro- gram here calls for making 32 barrels of beer but no ale, for a profit of $736. From this vertex an incident edge leads to still greater objective-function values. The simplex algorithm therefore pro- ceeds to the vertex (12,28) at the other end of the edge. Here the profit from brewing 12 barrels of ale and 28 barrels of beer is $800. At (12,28) all incident :dges lead in unfavorable directions; the algorithm therefore halts with a decla- -ation that the vertex (12,28) is optimal. In general the simplex method moves 110ng edges of a polytope from vertex to idjacent vertex, alWays improving the Ialue of the objective function. The pro— :edure can begin at any vertex, and it ialts when no adjacent vertex has a bet- er objective-function value than the :urrent one. This stopping rule is val- id only because the feasible region is convex: convexity guarantees that any locally maximum vertex is a globally maximum vertex. One need look only in the vicinity of each point to tell whether improvement is possible. How does one tell that an edge lead- 8 A—> ing away from a given vertex will im- prove the value of the objective func- tion? The key is the concept of a “mar— ginal value“ attributed to each resource at each vertex of the feasible polytope. For many-dimensional problems in lin- ear programming the algebraic method 4—9 CONVEXITY PROPERTY of linear programming states that for any two points to the feasi- ble region a line segment connecting the points must lie entirely within the region. Only the tea- sible region at the left is convex. The region in the diagram at the right is a set of discrete points. Convexity ensures that any local maximum of a linear objective function is a global maximum. 138 that is commonly employed to choose a path from vertex to vertex can be under- stood when one understands the signifi- cance of the marginal values. At the maximum vertex in the brew— cry problem there are 210 pounds of excess malt that are not utilized in the production program represented by that vertex. The addition or subtraction of a pound of malt from the initial supply of 1,190 pounds Would not change the ob taiuable profit. On the other hand, an- other ounce of hops would make possi— ble an increase of $2 in total profit. This increase is the marginal value of hops at the maximum vertex. it can be interpreted as the effect of "pushing" the constraint line for hops farther from the origin to reflect the extra avail- able ounce of hops [see illustration on opposite page]. Similarly, the marginal value of corn at this vertex is $1. Marginal prices have a natural eco- nomic interpretation. If the brewer could buy an additional ounce of hops. he could increase his profit by $2. If hops were available for less than $2 per ounce, it would be worthwhile to buy hops. 0n the other hand, if a price high- er than $2 per ounce were offered for hops. it would be worthwhile for the brewer to divert hops from the produc- tion of beer and ale and sell them on the market. This does not mean that the buying or selling of hops should contin- ue indefiniter at $2 per ounce, but for increases of about 19 ounces or decreas- es of up to 32 ounces, $2 remains the break-even price in this example. Marginal values are sometimes called shadow prices or imputed prices. They indicate the relative amount that each scarce resource contributes to the profit— ability of each item in production. For example, a barrel of ale requires five pounds of corn with an imputed price of $1 per pound, four ounces of hops with an imputed price of $2 per ounce and 35 pounds of malt with an imputed price of zero. The total imputed price of ale is equal to the profit of $13 per barrel. Suppose a new product, light beer. is proposed. Making a barrel of light beer requires two pounds of corn, five ounces of hops and 24 pounds of malt. How much profit per barrel must be obtained from light beer to justify diverting re- sources from the production of beer and ale? The imputed prices of the resources answer the question. The total imputed price of the ingredients in light beer is (2 X $1) + (5 X $2) + (24 X $0), which is equal to $12. This measures the prof- it that wo uld be lost by diverting resour- ces from the brewing of ale and beer to the production of one barrel of light beer. Thus for the brewing of light beer to be worthwhile it must yield a profit of at least $12 a barrel. The role of marginal prices in judging {0, 32.0?) the wisdom of introducing a new prod- uct can help to explain how the simplex method determines which edges of a polytope lead from a given vertex to a vertex with an improved value of the (0,32) objective function. Suppose the vertex currently under examination is the one i at (0,32) in the brewery problem. First one can determine the marginal price of each resource in this production pro- gram. Malt and hops are in excess sup- ply at this point and so their marginal " values are zero. Corn. however, is in short supply and so its marginal value is positive. Since only beer is being made at (0,32), the marginal value of the in- gredients in a barrel of beer should be equal to the profit associated with beer: $23. Since Only corn has nonzero mar- ginal value and since 15 pounds of corn are needed for each barrel of beer, the COFIN (11.9. 23.1) [26. 14) (34. D) marginal value of corn for this pro— ductiort program is $23 divided by 15 A—> pounds. or about $1.53 per pound. w. a) What is the marginal value of the in- gredients in ale measured at the (0.32) vertex? If this value is less than the profit that can be realized from selling a barrel of ale, the diversion of some resources (12.38. 2133} from beer production to ale would in- / crease the total profit of the brewer. Hops Making a barrel of ale requires five (25.57.1453) pounds of corn. Therefore the marginal / value of the ingredients in a barrel of ale c is (523/15) X 5. or about $7.67. As in the example of light beer. the marginal value represents the loss in profit that would result from diverting resources from beermaking to the brewing of ale. ‘ Because this value is less than $13 (the profit expected from selling each barrel of ale) it would be profitable to brew more ale than the production program at the point (0,32) specifies. Indeed, by keeping the total consumption of corn constant and diverting corn from beer to l ale, the brewer can increase his profit by $13 -- $7.67. or $5.33, for each barrel of ale. Since the amount of corn is held constant, producing more ale corre— sponds to moving along the edge of the feasible region that represents the con- MARGINAL INCREASES in the availabili- ty of scarce resources alter the potential prof- its of a hypothetical brewer in a predictable Way. If one additional pound of corn were available, the maximum profit would increase by $1, a change in the objective function that 0 is reflected in the movement of the maxi- mum vertex in the feasible region from the colored dot to the colored circle {upper graph). If there were an extra ounce of hops, the max- imnm profit would increase by $2 (middle .a graph). A change in the available malt would not alter the attainable profit: mall is already a surplus resource (lower graph). The changes in profit associated with changes in the avail- ability of each resource are known as shadow prices or imputed pric. They are used to dl- f [26.01 13.93] / (34.03, 0} rect the algorithm from vertex to vertex. The marginal changes are exaggerated for clarity. straint on the supply of com. This is pre- cisely how the simplex method recog- nizes that the objective function can be improved along this edge. In this case the edge connects the vertex (0,32) with the vertex (12,28) of the polygon. ...
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