Equivalent Linear Programs

Transcription

1 Appedix A Page 1 Equivalet Liear Programs There are a umber of problems that do ot appear at first to be cadidates for liear programmig (LP) but, i fact, have a equivalet or approximate represetatio that fits the LP framework. I these istaces, the solutio to the equivalet problem gives the solutio to the origial problem. This appedix describes the trasformatios that ca be used to covert a oliear problem to a liear program for the followig three situatios: (i) the objective is to maximize a separable, cocave oliear fuctio; (ii) the objective is to maximize the miimum of a set of liear fuctios; ad (iii) there are several prioritized objectives with specified goals. A.1 Noliear Objective Fuctio I some cases, liear programmig ca be used eve whe oliear terms are preset. Cosider the followig mathematical programmig model i compact algebraic form. Maximize z = f j (x j ) subject to a ij x j b i, i = 1,,m x j 0, j = 1,, The m costraits are liear but the objective cosists of oliear, separable terms f j (x j ), each a fuctio of a sigle variable oly. Whe the objective fuctio ca be writte i this maer ad each f j (x j ) is cocave (see below), the above maximizatio problem may be approximated with a liear model ad solved with a liear programmig algorithm. Whe the fuctios f j (x j ) are ot all cocave this approach will ot work. The absece of cocavity requires the developmet of a iteger programmig model, as described i Chapter 8. A aalogous situatio exists whe the objective is to miimize a separable, covex fuctio. A approximate liear programmig model ca be developed, but similarly, miimizig a cocave fuctio requires the use of iteger variables. Cocave Fuctios We first cosider the case i which the fuctio f j (x j ) is cocave. The solid lie i Fig. 1 depicts the graph of such a fuctio. Later i the book 5/29/02

2 2 Equivalet Liear Programs the defiitio of cocavity will be made precise but for ow we say that a cocave fuctio has the characteristic that a straight lie draw betwee ay two poits o its graph falls etirely o or below the graph. A fuctio that has a cotiuous first derivative is cocave if the secod derivative is everywhere opositive. f (x ) j j c 3 c 2 c d 1 d 2 d 3 x j Figure 1. Cocave fuctio with a piecewise liear approximatio The dotted lie i Fig. 1 represets a piecewise liear approximatio to the cocave fuctio. The approximatio idetifies r break poits alog the x j axis: d 1, d 2,,d r, ad r correspodig poits alog the f j axis: c 1, c 2,,c r. Now we have r pieces represetig the objective fuctio with the first startig at the origi. If f j (0) does ot equal 0, the fuctio f j (x j ) ca be replaced by f j (x j ) f j (0) without affectig the optimal solutio so we ca always put d 0 = c 0 = 0. The piecewise liear approximatio is implemeted i a liear programmig model by defiig ew variables x j1, x j2,,x jr to represet the pieces. The slope of the kth segmet is s jk = (c k c k 1 )/(d k d k 1 ) The piecewise liear approximatio to the jth term i the objective fuctio is r f j (x j ) s jk x jk. k=1 I each costrait, the variable x j is replaced by

3 Noliear Objective Fuctio 3 r x j x jk k=1 ad the ew variables must satisfy the followig bouds 0 x jk d k d k 1, k = 1,,r Of course, whe f j (x j ) = c j x j o substitutio is ecessary; otherwise, each origial variable x j must be replaced with r j ew variables. Here r j is the umber of break poits alog the x j -axis ad may vary from oe variable to the ext. Whe the appropriate substitutios are made, the approximate model becomes Maximize z = r j s jk x jk k=1 subject to r j a ij x jk b i, i = 1,,m k=1 0 x jk d k d k 1, k = 1,,r j, j = 1,, Thus to obtai a liear model, oe pays the price i terms of icreased problem size. The approximatio ca be made as accurate as desired by defiig eough break poits but with a correspodig icrease i dimesioality. The oly remaiig issue i the liearizatio process is whether a solutio to the ew problem is equivalet to a solutio to the origial problem. For x j replaced by Σ r k=1x jk, how ca we be sure that the pieces of the approximatio will be icluded i the solutio i the proper order? Evidetly, if x jk is greater tha 0, the solutio will ot be valid uless the variables for all the precedig pieces x jl, where l < k, are at their upper bouds. There are o explicit costraits i the model to guaratee this. Fortuately, whe we are maximizig ad the idividual fuctios f j (x j ) are cocave, the variables will eter i the proper order without explicit costraits. This is because the objective coefficiets s jk are decreasig with k. The goal of maximizatio will cause the pieces with the greatest slope to be selected first as the associated x jk variables take o positive values.

4 4 Equivalet Liear Programs Covex Fuctios Figure 2 depicts a typical covex fuctio which, by defiitio, has the property that a straight lie draw betwee ay two poits o its graph lies etirely o or above the graph. A fuctio that has cotiuous first ad secod derivatives is covex if the secod derivative is everywhere oegative. I geeral, if f j (x j ) is a covex fuctio the f j (x j ) is a cocave fuctio. Oe is a reflectio of the other aroud the horizotal axis. Whe the goal is to maximize the objective, the pieces of the liearized fuctio eter the solutio i exactly the reverse of the proper order. I this case, supplemetary biary variables must be used to eforce the correct sequece so the resultat model would o loger satisfy the liear programmig assumptios. f (x ) j j c 3 c 2 c d 1 d 2 d 3 x j Miimizatio Problems Figure 2. Piecewise liear approximatio of covex fuctio A simple but importat observatio i optimizatio worth repeatig is that miimizig a fuctio is equivalet to maximizig the same fuctio with its sig reversed. Because a covex term, f j (x j ), becomes cocave whe it is egated, we ca coclude that a miimizatio problem with covex, separable terms i the objective ca be approximated i the same way as a cocave fuctio whe the goal is to maximize. Reasoig as before, however, whe cocave terms appear i a miimizatio objective, liear programmig caot be used.

5 Noliear Objective Fuctio 5 Modelig Whe maximizig profit, reveue terms i the objective fuctio have a positive sig ad cost terms have a egative sig. Thus i order to use liear programmig to fid a solutio, all reveue terms must be cocave fuctios ad all cost terms must be covex fuctios. There are importat practical istaces where this is evidet. The reveue or beefit received from the sale of most commodities has a cocave shape because of the priciple of decreasig margial returs. There are may examples of price discouts to gai additioal sales, as illustrated by airlie ticket discout plas. O the other had, cost fuctios are ofte covex with respect to the quatities produced or purchased. For example, to icrease output i the ear term, it may be ecessary to pay overtime or use less efficiet meas of productio. These circumstaces promise a easy solutio to the problem because the model ca be approximated with piecewise liear terms ad solved with a liear programmig algorithm. Ufortuately, there are also may practical situatios whe liear programmig caot be used. These most ofte arise i problems ivolvig capacity expasio of facilities. The cost of buildig ad operatig a facility commoly ivolves ecoomies of scale; the larger the facility, the smaller the margial cost. This relatioship implies a cocave cost fuctio that caot be approximated with a liear programmig model. To obtai a solutio, it would be ecessary to develop a model that used piecewise liear approximatios as well as iteger variables. The computatioal effort to solve such a model would be sigificatly greater tha that required to solve a stadard liear program. Example The problem below ivolves the maximizatio of a cocave, separable quadratic fuctio over a set of liear costraits. We use a piecewise liear approximatio for the oliear terms i the objective to develop a LP model. 2 Maximize z = x 1 2 2x2 + 8x1 + 16x 2 subject to 0.9x x 2 < 5 x 1 3 x 1 0, x 2 0 The separable oliear terms of the objective are: 2 f 1 (x 1 ) = x 1 + 8x1

6 6 Equivalet Liear Programs f 2 (x 2 ) = 2x x2 We choose to use a piecewise approximatio with the itegers as breakpoits. Sice the secod costrait idicates that the upper boud o x 1 is 3 ad first costrait implies that the upper boud o x 2 is 4, we have r 1 = 3 ad r 2 = 4 givig the followig breakpoits: x 1 breakpoits x 2 breakpoits k k d k d k c k c k s 1k s 2k The liear programmig approximatio is the Maximize z = 7x x x x x x x 24 subject to 0.9x x x x x x x 24 5 x 11 + x 12 + x x ik 1 all i ad k Because all the oliear terms are cocave ad the objective is to maximize, the solutio to the LP model will yield a valid approximatio to the solutio of the origial model. Usig our Excel add-is, we fid that the optimum is x 11 = x 12 = 1, x 13 = 0, x 21 = x 22 = 1, x 23 = 0.667, x 24 = 0 Traslatig this solutio ito terms of the origial problem gives x 1 = 2, x 2 =

7 Maximizig the Miimum 7 A.2 Maximizig the Miimum There are a variety of situatios where maximizig total profit or miimizig total cost may ot be the preferred course of actio. Whe resources have to be distributed over more tha a sigle etity or orgaizatio, the goal might be to maximize the miimum profit that is realized by ay of the etities. Similarly, it might be optimal to set policy so that the maximum cost that is icurred by ay of the orgaizatios is miimized. I these scearios the decisio maker is implicitly hedgig agaist the worst possible outcome by specifyig what is respectively called a maximi or miimax strategy. This type of worstcase aalysis is commo whe the decisio maker is faced with a ucertai outcome ad is risk averse. A global optimum has o cocer for fairess so oe etity may be treated poorly i compariso to aother whe such a solutio is implemeted. For example, the best locatios for fire statios may ot be the solutio that miimizes total respose time, but rather the oe that miimizes the maximum respose time over a umber of eighborhoods. Oe objective for fittig a lie to a set of poits is to miimize the total deviatio betwee the poits ad the lie. A reasoable alterative would be to miimize the maximum deviatio over the set of poits. Similarly, the corporate problem of allocatig resources to decetralized divisios could be solved to maximize total profit; however, a alterative approach would be to maximize the miimum profit of the divisios. For problems whose costraits are otherwise liear, this kid of objective ca be modeled as liear program. Maximi Objective Let us assume that we have a liear programmig model defied by a set of costraits ad t objective fuctios of the followig form. z k = c 0k + c 1k x 1 + c 2k x 2 + +c k x = c 0k + c jk x j, k = 1,,t Each fuctio z k is a hyperplae. The goal is to fid a -dimesioal vector x = (x 1,,x ) that miimizes the fuctio f(x) give by f(x) = miimum{c 01 + c j1 x j, c 02 + c j2 x j,,c 0t + c jt x j } subject to the costraits of the problem. Although f(x) is ot give i explicit form, it ca be evaluated easily from the above equatio for ay real value of x. A useful property of f(x) is that it is piecewise liear ad cocave. This is illustrated i Fig. 3 for t = 4 ad x a scalar. The implicatio is that

8 8 Equivalet Liear Programs the problem of maximizig f(x) subject to liear costraits ca be trasformed ito a liear programmig, eve though f(x) is ot separable. f(x) z 1 = c 01 + c x 11 z = c + c x z = c + c x z = c 04 + c x 4 14 x Figure 3. Fuctio defied by miimum of several hyperplaes Miimax Objective The trasformatio is based o the observatio that f(x) is equal to the smallest umber, call it z, that satisfies z c 0k + c 1k x 1 + +c k x for all k. The equivalet optimizatio problem is the Maximize z subject to c jk x j z, k = 1,,t (M k ) plus the origial liear costraits. The decisio variables are z ad x j, j = 1,,, which may or may ot be restricted to be oegative. Thus i the trasformatio we have itroduced oe ew variable z to be maximized, ad t additioal liear costraits: (M 1 ), (M 2 ),...,(M t ). Reversig of the above case gives the objective to be miimized as The equivalet liear program is Maximum {z 1,z 2,...,z t }.

9 Maximizig the Miimum 9 Miimize z subject to c jk x j z, k = 1,,t (M k ) plus the origial costraits of the problem. The added costraits force all the liear fuctios, z k, to be less tha or equal to the variable z that is to be miimized. A Oe Dimesioal Locatio Problem At a promiet Texas uiversity, the various egieerig departmets are located i buildigs alog a sigle street, as show i Fig. 4. The distaces give i the figure are i feet from the assumed origi. The dea of egieerig wats to locate his office somewhere alog the street. All locatios are allowed (i.e., the street ca be cosidered as a cotiuum of possible locatios). The weekly umber of trips by faculty ad others betwee the Dea s office ad the departmets are listed i the trip table below. The followig three optimizatio criteria are beig cosidered. a. Miimize the total distace traveled. b. Miimize the maximum distace traveled from ay of the departmets. c. Miimize the total distace traveled, but o departmet is to be more tha 300 feet from the dea s office. CE EE IE PE CHE ME Figure 4. Map showig the locatio of egieerig departmets Trip table Departmet, i CE EE IE PE CHE ME Trips, w i Solutio Idea Each of the three problems will be solved i tur. To begi, defie the locatio of the dea's office as the decisio variable x. The distace from this office to a departmet is computed by cosiderig whether it is to the

10 10 Equivalet Liear Programs left or to the right of the departmet. For the PE departmet the appropriate equatio is x + L PE R PE = 350 where L PE is the distace that the dea's office is to the left of PE ad R PE is the distace the office is to the right of PE. The distace betwee the office ad the PE departmet is L PE + R PE. Logically, at most oe of these variables will be positive i ay solutio. Formal Model VARIABLE DEFINITIONS x = coordiate of the dea's office L i = distace from dea s office to departmet i whe the office is to the left R i = distace from dea s office to departmet i whe the office is to the right Let subscript i = 1,,6, have the followig associatio: 1 = CE, 2 = EE, 3 = IE, 4 = PE, 5 = CHE, 6 = ME. CONSTRAINTS DEFINING DISTANCES x + L i R i = a i for i = 1,,6 where a i is the x-coordiate of departmet i as show i Fig. 4. TOTAL DISTANCE TRAVELED D = 137(L 1 + R 1 ) + 160(L 2 + R 2 ) (L 6 + R 6 ) The distace traveled, D, is a fuctio of the left ad right variables, L i ad R i. It is determied by weightig each pair by the correspodig values w i i the trip table above, ad the summig each term. I a optimal solutio, at most oe of the variables i the pair (L i,r i ) will be positive. Criterio a. The goal here is to miimize the sigle objective of total distace traveled. This leads to the followig optimizatio problem.

11 Maximizig the Miimum 11 6 Miimize D = w i (L i + R i ) i=1 subject to x + L i R i = a i, i = 1,,6 x 0, L i 0, R i 0, i = 1,,6 Solvig gives the optimal locatio x * = 150 with the miimum total distace traveled D * = 86,100. Criterio b. We ow wish to miimize the maximum distace traveled from ay of the departmets. Let v deote this distace with the stipulatio that L i + R i v for i = 1,,6. The optimizatio problem is Miimize v subject to x + L i R i = a i, i = 1,,6 L i + R i v 0, i = 1,,6 x 0, L i 0, R i 0, i = 1,,6 The optimal locatio is x * = 300 with v * = 225, ad total distace D * = 95,325. The revised criterio has caused the total distace to icrease. Criterio c. For this part we use the costraits of part (b) ad the objective of part (a) with the additioal stipulatio that v 300 Miimizig total distace gives x * = 225, D * = 89,025, ad v * = 300. By specifyig a goal for the maximum distace, we have obtaied a itermediate value for total distace.

12 12 Equivalet Liear Programs A.3 Goal Programmig Most decisio-makig situatios do ot proceed from a sigle poit of view or admit a sigle objective. I fact, may decisios must be made i the face of competig iterests i a cofrotatioal eviromet. Cosider a local zoig commissio that must balace the desires of residets, small busiesses, developers, ad evirometalists; or a corporate maager who must allocate the aual budget to several operatig divisios; or a recet college graduate who must weigh salary, locatio, work eviromet, ad frige beefits of several job offers. The problem of the decisio maker is to balace the goals of the competitors i such a way that most are to some extet satisfied; i other words, to reach a compromise. Admiistrators are always lookig for the perfect compromise, the oe that satisfies everyoe, but of course this is rarely foud. Rather we rely o committees, commissios, electios, cotests, ad eve chace as ways to arrive at decisios that, at best, oly partially satisfy the participats. Up util this poit i the chapter, the models preseted have bee limited to a sigle optimizatio criterio. The methods of goal programmig exted our modelig capabilities by offerig ways to deal with more tha oe objective at a time. They do ot, however, provide the complete aswer because wheever there are competig goals, it is difficult if ot impossible to coduct a purely objective aalysis that yields the "best" decisio. There will always be a subjective compoet i the aalysis that reflects the decisio maker s prefereces. Nevertheless, the goal programmig approach does provide a orgaized way of cosiderig more tha oe objective at a time ad ofte yields compromise solutios that are acceptable to the protagoists. The basic idea is to establish specific umeric goals for each objective, ad the to seek a solutio that satisfies all the give costraits while miimizig the sum of deviatios from the stated goals. Frequetly, the deviatios are weighted to reflect the relative importace of each objective fuctio. Defiitios Objective fuctio (f k (x)): Oe of several fuctios of the decisio variables, x = (x 1,,x ), that evaluates the attaimet of some measure of effectiveess. Oly liear fuctios are cosidered here. f k (x) = c 1k x 1 + c 2k x c k x = c jk x j, k = 1,,t Lower Oe-Sided Goal: For the kth objective fuctio, a lower limit, L k, that the decisio maker does ot wat to fall below. It is desired to achieve a value of "at least" L k for the objective. Exceedig this value is permissible. The goal might be writte as a "greater tha or equal to" costrait:

13 Goal Programmig 13 f k (x) = c jk x j L k j =1 I the goal programmig methodology, this is ot a hard costrait so we allow solutios x such that f k (x) L k. Upper Oe-Sided Goal: For the kth objective fuctio, a upper limit, U k, that the decisio maker does ot wat to exceed. It is desired to achieve a value of "at most" U k for the objective. The goal might be writte as a "less tha or equal to" costrait: f k (x) = c jk x j U k Agai, this is ot a hard costrait so we allow solutios that yield values of f k (x) that exceed U k if these lead to the best compromise. Two-Sided Goal: Sets a specific target value, G k, for the kth objective so the value of f k (x) should be "equal to" some G k. The goal might be writte as a equality costrait: f k (x) = c jk x j = G k The solutio process will allow for deviatios from this goal i either directio. A particular objective will usually appear as either a lower oesided, upper oe-sided, or two-sided goal. I some cases, though, both upper ad lower goals may be specified for a objective with the rage betwee them defiig a regio of idifferece. Goal Costrait: The cetral costruct i goal programmig is the deviatio variable. Let y k + = positive deviatio or the amout by which the kth goal is exceeded y k - = egative deviatio or the amout by which the kth goal is uderachieved Oe of the followig three costraits is used i the liear programmig model to measure the deviatio from the goal.

14 14 Equivalet Liear Programs 1. c jk x j y + k + y - k = Lk 2. c jk x j y + k + y - k = Uk 3. c jk x j y + k + y - k = Gk As ca be see, each of these costraits has the same form. This might seem odd at first but should become clear whe we explai how objective fuctios are costructed. If more tha oe costrait were to be used i a model, say, to defie a regio of idifferece, it would be ecessary to distiguish each pair of deviatio variables. Pealty Weights (p k + ad p k - ): Costats that measure the per uit pealty for violatig goal costrait k. Let p k + = the pealty applied to the positive compoet. p k - = the pealty applied to the egative compoet. The three kids of goals are associated with the followig pealty assigmets: lower oe-sided goal: p k - > 0, pk + = 0, upper oe-sided goal: p k - = 0, pk + > 0, two-sided goal: p k - > 0, pk + > 0. Whe a lower boud is specified for the kth objective, for example, we set p k - > 0 because we wat to pealize the uderachievemet of the goal Lk. We do t wat to pealize its achievemet, though, so we set p k + = 0. Similar reasoig applies to the other two types of pealty assigmets. I the goal programmig model, the fuctio to be optimized comprises terms of the form z k = p k + y k + + p k - yk -. Nopreemptive Goal Programmig: I this approach, we put all the goals i the objective fuctio ad solve the liear program a sigle time. The objective for the problem is the weighted sum of the deviatio variables.

15 Goal Programmig 15 The pealties measure the relative importace of the goals. The objective is to ( ) Miimize z = z k = p k + y k + + p k y k k =1 k =1 Because the goals very ofte are measured o differet scales, the pealties play the double role of trasformig all goals to the same dimesioal uits as well as specifyig their relative importace. I this approach, the subjective step is the determiatio of the weights. Differet weights will ofte yield very differet solutios. Preemptive Goal Programmig: Here the goals are divided ito sets ad each set is give a priority; i.e., first, secod, ad so o. The assumptio is that a higher priority goal is absolutely more importat tha a lower priority goal. The solutio is obtaied by iitially optimizig with respect to the first-priority goals without regard to the values of lower priority objectives. The, holdig costat the value of the first-priority objective fuctio by addig the costrait z 1 (y + 1, y + 1 ) = z * 1, the optimal solutio is obtaied for the secod-priority goals. The feasible solutio space for this secod problem is the set of alterate optima for the first problem. The process cotiues util all priorities are cosidered. If o alterate optima exist at the ed of a particular stage, we have reached the ed of the computatios so we must be satisfied with the curret values of the lower priority objectives. If several goals have about the same priority we iclude all them i the set i the objective at the appropriate step of the process. The relative importace of the goals withi ay set are reflected by the specificatio of the pealty weights, as i the opreemptive case. The subjective part of this procedure is the divisio of the goals ito priority sets ad the selectio of pealties withi a priority set. Optimum Portfolio Problem A mutual fud maager has $200 millio to ivest ad is cosiderig five alterative ivestmets. A portfolio is defied by specifyig the umber of uits of each opportuity purchased. Each ivestmet has a fixed uit cost, but its aual retur is a radom variable. Therefore, its value is ot kow with certaity. The research departmet has determied that the expected retur ad variace per uit of ivestmet is proportioal to the umber of uits ivested i the opportuity. All the data are show i Table 1. Costs ad aual returs are give i millios of dollars. The total expected retur is the sum of the expected returs of the idividual ivestmets. Similarly, assumig idepedece, the total

16 16 Equivalet Liear Programs variace is the sum of the idividual variaces. The followig three goals have bee established for the portfolio ad are listed i priority order. Goal 1: The aual expected retur must be at least $45 millio. Goal 2: The total variace must be o more tha $150 millio 2. Goal 3: The amout ivested i opportuities 2 ad 4 should be equal. The $200 millio budget is a hard costrait. A preemptive goal programmig approach is to be used. Table 1. Uit data for ivestmet opportuities Ivestmet # Cost, $ Expected retur, $ Variace, $ Model For the liear programmig model we let x j be the umber of uits of ivestmet j purchased. The first techological costrait below limits the amout of moey ivested while the ext three reflect the three goals. We have added costraits to compute the value of the retur R ad variace V. Budget: 20x x x x x G1: 6x x 2 + 6x x 4 + 5x 5 y y - 1 = 45 G2: 40x x x x x 5 y y - 2 = 150 G3: 60x 2 65x 4 y y - 3 = 0 Retur: 6x x 2 + 6x x 4 + 5x 5 R = 0 Variace: 40x x x x x 5 V = 0 x j 0, j = 1,...,5; y + k 0 ad y - k 0, k= 1,2,3; R 0, V 0 G1 is a lower oe-sided goal so we adopt the pealties p 1 - = 1 ad p1 + = 0, ad solve the liear programmig problem Miimize z 1 = y R

17 Goal Programmig 17 subject to the above costraits. Although ot really ecessary, we have added the term ivolvig R to the objective fuctio so that the solutio will deliver the largest retur that satisfies G1. The solutio obtaied is x 1 =10, y 1 + = 15 ad y 2 + = 210, with all other variables 0. Goal 1 is satisfied with a retur of 60, larger tha the goal of 45 (y 1 + = 15). Goal 2 is ot satisfied because the variace is 400, much larger tha the goal of 150 (y 2 + = 250). Goal 3 is satisfied because both x 1 ad x 2 are zero. At the secod iteratio we add a costrait to keep the first goal at the value obtaied i the first iteratio: y 1 - = 0. Sice G2 is a upper oe-sided goal, we use the pealties p 2 - = 0 ad p2 + = 1, ad solve the liear program Miimize z 2 = y R V subject to the above costraits. We have added terms i R ad V to ecourage a large retur ad a small variace. This time the solutio does ot use x 1 but has x 2 =2.06 ad x 4 = The correspodig retur exactly meets the first goal of 45. The values of the secod set of deviatio variables are y 2 - = 11.8 ad y2 + = 0 which idicate that the secod goal is exceeded with a variace of For the third goal, the solutio y 3 + = idicates that there is a differece i the ivestmets i opportuities 2 ad 4. All other variables are 0. At the third iteratio we add aother costrait to keep the secod goal satisfied at its curret value; i.e., y 2 + = 0. G3 is a two-sided goal so we use equal pealties p 3 - = 1 ad p3 + = 1, ad solve the liear program Miimize z 3 = y y R V subject to the above costraits. Rouded to oe decimal poit, the solutio ow calls for ivestmet i three of the opportuities: x 1 =0.7, x 2 = 1.7, x 4 = 1.3. All deviatio variables are 0 except y 3 + = The goals for retur ad variace are exactly met (R = 45 ad V = 150) while the goal associated with the amouts ivested i opportuities 2 ad 4 is withi 13.3 of beig reached. This is the best solutio possible give the preemptive ature of the priorities. The results for iteratio 3 are summarized i Table 2.

18 18 Equivalet Liear Programs Table 2. Summary of results for fial iteratio Opportuity # Total Uits bought Cost, $ Expected retur, $ Variace, $

19 Fractioal Programmig 19 A.4 Fractioal Programmig A umber of situatios arise whe it is desirable to optimize the ratio of two fuctios. I productivity aalysis, for example, oe wishes to maximize the ratio of worker output to labor-hours expeded to perform a task. I fiacial plaig it is commo to maximize the ratio of the expected retur of a portfolio to the stadard deviatio of some measure of performace. Whe the two fuctios are liear, ad the decisio variables are defied over a polyhedral set, we get the followig fractioal programmig problem Maximize f(x) = c 0 + cx d 0 + dx subject to Ax = b, x 0 where c 0 ad d 0 are scalars, ad c ad d are -dimesioal row vectors of coefficiets Uder certai coditios, this optimizatio problem ca be trasformed ito a liear program. I particular, we will assume that x j is so restricted that the deomiator of the fractio is strictly positive ad that the maximum of f(x) is fiite; that is, d 0 + dx > 0 ad f(x) < for all x i {x : Ax = b, x 0}. To put the problem ito a more maageable form, we defie the variable t as ad write the objective fuctio as t 1 d 0 + dx f(x) = c 0 t + cxt. By assumptio, t > 0 for all feasible x j. We ow make the followig chage of variables. y j = x j t or i vector otatio, y = xt. Thus the trasformed model becomes the liear program Maximize c 0 t + cy subject to d 0 t + dy = 1 Ay bt = 0 y 0, t 0 Note that it is permissible to restrict t to be greater tha or equal to zero because of our assumptios. More geerally, whe f(x) = f 1 (x)/f 2 (x)the same kid of trasformatio ca

20 20 Equivalet Liear Programs be used to covert a fractioal program with cocave f 1 (x) ad covex f 2 (x) to a equivalet covex program.

21 The Complemetarity Problem 21 A.5 The Complemetarity Problem Whe ivestigatig the quadratic programmig problem i the oliear programmig methods chapter, we show that it ca be writte as a series of liear equatios i oegative variables subject to a set of complemetarity costraits of the form x j y j = 0 for all j. A specialized liear programmig algorithm ca the be used to fid a solutio. To describe the more geeral situatio, suppose we are give the vectors x = (x 1, x 2,,x ) ad y = (y 1, y 2,,y ). The complemetarity problem is to fid a feasible solutio for the set of costraits y = F(x), x 0, y 0 that also satisfies the complemetarity costrait x T y = 0 where F is a give vector-valued fuctio. The problem has o objective fuctio so techically it is ot a full-fledged oliear program. It is called the complemetarity problem because of the requiremets that either x j = 0 or y j = 0 (or both) for all j = 1, 2,,. A importat special case, which icludes the quadratic programmig problem, is the liear complemetarity problem (LCP) where F(x) = q + Mx. Here, q is a give colum vector ad M is a give matrix. Efficiet algorithms have bee developed for solvig the LCP uder suitable assumptios about the properties of the matrix M. The most commo approach ivolves pivotig from oe basic feasible solutio to the ext, much like the simplex method. Liear ad oliear applicatios of the complemetarity problem ca be foud i game theory, egieerig, ad the computatio of ecoomic equilibria.

22 22 Equivalet Liear Programs A.6 Exercises 1. For the oliear objective fuctio example i Sectio 3.1, chage the breakpoits for the liear approximatio as follows: breakpoits for x 1 are 0, 2 ad 3 ad breakpoits for x 2 are 0, 2 ad 4. Set up ad solve the resultat LP approximatio. 2. The operatios maager of a electroics firm wats to develop a productio pla for the ext six moths. Projected orders for the compay's products are listed below alog with the direct cost of productio i each moth. The pla must specify the mothly amout to produce so that all demad is met. Shortages are ot permitted. Ay amout produced i excess of demad ca be stored i ivetory for later use at a cost of $4/uit/mo. Iitial ad fial ivetories are 0. Demad Productio Moth (uits) cost ($/uit) I additio to the direct costs of productio ad ivetory, overhead costs must be charged for the maximum productio level obtaied ad the maximum ivetory level obtaied durig the 6-moth period. The followig iformatio should be used. (i) Overhead cost for productio = $300 (maximum productio level). (ii) Overhead cost for ivetory = $100 (maximum level of ivetory). These costs are charged oly oce durig the 6-moth period. Set up ad solve the liear programmig model that determies the miimum cost pla. 3. Cosider the situatio described i Exercise 2. Rather tha beig cocered about the overhead costs of productio ad ivetory, it is decided that the problem will be solved with a goal programmig approach. The followig goals have bee established. G1: The average productio cost is to be o more tha $109 per uit. G2: The maximum mothly productio level i the six moths is to be 1500 uits. G3: The maximum mothly ivetory level i the six moths is to be 100 uits. a. Assumig that the priority of the goals is i the order give, use the preemptive sequetial procedure to solve the problem.

23 Exercises 23 b. Assumig the priority of the goals is the reverse of the order give, use the preemptive procedure to solve this problem. 4. For each of the objective fuctios listed below, explai whether or ot a piecewise liear approximatio solved with a liear programmig code will yield a acceptable solutio. I all cases, the variables are restricted to be oegative. a. Maximize f (x) = 2x x 1 x 2 4x x 1 + 4x 2 b. Maximize f (x) = l(x 1 + 1) + l(x 2 + 1) c. Maximize f (x) = x x x 2 d. Miimize f (x) = x x x 2 e. Miimize f(x) = f j (x j ), where f j (x j ) = a j (x j ) b, with a j > 0 ad 0 < b < 1 f. Maximize f(x) = r j (x j ) c j (x j ) where r j = a j (1 exp( b j x j )) with a j > 0 ad b j > 0; c j = d j (x j ) b with d j > 0 ad b > 1 5. Develop a with a piecewise liear approximatio for the oliear objective fuctio i the problem give below ad solve with a LP code. Use the itegers as breakpoits. Maximize z = (x 1 4) 2 (x 2 4) 2 (x 3 4) 2 subject to x 1 + x 2 + x 3 1 x 1 + x 2 + x 3 6 x 1 0, x 2 0, x Cosider a problem of the form Miimize c j x j

24 24 Equivalet Liear Programs subject to a ij x j b i, i = 1,,m where the decisio variables x j are urestricted ad the cost coefficiets c j are all oegative. The objective fuctio comprises the sum of piecewise liear terms c j x j ad ca be show, with ot much difficulty, to be covex. a. Covert the above problem to a liear program by makig use of the ideas i the sectio o maximizig the miimum of several liear fuctios. b. Alteratively, covert the above problem to a liear program by makig use of the fact that ay urestricted variable ca be replaced with the differece of two oegative variables. That is, for x j urestricted, we ca make the substitutio x j x + j x- j, where x+ j 0 ad x- j 0. Note that to achieve the desired result, more tha a direct substitutio is required. 7. You are give m data poits of the form (a i, b i ), i = 1,,m, where a i is a - dimesioal row vector ad b i is a scalar, ad wish to build a model to predict the value of the variable b from kowledge of a specific vector a. I such a situatio, it is commo to use a liear model of the form b = ax, where x is a -dimesioal parameter vector to be determied. Give a particular realizatio of the vector x, the residual, or predictio error, associated with the ith data poit is defied as b i a i x. Your model should explai the available data as best as possible; i.e., produce small residuals. a. Develop a mathematical programmig model that miimizes the maximum residual. Covert your model to a liear program. b. Alteratively, formulate a model that miimizes the sum of the residuals. Covert this model to a liear program. 8. A govermet agecy has five projects that it wishes to outsource. After publishig a aoucemet cotaiig a request for proposals to perform the work, it received bids from three cotractors. The bids are show i the table below. The goal of the agecy is to miimize its total cost. Set up ad solve the liear programmig model uder the followig coditios. a. Each cotractor ca perform as may as two projects; all five projects must be doe. b. Each cotractor ca perform oly oe project ad as may projects as possible should be doe. c. There is o limit to the umber of projects a cotractor ca perform ad all projects must be doe.

25 Exercises 25 Cost of completig the differet projects ($1000) Project 1 Project 2 Project 3 Project 4 Project 5 Cotractor Cotractor Cotractor A productio schedulig problem must be solved over a 12-moth period. All quatities are to be determied o a mothly basis. Parameter defiitios ad related coditios are stated below. Idices o the parameters ad variables rage from 1 to 12. Demad i moth i is d i. This demad must be satisfied i each moth so shortages are ot allowed. Cost of productio i moth i is p i. The maximum productio i moth i is M i. Items produced may be either shipped to meet demad or held i ivetory to meet demad i a subsequet moth. The ivetory level at the ed of moth 12 must equal the iitial ivetory level at time zero (whe the time horizo begis). This quatity is to be determie i the solutio process. The maximum amout that ca be stored from oe moth to the ext is 16 uits. Problem statemet: Develop a model to determie the optimal productio quatity x i i each moth (i = 1,,12), ad the optimal amout to store i ivetory i each moth y i so that total cost is miimized. Write the model for the four cases below. Each should be aswered idepedetly of the others. a. The cost of ivetory is proportioal to the amout stored. The cost is h dollars per uit per moth. b. The cost of ivetory is a oliear fuctio of the amout stored. The total ivetory cost i moth i is h(y i ) = a(y i ) 2, where a is a costat ad has dimesios of dollars per moth. Use a piecewise liear approximatio with the breakpoits take as the powers of 2 (i.e., 0, 2, 4, 8, 16). c. The cost of ivetory depeds o how may moths a uit is stored, ad grows expoetially. The cost for oe moth is a dollars per uit, the cost for two moths is 4a per uit, the cost for three moths is 8a per uit, ad so o.

26 26 Equivalet Liear Programs d. Igore the cost of ivetory ad the limit o the maximum amout that ca be stored, ad miimize the maximum ivetory over the plaig period. 10. A compay maufactures two products, X ad Y, from a mix of chemicals. The products are sold by the poud. Up to 1000 pouds of X ca be sold for $12/lb but the price must be reduced to $9/lb for sales i excess of 1000 lb up to a maximum of 3000 lb i total sales. Product Y is sold for $18/lb for ay amout up to 2000 lb. If more tha 3000 lb of X or more tha 2000 lb of Y are produced, the excess must be discarded at a cost of $2/lb. After processig the mix, the products are withdraw i the followig proportios: 40% is X, 20% is Y, ad 40% is waste that must be discarded at a cost of $2/lb. Processig costs are $1.50/lb. The mix is made up of three raw materials idetified by the letters A, B ad C, ad must be at least 45% raw material A ad o more tha 30% C. Raw material C is free for up to 1500 lb. Material C costs $4.50/lb for amouts above 1500 lb. No more tha 3000 lb of material C is available at ay price. Material A costs $6/lb for ay amout. There is o limit to the amout of A that ca be purchased. Material B costs $3/lb up to 2500 lb ad $5.50/lb for additioal quatities up to a total of 4000 lb. Write out ad solve the liear programmig model that will determie the productio ad sales pla that maximizes profit. Defie all variables, describe each costrait, ad idicate the trasformatios used to liearize ay oliear fuctios.

27 Bibliography 27 Bibliography Bertsimas, D. ad J.N. Tsitsiklis, Itroductio to Liear Optimizatio, Athea Scietific, Belmot, MA, Goicoechea, A., D.R. Hase ad L. Duckstei, Multiobjective Decisio Aalysis with Egieerig ad Busiess Applicatios, Joh Wiley & Sos, New York, Igizio, J.P., Liear Programmig i Sigle- & Multiple-Objective Systems, Pretice Hall, Eglewood Cliffs, NJ, Keeey, R.L. ad H. Raiffa, Decisios with Multiple Objectives: Prefereces ad Value Tradeoffs, Cambridge Uiversity Press, Port Chester, New York, Murty, K.G., Liear Programmig, Joh Wiley & Sos, New York, Se, P., Multiple Criteria Decisio Support i Egieerig Desig, Spriger-Verlag, Lodo, Wikofsky, E.P., N.R. Baker ad D.J. Sweeey, A Decisio Process Model of R&D Resource Allocatio i Hierarchical Orgaizatios, Maagemet Sciece, Vol. 27, No. 3, pp , 1981.