CHAPTER V: LINEAR PROGRAMMING MODELING

Transcription

1 CHAPTER V: LINEAR PROGRAMMING MODELING CHAPTER V: LINEAR PROGRAMMING MODELING Introducton to Algebrac Modelng Defnng Indces and GAMS Sets Wrtng Equatons Unndexed Equatons Indexed Equatons Data Defntons Enterng Data Through Calculaton Specfcaton of Varables Equatons Wrtng Model Equatons Obectve Functon Equatons Constrant Equatons Specfyng Bounds Formng and Solvng a Model Report Wrtng Addtonal GAMS Topcs GAMS and Algebrac Modelng of the Resource Allocaton Problem Example and GAMS Implementaton Model Soluton Comments The Transportaton Problem Example Model Soluton Comments Det/Feed Mx/Blendng Problem Example Model Soluton B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 1

2 5.4.2 Comments Jont Products Example Model Soluton Comments Comments on GAMS and Algebrac Modelng Why Use Algebrac Modelng? Why Use GAMS to Support Modelng? Ad wth Intal Formulaton and Changes n Formulaton Automated Computatonal Tasks Facltates Documentaton and Later Use of Models Allows Use by Vared Personnel Supported by Lbrares of Applcatons References In ths chapter we ntroduce and subsequently rely heavly on the algebrac approach to LP modelng, GAMS usage, dualty mplcatons and some model specfcaton ssues. The chapter begns wth a presentaton of the general algebrac modelng approach wth GAMS. Subsequently, we organze the presentaton around commonly modeled stuatons. The frst problem presented s the classcal resource allocaton problem. Ths s followed by the transportaton and det/feed mx/blendng problems. Followng these s a formulaton that explctly ncorporates ont products. In presentng ths materal, we dentfy dfferent types of varables and constrants used n buldng models, as well as examples of modelng assumptons used when formulatng problems. Appled dualty s also treated. However, the dualty materal s not ntended to mply that the dual of every problem must be formed when modelng. Instead, we dscuss mplcatons for the shadow prces that arse due to prmal varable structures. The emprcal relatonshp between prmal and dual solutons wll also be exhbted, whch hopefully leaves no doubt that when solvng the prmal problem, the dual s smultaneously solved. A thrd theme heren s communcaton of the emprcal ssues nvolved n LP applcaton. Ths s attempted through the development of examples nvolvng coeffcent calculaton and deductve steps n modelng. We wll also demonstrate the lnk between algebrac representatons of problems and emprcal models. Ths dscusson s desgned to show readers the usefulness of algebrac models as a way of conceptualzng problems. Ths partcular chapter s not desgned to stand alone. Addtonal formulatons and algebrac/gams modelng technques are presented throughout the rest of the book. Furthermore, B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 2

3 reference to the GAMS manual (Brooke et al.) s essental to a thorough understandng of the materal. 5.1 Introducton to Algebrac Modelng Now we turn our attenton to the constructon of algebrac mathematcal programmng models usng summaton notaton and GAMS. Ths secton presents a bref ntroducton to a subset of GAMS. The GAMS manual (Brooke et al.) provdes a complete treatment. Ths secton s also supported by the materal n Appendx A Defnng Indces and GAMS Sets A crucal algebrac modelng element s the dentfcaton of ndces (also referred to mathematcally as a subscrpt, or n GAMS as a SET). The compactness wth whch one can specfy a model depends on the characterzaton of tems wth ndces. However, the readablty of the model depends on ndces beng dsaggregate enough to reveal essental features. Examples of the approprate specfcaton of ndces appear throughout the rest of the text. Defnton of ndces n GAMS nvolves the defnton of sets and set members. Namely, an ndex n summaton notaton s a SET n GAMS, and one specfes the range of the ndex by specfyng SET membershp. Examples of such set defntons are ncluded n the followng four GAMS statements. SETS I THE INDEX I /1,2,3,4,5/ CITIES CITY NAMES /BOSTON, PORTLAND / YEAR MODELS YEARS /1970*1974/ WEST WESTERN CITIES /PORTLAND/; The frst tem (CITIES) entered s the label SETS whch suggests that the followng materal contans SET defntons. Here four sets are beng defned and are called I, CITIES, YEAR and WEST. The set WEST s defned as a subset of set CITIES. Wthn each set defnton are three components; 1. the name of the set; 2. a verbal descrpton of what the set stands for (whch keeps a record for future users and helps descrbe the set n the output); and 3. a lst of the set members enclosed between a par of slashes wth multple set elements separated by commas (ths can also contan a defnton of the set element). When a set contans consecutvely numbered elements, an astersk (as n the YEAR set defnton) can be used to denote all elements n a numerc range (.e., 1970*1974 means nclude 1970, 1971, 1972, 1973, and 1974). Fnally, note that a semcolon s used to end the SET statement after all sets have been defned. Set elements can be defned usng numbers, characters, or a mxture of both. In GAMS, one can defne subsets. A subset such as WEST (CITIES) dentfes selected elements dentfyng such thngs as the western ctes. Brooke, et. al. present nformaton on more complex specfcatons. B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 3

4 5.1.2 Wrtng Equatons The obvous purpose of algebrac modelng s to wrte general algebracally based equatons whch encompass a wde varety of stuatons. Two fundamental classes of algebrac equatons exst. The frst class encompasses all sngle equatons whch are not defned wth respect to any knd of ndex. A scalar calculaton falls nto ths class. The second class nvolves equatons whch are defned accordng to ndces. Ths class nvolves famles of smultaneous equatons. For example, smlar equatons may be defned for each locaton, land class, and tme perod Unndexed Equatons A summaton notaton verson of an unndexed equaton s Z 3 a X Ths equaton depcts the sum across all the members of the ndex set where three tmes a s multpled tmes X and the resultant sum s placed nto the value of Z. A related example s Q r / y Notce n both equatons that all ndces are ncluded n the sum, and the result s a scalar number. Ths wll always be true wth unndexed or scalar equatons. There cannot be ndces or sets wthn the equaton whch are not summed over. The GAMS statements equvalent to the above two equatons are Z = SUM (J, 3*A(J)*X(J)); Q = SUM ((I,J), R(I,J)/Y(I,J)); 1 The elements, of these GAMS equatons, mert dscusson. Frst, both equatons equate a scalar to a sum. In dong ths, the set or sets to be summed over are specfed as the parameters to be used n formng the equaton. Note, after the summaton operator (SUM), the ndex or ndces are named, followed by a comma, wth the parameters separated by mathematcal operators. Parentheses must be balanced. Fnally, each expresson s ended by a semcolon. Although these examples are consstent wth the algebrac models, they are poor examples of GAMS modelng. Namely GAMS formulatons can be made much more self explanatory by utlzng set and set member names whch are up to ten characters long. For example, the followng two GAMS commands elaborate on those above, but are much easer to read; INCOME = SUM(PRODUCT, 3* PRICE (PRODUCT) * QUANTITY (PRODUCT)); TOTALQUAN = SUM ((SOURCE, DESTINATION), TOTALCOST (SOURCE, DESTINATION)/COSTPRUNIT(SOURCE, DESTINATION)); 1 Note that these GAMS statements show equatons for numercal calculatons, not equatons for ncluson as a constrant or obectve functon n a lnear programmng model. In such cases a slghtly dfferent syntax s utlzed, as defned n the next secton. B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 4

5 Here the labels help defne what the equaton represents Indexed Equatons The other class of equatons commonly used n algebrac modelng nvolves equatons defned over a famly of ndces. Consder for example : b r m k (f km a Q X k Z m ) for all for all m and where the "for all" statement defnes the famly of ndces for whch the equatons exst. Note that n these equatons the ndex mentoned n the "for all" clause s not summed over. Rather, the for all clause ndcates presence of these equatons for each of these ndces. The equvalent GAMS statements are B(ITEM) = SUM (J, A(ITEM,J)*X(J)) ; R(WHATISIT,J) = SUM (K, F(K,J,WHATISIT)*Q(I,J)) + Z(J,WHATISIT); Data Defntons When usng algebrac modelng, data tems are usually defned symbolcally. For example, a mght be defned as the usage of the th resource per unt of the th product manufactured. In the development of an applcaton, such tems need to be defned numercally. In GAMS, raw data are entered usng the SCALAR, PARAMETER and TABLE syntax. Suppose one wshes to enter sngle numbers whch are not referenced wth respect to a SET. In ths case, one uses the SCALAR syntax as follows: SCALARS LAND LAND AVAILABLE /100/ HOURS MACHINE TIME /50/ PRICESTEEL STEEL PRICE /2.25/; Agan the statements contan several parts. The keyword SCALAR (or SCALARS) ndcates unndexed data tems are beng defned. Then one enters each data tem name followed by a verbal descrpton and a numercal value enclosed n slashes. Ths can be followed by entres for as many scalars as necessary wth the last defnton followed by a sem-colon. Data are also entered usng the PARAMETER syntax. Parameter statements are used to enter ndexed (subscrpted) data n lst form. Examples are gven: PARAMETERS PRICE (ITEM) ITEM PRICES /LABOR 4.00, COAL 100/ B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 5

6 DISTANCE(I,J) INTER CITY DISTANCES /BOSTON.CHICAGO 20, BOSTON.CLEVELAND 15/; Ths syntax must begn wth the word PARAMETER (or PARAMETERS). Subsequently, the name of the data tem, ts ndces and a descrpton appears followed by a slash. Wthn the slashes the name of each data set ndex and ther values are gven. Commas separate set members and ther values. Ths sequence repeats for each set member to be specfed. In the examples, LABOR and COAL are members of the set ITEM wth ther assocated entres n the PRICE data takng on the values 4.00 and 100 respectvely. Smlarly, the set I ncludes BOSTON and the set J ncludes CHICAGO and CLEVELAND. Note, the syntax n the second expresson where BOSTON.CHICAGO ndcates ths s the data for a par of elements ( value then value). Fnally, a semcolon ndcates the PARAMETER statement end. Yet another data entry possblty nvolves the TABLE syntax. Ths syntax s most convenent for defnng mult-dmensonal parameters as llustrated by: TABLE DISTANCE(I,J) DISTANCE BETWEEN CITIES CHICAGO CLEVE BOSTON BALT 20 9 ; TABLE MODECOST(I,J,MODE) INTER CITY COST OF MOVING ITEMS TRUCK RAIL BOSTON.CHICAGO 10 5 BOSTON.CLEVE 8 7 BALT.CHICAGO 10 5 BALT.CLEVE 7 6 ; or TABLE MODECOST(I,J,MODE) INTER CITY COST OF MOVING ITEMS CHICAGO.TRUCK CHICAGO.RAIL CLEVE.TRUCK CLEVE.RAIL BOSTON BALT ; The sets appear n the table structure n the order they are lsted. Thus, for the above table the two dmensonal tem DISTANCE (I,J) s defned over the sets I and J wth the names of the members of the set I appearng vertcally down the left-hand sde of the table whle the member names for J are dentfed horzontally. Data values appear underneath the names of the members of the J set. The second and thrd examples present alternatve ways of enterng a set wth more than two dmensons. In the frst case the frst two sets are handled n the vertcal column, and n the second B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 6

7 case the last two sets are handled across the top. In both cases, the two sets are handled by dentfyng the member of the frst ndexed set followed by a perod then the member name from the second set Enterng Data Through Calculaton One may also calculate data. In ths case, the set dependence of a parameter and a verbal descrpton s specfed wthout data. Then an equaton s specfed to defne the data. For example, suppose the cost of transportng tems was a known lnear functon of mode and dstance. A procedure to mplement ths would be as follows: PARAMETER MODECOST (I,J, MODE) TABLE COSTUNIT (TERM, MODE) COMPUTED COST OF SHIPPING BY MODE; COST DATA FOR SHIPPING MODE TRUCK RAIL CONS PERMILE.5.25 ; MODECOST (I, J, MODE) = COSTUNIT ("CONS",MODE) + COSTUNIT ("PERMILE",MODE)*DISTANCE (I,J); DISPLAY MODECOST; In ths example the PARAMETER statement does not contan data. Rather, calculatons defne the varable MODECOST based on data n the tables DIST and COSTUNIT from the prevous page whch gve dstance and the dependence of transport cost on dstance. After MODECOST s calculated, t s coped to the output usng a DISPLAY statement. The data calculaton feature s powerful as one can enter raw data and then do calculatons of model parameters, and record the assumptons made n calculatng the data. One also can do smple replacements. For example, the statement MODECOST(I,J,MODE) = 5.; would result n havng all elements of MODECOST equal to Specfcaton of Varables In lnear programmng models, the varables wll have nonnegatvty or other sgn condtons. In GAMS ths s dentfed by denotng POSITIVE VARIABLES, NEGATIVE VARIABLES, VARIABLES whch are free varables, and later BINARY or INTEGER VARIABLES. For example POSITIVE VARIABLES VOLUME (ITEM) QUANTITY (CITY1, CITY2) AMOUNT PRODUCED BY ITEM AMOUNT SHIPPED BETWEEN CITIES; B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 7

8 NEGATIVE VARIABLES LOSS AMOUNT OF MONEY LOST; VARIABLES OBJ OBJECTIVE FUNCTION VALUE; Note that the varable and ts sgn restrcton are smultaneously defned. A POSITIVE specfcaton means that the varable s restrcted to be greater than or equal to zero, NEGATIVE means less than or equal to zero, and the word VARIABLE means unrestrcted n sgn. After these statements, the name of each varable along wth ts set dependence appears, and may be followed by a short descrpton of the varable. The statement lsts each varable wth the partcular sgn restrcton, followed by a semcolon. Every varable used n a GAMS model must be defned and the model must contan at least one unrestrcted varable whch s the tem optmzed Equatons The set of equatons whch appear n the model are specfed through the EQUATIONS statement. EQUATIONS OBJECTIVE OBJECTIVE FUNCTION AVAILABLE (RESOURCES) RESOURCE AVAILABILITY CONSTRAINTS SUPPLY (PLANT) SUPPLY AVAILABLE AT A PLANT LIMIT (RESOURCES, PLANT) RESOURCES AVAILABLE BY PLANT; These expressons nclude the name of the equaton, an ndcaton of the sets for whch ths equaton exsts and a verbal descrpton. Entres appear for each of the model equatons followed by a semcolon Wrtng Model Equatons After the model equatons have been defned, then they are defned algebracally. The algebrac specfcaton dffers somewhat dependng on whether the obectve functon or the constrants are beng entered Obectve Functon Equatons The obectve functon equaton s typcally of the form: Max c x However, GAMS forces the modeler to rewrte the equaton so that the obectve functon expresson s set equal to an unrestrcted varable, and that varable s named n the solve statement. Thus we alter the problem to become B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 8

9 Max S.t. Z Z c x where Z s a varable whch s unrestrcted n sgn (named n the GAMS VARIABLES lst). Specfyng the obectve functon equaton n GAMS requres the specfcaton of an equaton name followed by an algebrac representaton. Suppose we label the obectve functon equaton OBJECTIVEF. The resultant GAMS statement s : Obectvef.. Z =E= SUM (J, c(j)*x(j)); Note the structure, frst the equaton s named followed by two perods. Then the algebrac statement appears where the unrestrcted varable that s maxmzed s set equal to an expresson for the obectve functon followng the rules for equaton wrtng as dscussed above wth one excepton. That s, here the equalty s wrtten =E=. Fnally, the expresson s followed by a semcolon Constrant Equatons The constrant equatons are entered n essentally the same form as the obectve functon. Frst the equaton s named. In ths namng any ndces(sets) over whch the equatons are defned are named. Then ths s followed by two perods and the algebrac equaton for the constrant. Subsequently, an ndcaton of the form of the nequalty appears (=L= for less than or equal to; =G= for greater than or equal to; and =E= for equal to). Fnally, ths s followng by the expresson for the rght-hand-sde and a sem-colon. Consder for example the constrants: f Q f mq Z m 0 m k m a Z X kp Z m for all m B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 9 b d p 4 for all for all p where the "for all" statement defnes the famly of ndces for whch the equatons exst. Suppose we label the frst constrant RESOURCE ndexed over the set ITEM. Smlarly, we call the second constrant DEMAND ndexed over PLACE, the thrd constrant SOMEITEM ndexed over WHAT and the fourth ADDIT. The resultant GAMS statements are: RESOURCE(ITEM).. SUM (J, A(ITEM,J)*X(J)) =L= B(ITEM); DEMAND(PLACE).. SUM (K, Z(K,PLACE)) =G= D(PLACE); sometem(what).. Sum(,sum(,f(,,what)*q(,))+z(,what)) =e=4; addt.. Sum((m,),sum(,f(,,what)*q(,))+z(,what)) =e=0; The equatons follow the equaton wrtng conventons dscussed above and n Appendx A. The

10 only excepton s that one may mx constants and varables on ether sde of the equaton. Then one could wrte the frst equaton as: RESOURCE(ITEM).. B(ITEM) =G= SUM (J, A(ITEM,J)*X(J)); Specfyng Bounds Upper and lower bounds on ndvdual varables are handled n a dfferent fashon. For example, f the varables VOLUME and QUANTITY had been defned, and were to be bounded, then the bounds would be defned through statements lke the followng VOLUME.UP (ITEM) = 10.; QUANTITY.LO (CITY1, CITY2) = MINQ (CITY1, CITY2); VOLUME.UP ("TREES") = LIMIT ("TREES"). In the frst example, upper bounds are specfed for the varable VOLUME across all members of the set ITEM equalng 10. In the second example, the bound s mposed through a prespecfed parameter data. In the thrd example, the bound s calculated Formng and Solvng a Model After specfyng sets, varables, data, and other approprate nput, one enters statements whch defne the model and nvoke the solver. The MODEL defnton can be of two forms. In the frst form, the expresson MODEL RESOURCE /ALL/ ; gves the model a name (RESOURCE), and specfes that ALL equatons are ncorporated. Alternatvely, the statement MODEL COSTMIN /OBJECTIVE,SUPPLY,DEMAND/; gves the model name and the names of the equatons to nclude. In ths case the model name s COSTMIN and the equatons ncluded are OBJECTIVE, SUPPLY and DEMAND. In turn, the SOLVE statement s SOLVE COSTMIN USING LP MINIMIZING OBJ; where the general syntax s SOLVE "modelname" USING LP MAXIMIZING [OR MINIMIZING] "obectve functon varable name." The obectve functon varable name s the varable added and set equal to the mathematcal expresson for the obectve functon one may also solve mxed nteger problems by alterng the usng phrase to "usngmp" and nonlnear problems wth "usng NLP". B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 10

11 Report Wrtng GAMS has useful features whch allow soluton results to be used n calculatons to compute varous tems of nterest or do varous result summares. One can, at any tme after the SOLVE statement, use: a) a varable name followed by a.l to obtan the value of the soluton for that varable; b) a varable name followed by a.m to get the reduced cost assocated wth that varable; c) an equaton name followed by.l to get the value of the left hand sde of that equaton; and d) an equaton name followed by.m to get the shadow prce assocated wth the equaton. Slack varables can be computed by takng an equaton rght hand sde mnus the assocated.l value. The DISPLAY statement can be used to prnt out calculaton results Addtonal GAMS Topcs Two addtonal GAMS topcs are worth mentonng. These nvolve the use of condtonal statements and the use of loops. Condtonal statements are utlzed to ndcate that there are cases where partcular tems should not be defned when dong a calculaton and nvolve a syntax form usng a $. Two examples are DEMAND(REGION)$DEMANDQ(REGION).. SUM (SUPPLY, QUAN(SUPPLY, REGION)) =G= DEMANDQ(REGION); OBJECTIVE.. OBJ = SUM((I,J)$(DIST(I,J) LT 20), COST(I,J)*QUAN(I,J)); In the frst case, the notaton $DEMANDQ(REGION) tells GAMS to generate ths equaton only f the DEMANDQ term s nonzero resultng n a DEMAND equaton beng generated for only those members of REGION whch have nonzero demand. In the second expresson, the $(DIST(I,J) LT 20) clause ndcates that one should add (,) pars nto the sum only f the DISTANCE between the par s less than 20. In general, use of the $ notaton allows one to mpose condtons wthn the model setup and data calculaton procedures. LOOP statements are also worth mentonng as they can be used to repeatedly execute a set of GAMS statements. Examples appear n the DIET example on the dsc where the LOOP s utlzed to vary a prce and solve over and over agan. Smlarly, the EVPORTFO example uses a LOOP to solve for alternatve rsk averson parameters. 5.2 GAMS and Algebrac Modelng of the Resource Allocaton Problem The classcal LP problem nvolves the allocaton of an endowment of scarce resources among a number of competng products so as to maxmze profts. In formulatng ths problem algebracally let us pursue the logcal steps of: a) ndex dentfcaton, b) varable dentfcaton, c) constrant dentfcaton and d) obectve functon dentfcaton. B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 11

12 Parameter dentfcaton s mplct n steps c and d. The ndces necessary to formulate ths problem are nherent n the problem statement. Snce there are "competng products" we need an ndex for products whch we defne here as. The ndex may take on the values of 1 through n ndcatng that there are n dfferent competng products. The problem statement also mentons "scarce resources", so we defne the ndex to represent each of m possble resources (=1,2,...,m). Now suppose we turn our attenton to varables. Two potental varables are mplct n the above statement. The frst type nvolves varables ndcatng the allocaton of resources and the second type s assocated wth the competng products. In structurng ths problem we would have to know the exact amount of each resource that s used n makng each product. Thus, when we know how much of a product s made, we know the quantty of each resource allocated to that product. Ths means the varable n ths problem nvolves the amount of each product to be made. Suppose we defne a varable X whch s the number of unts of the th product made. We may now defne constrant equatons. In ths case, a constrant s needed for each of the scarce resources. The constrant forces resource usage across all producton possbltes to be less than or equal to the resource endowment. An algebrac statement of ths restrcton requres defntons of parameters for resource usage by product and the resource endowment. Let a depct the number of unts of the th resource used when producng one unt of the th product and let b depct the endowment of the th resource. We now can construct an algebrac form of the constrant. Namely we need to sum up the total usage of the th resource. The total usage of resource s the sum of the per unt resource usage tmes the number of unts produced (a X ). The approprate algebrac statement of the th constrant s a X b wth a constrant defned for each of the resources () n the problem. A condton s also needed whch states only zero or postve producton levels are allowed for each producton possblty. X 0, for all. Turnng to the obectve functon, an equaton for profts earned s also needed. Ths nvolves the defnton of a parameter for the proft contrbuton per unt of the th product (c ). Then, the algebrac statement of the obectve functon s the sum of the per unt proft contrbuton tmes the amount of each product made or c X whch s to be maxmzed. Algebracally, the LP formulaton of the problem s B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 12

13 Max s.t. c X a X X b 0 for all for all where s ndexed from 1 to n, s ndexed from 1 to m, c s the proft per unt of actvty, X s the number of unts of actvty produced, a s the number of unts of resource requred per unt of product, and b s the endowment of resource. Ths formulaton arose early n the development of LP. Whle the exact problem s not stated as above, there are problems very close to t n Koopman's 1949 symposa report. The formulaton explctly appears n Dorfman's 1951 book. Kantorovch also presents an early example. Over tme, ths partcular formulaton has been wdely used. In fact, t provdes the frst example n vrtually all textbooks Example and GAMS Implementaton Suppose that E-Z Char Makers are tryng to determne how many of each of two types of chars to produce. Further, suppose that E-Z has four categores of resources whch constran producton. These nvolve the avalablty of three types of machnes: 1) large lathe, 2) small lathe, and 3) char bottom carver; as well as labor avalablty. Two types of chars are produced: functonal and fancy. A functonal char costs $15 n basc materals and a fancy char $25. A fnshed functonal char sells for $82 and a fancy char for $105. The resource requrements for each product are shown n Table 5.1. The shop has flexblty n the usage of equpment. The chars may have part of ther work substtuted between lathes. Labor and materal costs are also affected. Data on the substtuton possbltes are gven n Table 5.2. Assume the avalablty of tme s 140 hours for the small lathe, 90 hours for the large lathe, 120 hours for the char bottom carver, and 125 hours of labor. Ths problem can be cast n the format of the resource allocaton problem. Sx dfferent char/producton method possbltes can be delneated. Let X 1 = the number of functonal chars made wth the normal pattern; X 2 = the number of functonal chars made wth maxmum use of the small lathe; X 3 = the number of functonal chars made wth maxmum use of the large lathe; X 4 = the number of fancy chars made wth the normal pattern; X 5 = the number of fancy chars made wth maxmum use of the small lathe; X 6 = the number of fancy chars made wth maxmum use of the large lathe. The obectve functon coeffcents requre calculaton. The basc formula s that profts for the th actvty (c ) equal the revenue to the partcular type of char less the relevant base materal costs, less any relevant cost ncrease due to lathe shfts. Thus, the c for X 1 s calculated by subtractng 15 from 82, yeldng the entered 67. The constrants on the problem mpose the avalablty of each of B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 13

14 the four resources. The techncal coeffcents are those appearng n Tables 5.1 and 5.2. The resultant LP model n algebrac form s Max 67X X X X X X 6 s.t. 0.8X X X X X X 6 < X X X X X X 6 < X X X 3 + X 4 + X 5 + X 6 < 120 X X X X X X 6 < 125 X 1, X 2, X 3, X 4, X 5, X 6 > 0 Ths problem can be cast nto an algebrac modelng system lke GAMS n numerous ways. Two approaches are presented here. The frst s fathful to the above algebrac formulaton and the second s more talored to the problem. Consder the frst formulaton as shown n Table 5.3 and the fle called RESOURCE. The approach we employ n settng up the GAMS formulaton nvolves a more extensve defnton of both varables and constrants than n the algebrac format. The ndces are defned as GAMS sets, but nstead of usng the short names and longer names are used. Thus, n statement one the set (whch stands for the dfferent products that can be produced) s named PROCESS and the defnton of the elements of (rather than beng 1-6) are mnemoncs ndcatng whch of the products s beng produced and whch lathe s beng used. Thus, FUNCTNORM stands for producng functonal chars usng the normal process whle the mnemonc FANCYMXLRG stands for producng fancy chars wth the maxmum use of the large lathe. Smlarly, n lne four, the subscrpt s named RESOURCE. In turn, the parameters reflect these defntons. The avalablty of each resource (b ) s specfed through the RESORAVAIL parameter n lnes The per unt use of each resource by each producton process (a ) s entered through the RESOURSUSE defnton n lnes The obectve functon proft parameter (c ) s not drectly defned but rather s calculated durng model setup. The nputs to ths calculaton are defned n the data wth the per unt prces of the chars defned by process n PRICE lnes 8-9 and the producton cost defned n lnes In turn, the parameters are used n computng the obectve functon n lne 39 where prce mnus producton cost s computed. The varables (X ) are defned n lnes where the POSITIVE specfcaton means greater than or equal to zero. Note that a varable named PROFIT s defned n lne 33. Ths varable does not have a counterpart n the algebrac model. Ths unrestrcted varable s requred by GAMS and wll be the quantty maxmzed. Subsequently, the two equatons are specfed. The equaton for the obectve functon s not ndexed by any sets and s defned n lne 35. The resource constrant equaton ndexed by the set RESOURCE appears n lne 36. In turn then, the algebrac statement of the obectve functon appears n lnes whch sums net profts over all possble producton processes by computng prce mnus the producton cost tmes the level produced. The resource constrants sum resource use over all the processes and hold ths less than or equal to the resource avalablty (lnes 42-44). The fnal step n the GAMS mplementaton s to specfy the model name whch contans all the equatons (lne 46) and then a request to solve the model (lne 47). Another GAMS formulaton of the problem appears n Table 5.4 and fle RESOURC1. Ths s less fathful to the algebrac formulaton, but s presented here snce t shows more about GAMS. B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 14

15 Here, the subscrpt defnton s broken nto two parts. One of whch reflects the type of CHAIR beng made and the second the type of PROCESS beng utlzed. Thus, FUNCT.NORMAL refers to a functonal char usng the normal process. In turn, ths allows PRICE and BASECOST to be specfed by CHAIR, but the EXTRACOST from usng the addtonal processes needs to be specfed n terms of both CHAIR and PROCESS. The resource usage matrx now has three dmensons: one for RESOURCE, one for CHAIR, and one for PROCESS. Also, the dmensons of the RESOURCUSE array are changed accordngly (lne 20). The model then proceeds bascally as before wth the PRODUCTION varable now havng two ndces, CHAIR and PROCESS (lne 33). The specfcaton also means a slghtly more complex expresson n the obectve functon n whch the net return to a partcular char, usng a partcular process s calculated as char prce mnus producton cost, mnus extra cost. In turn, that s multpled by the producton level and summed (lnes 41-42). Examnng and contrastng these formulatons shows some of the power of GAMS. Namely, n both formulatons, one can put n raw nformaton such as char prces and costs and then computatons to setup the model. Ths leaves a record of how the numbers were calculated rather than havng model coeffcents computed elsewhere. In addton, the use of longer names (up to 10 characters) n specfyng the model, means that the GAMS nstructons can be wrtten n suffcent detal so that they are easly understood. For example, look at the obectve functon equaton n Table 5.4. Note, that t contans producton level tes, the prce of the char mnus the cost of the char mnus the extra cost by a process. Ths s a much more readly accessble formulaton than exsts n many computatonal approaches to lnear programmng (see the materal below or treatments lke that n Beneke and Wnterboer) Model Soluton The resultant soluton under ether formulaton gves an optmal obectve functon value of 10, The optmal values of the prmal varables and shadow prces are shown n Table 5.5. Ths soluton ndcates that functonal chars and fancy chars should be made usng the normal process, whle 5.18 fancy chars should be made usng maxmum use of a large lathe. Ths producton plan exhausts small and large lathe resources as well as labor. The dual nformaton ndcates that one more hour of the small lathe s worth $33.33, one more hour of the large lathe $25.79, and one more hour of labor $ The reduced cost valuaton nformaton also shows, for example, that functonal char producton wth maxmum use of a small lathe would cost $11.30 a char. Fnally, there s excess capacty of hours of char bottom carvng Comments The resource allocaton problem s the most fundamental LP problem. Ths problem has many uses; however, most uses nvolve slghtly more complex problem structures whch wll be dscussed n the remander of the book. A number of modelng assumptons are mplct n ths formulaton. Frst, the prce receved for producton of chars s ndependent of the quantty of chars produced. The frm would receve the same prce per unt whether t produced 5 or 500 mllon chars. Ths may be unrealstc for large frms. Increased producton may brng ether an ncreasng or decreasng prce. Representaton of B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 15

16 decreasng returns s presented n the nonlnear approxmaton chapter and n the prce endogenous modelng chapter. The nteger programmng chapter contans a formulaton for the case where prces ncrease wth sales. A second assumpton of the above formulaton s that the fxed resource avalablty does not change regardless of ts value. For example, the E-Z char problem soluton placed a value on labor of $27 an hour. However, the frm may feel t can afford more labor at that prce. Consequently, one may wsh to extend the model so that more of the resources become avalable f a suffcently hgh prce would be pad. Ths general topc wll be covered under purchase actvtes whch are ntroduced n the ont products problem below. It s also covered n the nonlnear approxmaton and prce endogenous chapters. Fnally, consder an assumpton that does not characterze the above formulaton. Many people ordnarly regard the resource allocaton problem as contanng a fxed coeffcent producton process where there s only a sngle way of producng any partcular product. However, we ncluded multple ways of producng a product n ths example problem to show that a LP model may represent not only one way of producng a product, but also dfferent ways nvolvng dfferent combnatons of nputs. 5.3 The Transportaton Problem The second problem covered s the transportaton problem. Ths problem nvolves the shpment of a homogeneous product from a number of supply locatons to a number of demand locatons. Ths problem was orgnally formulated by Kantorovch n 1939 and Htchcock n Koopmans (1949) restated the model and spurred research n the area. Settng ths problem up algebracally requres defnton of ndces for: a) the supply ponts whch we wll desgnate as, and b) the demand locatons whch we wll desgnate as. In turn, the varables ndcate the quantty shpped from each supply locaton to each demand locaton. We defne ths set of varables as X. There are three general types of constrants, one allowng only nonnegatve shpments, one lmtng shpments from each supply pont to exstng supply and the thrd mposng a mnmum demand requrement at each demand locaton. Defnton of the supply constrant requres specfcaton of the parameter s whch gves the supply avalable at pont, as well as the formaton of an expresson requrng the sum of outgong shpments from the th supply pont to all possble destnatons ( ) to not exceed s. Algebracally ths expresson s X s Defnton of the demand constrant requres specfcaton of the demand quantty d requred at demand pont, as well as the formaton of an expresson summng ncomng shpments to the th demand pont from all possble supply ponts ( ). Algebracally ths yelds X d. Fnally, the obectve functon depcts mnmzaton of total cost across all possble shpment B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 16

17 routes. Ths nvolves defnton of a parameter c whch depcts the cost of shppng one unt from supply pont to demand pont. In turn, the algebrac formulaton of the obectve functon s the frst equaton n the composte formulaton below. Mn s.t. cx X X X s d 0 for all for all for all, Ths partcular problem s a cost mnmzaton problem rather than a proft maxmzaton problem. The transportaton varables (X ) belong to the general class of transformaton varables. Such varables transform the characterstcs of a good n terms of form, tme, and/or place characterstcs. In ths case, the transportaton varables transform the place characterstcs of the good by transportng t from one locaton to another. The supply constrants are classcal resource avalablty constrants. However the demand constrant mposes a mnmum level and consttutes a mnmum requrement constrant. Suppose we address dualty before turnng to GAMS and an example. The dual of the transportaton problem establshes mputed values (shadow prces) for supply at the shppng ponts and demand at the consumpton ponts. These values are establshed so that the dfference between the value of demand and the cost of supply s maxmzed. The dual problem s gven by 2 Max s.t. - - s U U U, d V V V c 0 for all, for all, where U s the margnal value of one unt avalable at supply pont, and V s the margnal value of one unt demanded at demand pont. The dual problem s best explaned by frst examnng the constrants. The constrant assocated wth X requres the value at the demand pont (V ) to be less than or equal to the cost of movng the good from pont to pont (c ) plus the value at the supply pont (U ). Consequently, the model requres the margnal value of supply at the supply pont plus the transportaton cost to be no smaller than the value at the demand pont. Ths also requres the dfferences n the shadow prces between demand and supply ponts to be no greater than the transport cost. Ths requrement would arse n a hghly compettve market. Namely f someone went nto the transportaton busness quotng delvery and product acquston prces, one could not charge the demanders of the good more than the prce pad for a good plus the cost of movng t, or compettors would enter the market takng away busness. Ths also shows the general nature of 2 Ths formulaton follows after the supply equaton has been multpled through by -1 to transform t to a greater-than constrant. B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 17

18 the dual constrant mposed by a prmal transformaton varable. Namely, a restrcton s mposed on the dfference n the shadow prces. The dual obectve functon maxmzes profts whch equal the dfference between the value of the product at the demand ponts (d V ) and the cost at the supply ponts (s U ) Example ABC Company has three plants whch serve four demand markets. The plants are n New York, Chcago and Los Angeles. The demand markets are n Mam, Houston, Mnneapols and Portland. The quantty avalable at each supply pont and the quantty requred at each demand market are Supply Avalable Demand Requred New York 100 Mam 30 Chcago 75 Houston 75 Los Angeles 90 Mnneapols 90 The assumed dstances between ctes are Portland 50 Mam Houston Mnneapols Portland New York Chcago Los Angeles Assume that the frm has dscovered that the cost of movng goods s related to dstance (D) by the formula -- Cost = 5 + 5D. Gven these dstances, the transportaton costs are Mam Houston Mnneapols Portland New York Chcago Los Angeles The above data allow formulaton of an LP transportaton problem. Let denote the supply ponts where =1 denotes New York, =2 Chcago, and =3 Los Angeles. Let represent the demand ponts where =1 denotes Mam, =2 Houston, =3 Mnneapols, and =4 Portland. Next defne X as the quantty shpped from cty to cty ; e.g., X 23 stands for the quantty shpped from Chcago to Mnneapols. A formulaton of ths problem s gven n Table 5.6. Ths formulaton may also be presented n a more compact format as Mnmze < 100 B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 18

19 < < > > > > 50 Ths shows the common ncdence n LP formulatons of sparsty. Although there s room for 84 coeffcents n the body of the constrant matrx only 24 of these are non-zero. Thus, the problem s only 24/84ths dense, revealng a large number of zeros wthn the body of the matrx. The rght hand sdes llustrate endowments of supply through the frst three constrants, and mnmum requrements n the next four constrants. The varables nvolve resource usage at the supply pont and resource supply at the demand pont. These actvtes transform the place utlty of the good from the supply pont to the demand pont. The GAMS mplementaton s presented n Table 5.7 and n the fle TRNSPORT. Two sets are defned (lnes 1-4) one dentfes the supply plants, the other defnes the demand markets. Subsequently, supply avalablty and demand requrements are specfed through the parameter statements n lnes The dstance between the plants s specfed n lnes Followng ths, unt transport costs are computed by frst defnng the parameter for cost n lne 20, and then expressng the formula of $5.00 fxed cost plus $5.00 tmes the dstance n lne 21. Next, the nonnegatve shpment varables are specfed (X ). The unrestrcted varable called TCOST equals the obectve functon value. Then, three equatons are specfed, one equates the obectve value to TCOST and the other two mpose the supply and demand lmtatons as n the algebrac model. These n turn are specfed n the next few statements, then the model s specfed usng all the equatons and solved mnmzng the TCOST varable Model Soluton The soluton to ths problem s shown n Table 5.8. The optmal value of the obectve functon value s 7,425. The optmal shppng pattern s shown n Table 5.9. The soluton shows twenty unts are left n New York's potental supply (snce constrant 1 s n slack). All unts from Chcago are exhausted and the margnal value of addtonal unts n Chcago equals $15 (whch s the savngs realzed f more supply were avalable at Chcago whch allowed an ncrease n the volume of Chcago shpments to Mnneapols and thereby reducng New York-Mnneapols shpments). The soluton also shows what happens f unused shppng routes are used. For example, the antcpated ncrease n cost that would be necessary f one were to use the route from New York to Portland s $75, whch would ndcate a reshufflng of supply. For example, Los Angeles would reduce ts shppng to Portland and ncrease shppng to somewhere else (probably Houston). The assocated dual problem s: B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 19

20 Max -100U 1-75U 2-90U V V V V 4 s.t. -U 1 + V 1 < 20 -U 1 + V 2 < 40 -U 1 + V 3 < 35 -U 1 + V 4 < 120 -U 2 + V 1 < 50 -U 2 + V 2 < 60 -U 2 + V 3 < 20 -U 2 + V 4 < 70 -U 3 + V 1 < 90 -U 3 + V 2 < 35 -U 3 + V 3 < 70 -U 3 + V 4 < 40 The constrants of ths problem are wrtten n the less than or equal to, for requrng the value at demand pont to be no more than the value at the shppng pont plus the transportaton cost. The soluton to the dual problem yelds an obectve functon of 7,425 wth the optmal values for the varables shown n Table A comparson of Table 5.8 and Table 5.10 reveals the symmetry of the prmal and dual solutons. The values of the X n the prmal equal the shadow prces of the dual, and vce versa for the shadow prces of the prmal and the U and V n the dual Comments The transportaton problem s a basc component of many LP problems. It has been extended n many ways and has been wdely used n appled work. A number of assumptons are contaned n the above model. Frst, transportaton costs are assumed to be known and ndependent of volume. Second, supply and demand are assumed to be known and ndependent of the prce charged for the product. Thrd, there s unlmted capacty to shp across any partcular transportaton route. Fourth, the problem deals wth a sngle commodty or an unchangng mx of multple commodtes. These assumptons have spawned many extensons, ncludng for example, the transshpment problem (Orden), wheren transshpment through ntermedate ctes s permtted. Another extenson allows the quantty suppled and demanded to depend on prce. Ths problem s called a spatal equlbrum model (Takayama and Judge(1973)) and s covered n the prce endogenous chapter. Problems also have been formulated wth capactated transportaton routes where smple upper bounds are placed on the shpment from a supply pont to a demand pont (.e., X less than or equal to UL ). These problems are generally n the purvew of network theory (Bazarra et. al., Kennngton). Mult-commodty transportaton problems have also been formulated (Kennngton). Cost/volume relatonshps have been ncluded as n the warehouse locaton model n the second nteger programmng chapter. Fnally, the obectve functon may be defned as contanng more than ust transportaton costs. Ordnarly one thnks of the problem wheren the c s the cost of B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 20

21 transportng goods from supply pont to demand pont. However, modelers such as Barnett et al. (1984) have ncluded the supply cost, so the overall obectve functon then nvolves mnmzng delvered cost. Also the transport cost may be defned as the demand prce mnus the transport cost mnus the supply prce, thereby convertng the problem nto a proft maxmzaton problem. The transport model has nterestng soluton propertes. The constrant matrx possesses the mathematcal property of unmodularty (see Bazaraa et al. (1984) for explanaton) whch causes the soluton to be nteger-valued when the rght hand sdes are ntegers. Also, when the sum of the supples equals the sum of the demands, then the problem s degenerate, and the dual wll have multple optmal solutons. That s, the supply and demand prces wll not be unque. The transportaton problem has been the subect of consderable research and applcaton. The research has led to specal soluton approaches ncludng the Ford and Fulkerson out of klter algorthm, and specalzatons of the prmal smplex algorthm. Dscusson of such topcs can be found n Dantzg (1951), Bradley et al., and Glover et al. (1974). It s worthwhle to pont out that the algorthms allow the problem to be solved at a factor of 100 tmes faster than the use of the general smplex method. Readers attemptng to solve transportaton problems should consult the network lterature for effcent soluton procedures. There have been many applcatons of dfferent versons of the transportaton problem. For example, t has been used to study the effect of ralroad regulatory polcy (Baumel et al.), gran subtermnal faclty locaton (Hlger et al.), and port locaton (Barnett et al.(1984)). The assgnment and/or contract award problems are transport problems whch arose early n the development of LP (see the assgnment and contract awards sectons n Rley and Gass). There also are related formulatons such as the caterer problem. 5.4 Det/Feed Mx/Blendng Problem One of the earlest LP formulatons was the det problem along wth the assocated feed mx and blendng varants. The det context nvolves composng a mnmum cost det from a set of avalable ngredents whle mantanng nutrtonal characterstcs wthn certan bounds. Usually, a total detary volume constrant s also present. Stgler studed the problem before LP was developed. However, he noted that tastes and preferences cause a dsparty between observed and mnmum cost human dets. Bascally the human det form of ths problem takes a largely expostory role wth few applcatons. However, Waugh appled LP to the lvestock feed formulaton problem, and t has become one of the most wdely used LP applcatons. The formulaton has also been appled to the blendng of ce cream, sausage, and petroleum (Rley and Gass). The basc model s formulated as follows. Defne ndex ( ) representng the nutrtonal characterstcs whch must fall wthn certan lmts (proten, calores, etc.). Defne ndex ( ) whch represents the types of feedstuffs avalable from whch the det can be composed. Then defne a varable ( F ) whch represents how much of each feedstuff s used n the det. The constrants of the problem nclude the normal nonnegatvty restrctons plus three addtonal constrant types: one for the mnmum requrements by nutrent, one for the maxmum requrements by nutrent and one for the total volume of the det. In settng up the nutrent based constrants parameters are needed whch tell how much of each nutrent s present n each feedstuff B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 21

22 as well as the detary mnmum and maxmum requrements for that nutrent. Thus, let a be the amount of the th nutrent present n one unt of the th feed ngredent; and let UL and LL be the maxmum and mnmum amount of the th nutrent n the det. Then the nutrent constrants are formed by summng the nutrents generated from each feedstuff (a F ) and requrng these to exceed the detary mnmum and/or be less than the maxmum. The resultant constrants are a F UL a F LL A constrant s also needed that requres the ngredents n the det equal the requred weght of the det. Assumng that the weght of the formulated det and the feedstuffs are the same, ths requrement can be wrtten as F 1. Fnally an obectve functon must be defned. Ths nvolves defnton of a parameter for feedstuff cost (c ) and an equaton whch sums the total det cost across all the feedstuffs,.e., c F The resultng LP formulaton s Mn s.t. a a c F F F F F UL LL 1 0 for all for all for all Ths formulaton depcts a cost mnmzaton problem. The F actvtes provde an example of purchase varables, depctng the purchase of one unt of feed ngredents at an exogenously specfed prce whch, n turn, provdes the nutrent characterstcs n the mxed det. The constrants are n the form of resource lmts and mnmum requrements. The dual of ths problem contans varables gvng the margnal value of the nutrent upper and lower lmts, as well as the value of the overall volume constrant. The dual s B.A. McCarl and T.H. Spreen, 2013 LINEAR PROGRAMMING MODELING 22