Sintassi di LINGO. Model: MAX = 1Drop + 1.5Deco; Drop <= 400; Deco <= 200; 1/60Drop + 3/60Deco <=16; end

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Myles Green
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1 Sntass d LINGO Model: MAX = *Drop +.5*Deco; Drop <= 400; Deco <= 200; /60*Drop + 3/60*Deco <=6; Comments n the model are ntated wth an eclamaton pont (!) and appear n green tet. LINGO specfed operators and functons appear n blue tet. All other tet s shown n black. Each LINGO statement must n a sem-colon (;). Varable names are not case-senstve and must begn wth a letter (A-Z). Other characters n the varable name may be letters, numbers (0-9), or the underscore character (_) Varable names can be up to 32 characters n length. SETS LINGO allows you to group many nstances of the same varable nto sets. For eample, f a model nvolved 27 delvery trucks, then these 27 trucks could be descrbed more smply as a sngle set. Sets may also nclude attrbutes for each member, such as the haulng capacty for each delvery truck. Sets may be ether prmtve or derved. A prmtve set s one that contans dstnct members. A derved set, however, contans other sets as ts members. To use sets n a model, the specal secton called the SETS secton must be defned before any of the set members are used n the model s constrants. Ths secton begns wth the tag and s wth the tag. A prmtve set could be defned as follows: Trucks/TR..TR27/:Capacty; Ths set s gven the setname Trucks and contans 27 members, dentfed by TR TR27. The attrbutes for each member are called Capacty. The derved set s defned smlarly, but must also nclude the parent set lst. An eample of a derved set could be: Product/X Y/; Machne/L M/; Make(Product Machne)/X L, X M, Y M/; Ths set declaraton defnes two prmtve sets, Product and Machne, and one derved set called Make. The Make set s derved from the parent sets Product and Machne. Members are specfed as shown. Notce that a fourth Product-Machne combnaton, Y L, could be theoretcally possble. Ths eample does not allow for such a combnaton. If all combnatons of the parent sets are possble, then no member set need be defned. An attrbute lst for the derved set can also be ncluded n the same way as for a prmtve set. Several set loopng functons are also avalable for use n LINGO. These functons are as generates constrants over members of a sums an epresson over all members of the computes the mnmum of an epresson over all members of the set.

2 computes the mamum of an epresson over all members of the set. Each of the above loopng functons has a smlar form of synta and the loopng functons can even be nested. Eamples of epressons usng each type of loopng functon are as follows: statement sets the haulng capacty for all 27 delvery trucks n the Trucks set to at most 3000 Capacty(T)<=3000); statement calculates the total haulng capacty from the ndvdual trucks: TOTAL_HAUL=@SUM(Trucks(J): Capacty(J)); statements fnd the etreme haulng capacty levels from the ndvdual delvery trucks: MIN_HAUL Capacty(J)); MAX_HAUL Capacty(J)); Data LINGO provdes a separate secton called the DATA secton n whch values can be defned for dfferent varables. Set members can be ntalzed n ths secton, attrbutes of the sets can be defned, or scalar varable parameters can be assgned values as well. The DATA secton s defned after the SETS secton s defned n the model. The secton begns wth the tag and s wth the tag. Statements wthn the DATA secton follow the synta: object_lst = value_lst; The object lst contans the names of the attrbutes or of the set whose values are beng ntalzed. The value lst assgns the values to the specfed members of the object lst. The followng eamples show two ways to use the DATA secton n LINGO. In each eample, the X and Y attrbutes of SET are beng ntalzed to [, 2, 3] and [4, 5, 6], respectvely. The frst eample defnes values for each attrbute separately: SET /A, B, C/: X, Y; X =, 2, 3; Y = 4, 5, 6; The net eample shows how one statement can be used to assgn values to the two attrbutes smultaneously. Each row assgns dfferent values to the X, Y par: SET /A, B, C/: X, Y; X, Y =, 4, 2, 5, 3, 6; When parameters or attrbutes are defned n the DATA secton of a model, a feature called Whatf Analyss can be used to eamne the effects of varyng the value of the parameter or attrbute. For eample, f the nflaton rate s most lkely gong to fall between 2% and 6%, the parameter can be defned as follows n the DATA secton: INFLATION_RATE =?; When LINGO encounters the? n the DATA secton, t wll prompt the user to enter a value for the parameter. The user can then enter values between 2% and 6%, and LINGO wll solve the

3 3 model usng that what-f value. All the elements of an attrbute can be ntalzed to a sngle value usng the DATA secton as well. The followng eample shows how to assgn the value of 20 to all seven members of the NEEDS attrbute and 00 to all seven members of the COST attrbute: DAYS / MO, TU, WE, TH, FR, SA,SU/: NEEDS, COST; NEEDS, COST = 20, 00; Data values can also be omtted from the DATA secton of a LINGO model to ndcate that LINGO s free to determne the values of those attrbutes tself. The followng eample shows that the frst two values of the attrbute CAPACITY have been ntalzed to 34, but the last three varables have not been defned: YEARS /..5/: CAPACITY; CAPACITY = 34, 34,,, ; Varable Types All varables n a LINGO model are consdered to be non-negatve and contnuous unless otherwse specfed. LINGO s four varable doman functons can be used to overrde the default doman for gven varables. These varable doman functons any postve nteger a bnary value (e, 0 or any postve or negatve real any value wthn the specfed bounds Smlar synta s used varable doman functons. The general form for the declaraton of a varable usng any of these functons functon has a slghtly modfed synta, whch ncludes the upper and lower bounds for the acceptable varable values. The general form for the declaraton of a varable between a lower bound and an upper bound s gven X, upperbound); LINGO Operators and Functons LINGO provdes a vast array of operators and functons, makng t a useful problem-solvng tool. A selecton of the prmary operators and functons s gven below. There are three types of operators that LINGO uses: arthmetc, logcal, and relatonal operators. The arthmetc operators are as follows: Eponentaton: ^ Multplcaton: * Dvson: / Addton: + Subtracton: - The logcal operators are used n set loopng functons to defne true/false condtons: #LT#: TRUE f the left argument s strctly less than the rght argument, else FALSE #LE#: TRUE f the left argument s less-than-or-equal-to the rght argument, else FALSE #GT#: TRUE f the left argument s strctly greater than the rght argument, else FALSE

4 4 #GE#: TRUE f the left argument s greater-than-or-equal-to the rght argument, else FALSE #EQ#: TRUE f both arguments are equal, else FALSE #NE#: TRUE f both arguments are not equal, else FALSE #AND#: TRUE only f both arguments are TRUE, else FALSE #OR#: FALSE only f both arguments are FALSE, else TRUE #NOT#: TRUE f the argument mmedately to the rght s FALSE, else FALSE The relatonal operators are used when defnng the constrants for a model. They are as follows: The epresson s equal: = The left sde of the epresson s less than or equal to the rght sde: <= The left sde of the epresson s greater than or equal to the rght sde: >= The followng lst contans a samplng of mathematcal functons that can be used n returns the absolute value of returns - f X s negatve and + f X s calculates calculates the natural log of returns the sne of X, where X s the angle n returns the cosne of returns the tangent of X LINGO also contans a plethora of fnancal, probablty, and mport/eport functons. These are commonly used n more advanced models, whch are beyond the nted scope of ths tutoral.

5 5 Alcun esemp I soldatn ma st () (2) 2 5 (3) 0 =, 2 Model: Ma=7*+3*2; 25*+20*2<=480; Donne sposate mn +.62 st , 0 nter 2 In forma parametrca: 2 mn c = 2 st.. a b =,...,4 () j= j 0 nteger =,...,4 (2) dove: c = [.6] A = b = [ ] model: sets: ctyp /..2 /: c, ; ctyp/..4/:b; matr(ctyp, ctyp) : a; sets data: c=,.6; a=0.3,0.3, 0.,0.2, 0.,0.3, 0.,0.5; b=50,0,20,00; data mn >=b() Comment al modello * ) occorre ndcare come comporre gl add (moltplcando c e ) e l nseme d appartenenza de ndca che occorre esegure rpetutamente un operazone; quante volte è dato dall nseme ctyp (da a 4 a pass d ), utlzzando una varable che assume ordnatamente valor da a 4. Vene coè realzzata la parte destra del vncolo ()

6 s ndcano gl add (n modo dettaglato per evtare possbl ambgutà e l loro nseme d appartenenza. Fabbsogno: q. da noleg. a partre dal mese per una q d mes par a α α Posto l parametro c = costo d noleggo per mes, c = [ 400, 700,900] e posto d = domanda nel mese, d = [ 9,5, 7,9,0,5], la funzone obettvo può essere abbozzata come: ( ) ( ) 400* * *.... Formalmente: 3 6 mn c j j= = 3 st.. d =,..,6 j= = 0 = 0 = model: sets: rows/..6/; cols/..3/:c; matr(rows,cols):; dset/..6/:d; sets data: c=400,700,900; d=9,5,7,9,0,5; data (5,3)=0; @gn((,j)))); Le mnere ma 0* 4 = = j= = j= st.. l =,..,4; j=,..,5 () Mz =,..,4; j =,..,5 (2) 3 j =,..,5 (3) z y =,..,4; j =,..,5; k =,.., j (4) 4 4 q = d j=,..,5 (5) j = = z 0 k r y { } y, z 0, Il codce è sotto per motv d spazo model: sets: type/..4/:l,royaltes,qualty; typej/..5/:demanded; mne(type,typej):,z,y; sets data: l=2,2.5,.3,3; royaltes=5,4,4,5;

7 qualty=,.7,.5,.5; demanded=.9,.8,.2,.6,; bgm=4; 7

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1) Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A