The logistic decision making in management accounting with genetic algoritms and fuzzy sets

Ma hwa e & So Compu ing 7 (2000) 229-241 229 The Logis ic Decision Making in Managemen Accoun ing wi h Gene ic Algo i hms and Fuzzy Se s E. López-González, M.A. Rod íguez-Fe nández and C. Mendaña-Cue o Dep . o Economy and Business Managemen , Uni . o León Campus de Vegazana, s/n. (24071) León, Spain dde{elg/m /cmc}@unileon.es Abs ac The logis ics p oblems in business en i onmen s deal wi h assigna ion om a numbe o sou ces o a numbe o des ina ions. Each sou ce o e s amoun s o goods, while each des ina ion demands quan i ies o hese goods. The objec is o ind he cheapes anspo ing schedule ha sa is ies he demand wi hou iola ing supply es ain s. In his pape we p opose o use Fuzzy Se s o ep esen s he p e isional in o ma ion ela ed o cos s, demands and o he a iables. Mo eo e , we sugges including he p oblem o sho es ou e o he dis ibu ion ehicles. Finally, o sol e his complex p oblem we p opose o use a Gene ic Algo i hm wi h a Fuzzy Fi ness Func ion. Keywo ds: anspo a ion p oblem, sho es ou e p oblem, logis ic, minimising cos s, imp ecise in o ma ion, uzzy se s and gene ic algo i hms. 1 In oduc ion Many di e en op imisa ion p oblems can be o mula ed on ne wo ks: Fo ins ance, we migh be in e es ed in inding he sho es pa h om one node o ano he , inding he cheapes way o connec all he nodes, o inding he bes way o mo e objec s h ough a ne wo k. Ne wo ks op imisa ion ep esen s one o he mos impo an a eas o managemen science o se e al easons [3] [5]. Fi s , ne wo ks can be used o model many di e se applica ions such as anspo a ion sys ems, communica ion sys ems, ehicle ou ing p oblems, p oduc ion planning and cash low analysis. Second, manage s accep ne wo ks mo e eadily because hey p o ide a isual pic u e o he p oblem unde s udy. Finally, ne wo ks ha e ce ain ma hema ical E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o 2 3 0 p ope ies ha allow managemen scien is s o de elop special algo i hms ha a e able o sol e much la ge p oblems han o he op imisa ion me hods. In his pape we p e end o show a new app oach o a classical ne wo k op imisa ion p oblem: he anspo a ion p oblem. Fi s ly, allowing he ehicles ha can es ablish ou es, and, also, pe mi ing o manage wi h imp ecise o ague in o ma ion. We p opose o use he Fuzzy Se s Theo y [15] [8] [16], wi h he aim o being able o handle he unce ain y, which is a cha ac e is ic o decision-making p ocesses in dis ibu ion p oblems. Mo eo e , o op imise he dis ibu ion ne wo k we sugges o use a Gene ic Algo i hm (GA) [4] [1] [10] [6]. The main eason o his is ha he GAs a e heu is ic op imisa ion me hods which don’ impose es ic ions o he posing o he p oblem. In his s udy, he algo i hm is cha ac e ised by i s use o a Fuzzy Fi ness Func ion ha allows he e alua ion o imp ecise in o ma ion. In he ligh o he abo e, he nex sec ion shows a desc ip ion o he p oblem o be sol ed. Thi d sec ion p esen s he uzzy app oach o his p oblem. In ou h sec ion we in oduces he gene ic algo i hm used o sol e he a o emen ioned app oach. Fi h sec ion o e s a p ac ical example o he dis ibu ion p oblem and, inally, we include some concluding ema ks. 2 The Logis ic Business P oblem We conside he p oblem o inding he leas cos means o shipping supplies om se e al o igins o se e al des ina ions. O igin poin s o sou ces can be ac o ies, wa ehouses, o any o he poin s om which goods a e shipped. Des ina ions a e any poin s ha ecei e goods. To sol e his p oblem we mus ind how many goods a e supplied om each sou ce o each des ina ion and wha is he ou e ha he dis ibu ion ehicle mus ollow in o de o minimize he o al dis ibu ion cos s. T adi ionally his p oblem has been analysed in wo di e en decisions. Fi s ly, ob aining he quan i ies o be shipped om each o igin o each des ina ion minimising he cos s and sa is ying he demand, o T anspo a ion P oblem (Figu e 1). On he o he hand, inding he sho es ou e ha one ehicle mus ollow o each all he des ina ions, o Sho es Rou e P oblem, so known as T a elling Salesman P oblem. In his pape we sugges o combine bo h p oblems in o de o ob ain an op imal solu ion ha minimises he dis ibu ion cos s o he quan i ies supplied and he ou es ollowed. A logis ic p oblem has a e y la ge, some imes in ini e, numbe o easible solu ions. T adi ionally, his p oblem has pa ially educed no pe mi ing shipmen s among des ina ions. This kind o p oblems can be sol ed wi h Linea P og amming and o he me hods as S epping S one, MODI, Hou h ake Me hod o Hamme Balas Me hod. Bu he global app oach, conside ing he ehicles dis ibu ion ou es is poo ly s udied [13] [5]. The Logis ic Decision Making in Managemen Accoun ing.... 2 3 1 Figu e 1. T anspo a ion P oblem Mo eo e , some o he in o ma ion conside ed is imp ecise. Cos s, supplies and demands may be no known in a c isp nume ical way [2]. The e o e, in his pape we sugges o use Fuzzy Se s o ep esen hei ague knowledge. 3 Fuzzy Logis ic Model Wi h his me hod he companies mus know how many commodi ies supply om each o igin o each des ina ion when he in o ma ion on held is no p ecise and a uzzy ep esen a ion in mo e accu a e. As we show below, he in o ma ion necessa y is ela ed o: • Des ina ion o clien demands. The company mus sa is y he clien demands. Some imes his in o ma ion is ague; he e o e we conside a mo e ealis ep esen a ion using uzzy se s. So, o n des ina ions we could ha e: { } n CCC C ~ , , ~ , ~ ~ 2 1  = • Sou ces numbe and i s supply capaci y. We suppose ha each sou ce has a supply capaci y, no mally known in a s ic way. So o m sou ces we could ha e: { } m FFF F , , , 2 1  = • Vehicle Capaci y. We conside ha in each sou ce he company has a ehicle used o dis ibu e he commodi ies. The e o e, we mus ake in o accoun he ehicle capaci y, because he numbe o a els will depend o i . Fo he a o emen ioned m sou ces, he ehicle capaci ies could be: { } m VVVV , , , 2 1  = • T anspo ing cos s o he emp y ehicles. Some imes, he ehicles mus e u n o hei espec i e sou ces due o ha each one has dis ibu ed hei capaci ies. Mo eo e , he las a el is made be ween he las clien supplied and he sou ce o he ehicle. Thus, we mus ake in o accoun wha is he shipping cos when E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o 2 3 2 he ehicle is emp y. This, no mally depends o he dis ance ( ou ed kilome es). So, o m sou ces, he anspo ing cos o emp y ehicles could be: { } m c c c c , , , 2 1  = • Inc easing cos o each uni o commodi ies shipped. When he ehicles dis ibu e he commodi ies he anspo ing cos inc ease wi h each uni shipped. As well, his inc easing cos depends o he dis ance a elled. Mo eo e , he knowledge o his cos could no be p ecise and hen ep esen ing i using uzzy se s could imp o e he esul s ob ained. Thus, o m ehicles he inc easing cos could be: { } m c c c c ∆∆∆=∆ ~ , , ~ , ~ ~ 2 1  • Dis ance among sou ces and des ina ions. The dis ibu ion is made om he sou ces o he des ina ions o clien s. So, i is necessa y o know he dis ance among hem.               = mnmm n n DDD DDD DDD D , , , , , , , , , 2 1 2 22 21 1 12 11     • Dis ance among des ina ions. As we men ioned be o e, he ehicles can es ablish ou es om he sou ce o he se e al des ina ions ha his sou ce supplies. Thus, i is necessa y o know he dis ances among he des ina ions               − − − = , , ' , ' ' , , , ' ' , , ' , ' 2 1 2 21 1 12     nn n n DD DD DD D Acco ding wi h his app oach, we a e ying o each he eal dis ibu ion p oblem. The ea e , we need some ool capable o sol e his complex p oblem o able o gi e a easonable solu ion. In his pape we p opose o use a Gene ic Algo i hm wi h a Fuzzy Fi ness Func ion. Al hough we desc ibe o use uzzy se s, we sugges ep esen ing all he uzzy in o ma ion as T apezoidal Fuzzy Numbe s (TFN). 4 Gene ic Algo i hms and Fuzzy Logis ic P oblems Some au ho s ha e applied Gene ic Algo i hms o anspo a ion p oblems [14] [12]. In many cases hey sol e he p oblem educing hei complexi y o using p ecise knowledge o he a iables. In ou app oach, we sugges a uzzy ep esen a ion (TFN) o he in o ma ion on held and a less es ic i e model. Due o his, he GA mus de e mine he quan i ies supplied om each sou ce o each des ina ion and he ou es a elled o he ehicles. Fo bo h objec i es he c i e ia is he same: minimising he o al dis ibu ion cos s. The Logis ic Decision Making in Managemen Accoun ing.... 2 3 3 4.1. Gene ic Algo i hms GAs a e sea ch algo i hms which use p inciples inspi ed by na u al gene ics o e ol e solu ions o p oblems [7]. The basic idea is o main ain a popula ion o ch omosomes, which ep esen s candida e solu ions o he conc e e p oblem being sol ed, which e ol es o e ime h ough a p ocess o compe i ion and con olled a ia ion. GAs ha e go a g ea measu e o success in sea ch and op imisa ion p oblems. A GA s a s o wi h a popula ion o andomly gene a ed ch omosomes (solu ions), and ad ances owa d be e ch omosomes by applying gene ic ope a o s modelled on he gene ic p ocesses occu ing in na u e. Du ing successi e i e a ions, called gene a ions, ch omosomes in he popula ion a e a ed o hei adap a ion as solu ions, and on he basis o hese e alua ions, a new popula ion o ch omosomes is o med using a selec ion mechanism and speci ic gene ic ope a o s such as c osso e and mu a ion. An e alua ion o i ness unc ion mus be de ised o each p oblem o be sol ed. Gi en a pa icula ch omosome, a possible solu ion, he i ness unc ion e u ns a single nume ical i ness, which is supposed o be p opo ional o he u ili y o adap a ion o he solu ion ep esen ed by ha ch omosome. GAs may deal success ully wi h a wide ange o p oblem a eas, pa icula ly in managemen applica ions. The main easons o his success a e: 1) GAs can sol e ha d p oblems quickly and eliably, 2) GAs a e easy o in e ace o exis ing simula ions and models, 3) GAs a e ex endible and 4) GAs a e easy o hyb idise. All hese easons may be summed up in only one: GAs a e obus . GAs a e powe ul in di icul en i onmen s whe e he space is usually la ge, discon inuous, complex and poo ly unde s ood. They a e no gua an eed o ind he global op imum solu ion o a p oblem, bu hey a e gene ally good a inding accep ably good solu ions o p oblems quickly. These easons ha e been behind he ac ha , du ing he las ew yea s, GAs applica ions ha e g own eno mously in many ields. The basic p inciples o GAs we e i s laid down igo ously by Holland [7], and a e well desc ibed in many books, such as [4] [11]. 4.2 A Gene ic Algo i hm o Fuzzy Logis ic P oblems To sol e he uzzy logis ic p oblem we p opose o use a GA wi h he ollowing componen s: Gene ic Rep esen a ion The solu ions o he p oblems a e a se o dis ibu ion quan i ies o be shipped om each sou ce o each des ina ion and he ou es ollowed by he ehicles o each sou ce. To codi y his solu ions we p opose o use wo ma ixes: one ha con ains he quan i ies dis ibu ed om each o igin o each des ina ion, and, o he , ha ep esen s he pa h ollowed o he se e al ehicles. The codi ica ion o quan i ies ma ix, o an example o i e sou ces ha mus supply o ou des ina ions could be: E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o 2 3 4 1 1 SSou ce 1 Sou ce 2 Sou ce 3 Sou ce 4 Sou ce 5 To al Des ina ion 1 15 0 0 70 0 85 Des ina ion 2 0 40 0 0 0 40 Des ina ion 3 25 10 40 0 50 125 Des ina ion 4 0 0 60 0 5 65 To al 40 50 100 70 55 315 ha indica es he quan i ies supplied om each sou ce o each des ina ion. On he o he hand, he codi ica ion o ehicles ou ed could be: 2 1 SSou ce 1Sou ce 2Sou ce 3Sou ce 4Sou ce 5 Des ina ion 1 1 2 1 4 3 Des ina ion 2 4 3 2 3 2 Des ina ion 3 2 1 3 2 1 Des ina ion 4 3 4 4 1 4 ha indica es he posi ion o each des ina ion in he ehicle ou es. Mo eo e , in o de o gene a e use ulness solu ions, he company mus decide he co e ing le el o each clien uzzy demand. The TFN can be iden i ied as possibili y dis ibu ions. So, he company es ablish he isk o no co e demand o each clien . Acco ding o his, he quan i y ma ix ep esen s a ailable solu ions o he decision. Fuzzy Fi ness Func ion The i ness unc ion mus assign mo e alue o hese solu ions ha a e good solu ions o he dis ibu ion p oblem. To his we p opose o calcula e he o al dis ibu ion cos ha he wo ma ixes suppose. The s eps a e as ollows: A he beginning, each ehicle depa s om i s sou ce ull o commodi ies. The i s des ina ion, es ablished by he ou es ma ix, de e mines he cos o his i s a el. Once he i s des ina ion is sa is ied (wi h one o mo e a els) he second des ina ion is a emp ed. I he ehicle is emp y du ing he ou e, i e u ns emp y o he sou ce o ull i s capaci y. As well, when all he des ina ions a e sa is ied hen he ehicle go back o i s o igin, comple ing he ou e. The sum o all hese anspo ing cos s (depa om o igin ull, a emp i s des ina ion, a emp second, go back o o igin when emp y, e c.) would be a pa o he solu ion. To ob ain he o al cos we mus add he cos o he o he dis ibu ion ou es con ained in he ma ixes and es ablished o o he ehicles. Due o he ac ha some o he in o ma ion can be ep esen ed by uzzy se s, we sugges o use he ope a ions designed o hem [3]. Mo eo e , in he cases ha he ope a ions could no p oduce a TFN, we app oxima e he esul as one o his [9], assuming a li le e o . To se up a hie a chy among he solu ions, he p oposal is o use he uzzy dis ance [9], based on Hamming Dis ance. Doing i , we calcula e he dis ance om he o igin B ~ (single on 0) o each solu ion uzzy cos , which is de ined as ollows: α α αααα dBABABA d ) ( ) ~ , ~ ( 1 0 2 2 1 1 ∫−+−= = The Logis ic Decision Making in Managemen Accoun ing.... 2 3 5 whe e [ ] 2 1 , αα AA is con idence in e al o A ~ a he signi ica ion le el α . The mo e accu a e solu ions will ha e a lowe dis ance. Thus o ob aining he i ness alue we p opose o calcula e he in e se o he dis ance o each solu ion. Selec ion p ocess We p opose o use Roule e Wheel Ranking [4] o selec he indi iduals ha will be he “pa en s” o he nex gene a ion. C osso e ope a o T adi ional c osso e s canno be used, because he solu ions a e wo ma ixes ha ha e o accomplish some cons ain s. Due o his we p opose wo di e en c osso e s o each ma ix, as ollows: ♦ Dis ibu ion quan i ies ma ixes. To combine he in o ma ion o hese ma ixes ob ained om wo “pa en s” we ha e used he c osso e p oposed o Vignaux and Michalewicz o T anspo a ion P oblem [14]. ♦ Vehicle ou es ma ixes. As well, we mus c oss he ou e ma ixes o bo h “pa en s”. To his, we selec one sou ce and in e change be ween he “pa en s” he ou es ha he ehicle o his o igin mus ollow. I we ha e o c oss he ollowing ou e ma ixes o he a o emen ioned p oblem, i s we andomly selec one sou ce (sou ce 2): 2 1 SSou ce 1 Sou ce 2 Sou ce 3Sou ce 4Sou ce 5 Des ina ion 1 1 2 1 4 3 Des ina ion 2 4 3 2 3 2 Des ina ion 3 2 1 3 2 1 Des ina ion 4 3 4 4 1 4 2 2 SSou ce 1 Sou ce 2 Sou ce 3Sou ce 4Sou ce 5 Des ina ion 1 3 4 4 1 2 Des ina ion 2 4 3 2 2 4 Des ina ion 3 1 2 3 3 3 Des ina ion 4 2 1 1 4 1 A e ha , we in e change he lis o ou es ollowed, being he ma ixes esul ing. '2 1 SSou ce 1Sou ce 2Sou ce 3Sou ce 4Sou ce 5 Des ina ion 1 1 4 1 4 3 Des ina ion 2 4 3 2 3 2 Des ina ion 3 2 2 3 2 1 Des ina ion 4 3 1 4 1 4 E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o 2 3 6 '2 2 SSou ce 1Sou ce 2Sou ce 3Sou ce 4Sou ce 5 Des ina ion 1 3 2 4 1 2 Des ina ion 2 4 3 2 2 4 Des ina ion 3 1 1 3 3 3 Des ina ion 4 2 4 1 4 1 Mu a ion ope a o The in en ion o his ope a o is o in oduce di e si y in o he solu ions. T ying o his we mus make di e ences among he wo ypes o ma ixes. • Dis ibu ion quan i ies mu a ion. We p opose o use a special mu a ion me hod designed o keep he solu ions esul ing as easible ones. The s eps a e as ollow: • S ep 1. We selec one sou ce (Sou ce A) and one des ina ion (Des ina ion A) o he ma ix. This des ina ion mus ecei e a non-ze o quan i y om he sou ce selec ed. Fo he example, i we mu a e he i s ma ix ob ained a e he c ossing p ocess, he sou ce and des ina ion could be: 1 1 SSou ce 1 Sou ce 2 Sou ce 3 Sou ce 4 Sou ce 5 To al Des ina ion 1 8 0 25 35 17 85 Des ina ion 2 0 40 0 0 0 40 Des ina ion 3 15 5 45 35 25 125 Des ina ion 4 17 5 30 0 13 65 To al 40 50 100 70 55 315 • S ep 2. We gene a e a andom numbe be ween 0 and he quan i y assigned o his des ina ion. Fo he example could be 20. • S ep 3. We sea ch in he emaining sou ces one quan i y assigned o a di e en des ina ion which amoun is bigge han he andom numbe gene a ed be o e (Sou ce B and Des ina ion B). Fo he example could be: 1 1 SSou ce 1 Sou ce 2 Sou ce 3Sou ce 4Sou ce 5 To al Des ina ion 1 8 0 25 35 17 85 Des ina ion 2 0 40 0 0 0 40 Des ina ion 3 15 5 45 35 25 125 Des ina ion 4 17 5 30 0 13 65 To al 40 50 100 70 55 315 • S ep 4. We discoun he andom numbe o he assigned amoun s om he Sou ce A o Des ina ion A and om he Sou ce B o Des ina ion B. A e ha , we add he andom numbe o he assigned amoun s om Sou ce A o Des ina ion B and om Sou ce B o Des ina ion A. Acco ding o his, o he example, he mu a ed ma ix could be: The Logis ic Decision Making in Managemen Accoun ing.... 2 3 7 '1 1 SSou ce 1 Sou ce 2 Sou ce 3 Sou ce 4 Sou ce 5 To al Des ina ion 1 8 0 25 35 17 85 Des ina ion 2 0 20 20 0 0 40 Des ina ion 3 15 25 25 35 25 125 Des ina ion 4 17 5 30 0 13 65 To al 40 50 100 70 55 315 Finally, wi h he a o emen ioned p ocess we can mu a e he quan i y ma ixes main aining hem as easible ones. • Vehicles ou es mu a ion. On he o he hand, o mu a ing he ou e ma ix we p opose and in e change me hod. Fi s , we selec andomly one sou ce. A e ha , we selec wo des ina ions and in e change hei posi ion in he ou e ollowed o he sou ce ehicle. Hal c i e ia o he bes solu ion sea ch The p oposal is o he algo i hm o go h ough a numbe o gene a ions speci ied by he use un il he bes solu ion is ound. Mo eo e , in o de no o lose good solu ions, he cha ac e is ic e med eli ism [4] has been in oduced. This p ocedu e consis s o keeping he bes indi idual om a popula ion in successi e gene a ions unless and un il some o he indi idual succeeds in doing be e in espec o sui abili y. In his way, he bes solu ion o a p e ious popula ion is no los un il ou classed by a mo e sui able solu ion. As explained, applica ion o he model p oposed he e allows sol ing he dis ibu ion p oblem unde unce ain en i onmen al condi ions. 5 Expe imen : an Example o P ac ical Applica ion In his sec ion we p esen an example ha deals wi h he logis ic p oblem o a p oduc ion u ni u e company. To do ha we di ide his sec ion in subsec ions: one wi h he posing o he p oblem and o he wi h he solu ion using he GA p oposed. 5.1 In oduc ion o he p oblem Le i be imagined ha a u ni u e company supplies desks o se e al clien s. To ob ain he desks ha e ou ac o ies and each one wi h i s own dis ibu ion ehicle. In Cha 1 we show he p oduc ion capaci y o each ac o y and he capaci y, anspo ing cos emp y and inc ease o anspo ing uni o hei espec i e ehicles. Fac o y P oduc ion capaci y Vehicle capaci y T anspo ing cos emp y ($/Km.) T anspo ing cos Inc easing o p oduc uni ($/Km.) A2000 1000 60 (4; 8; 8; 10) B4000 1400 40 (8; 10; 10; 12) C6000 200 80 (2; 4; 4; 6) D2000 400 20 (6; 6; 6; 6) Cha 1