Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons MAKING A DECISION WHEN DEALING WITH UNCERTAIN CONDITIONS. Introducton by Luca Cabulea and Mhaela Aldea The decson theory offers the opportunty of choosng the best alternatve from more avalable possbltes offerng a logcal process of makng decsons. For makng the best decson the followng must be mentoned:. All the alternatves or possble varants must be dentfed, that s the ways n whch the decson maker can act. 2. All possble natural states must be specfed but defned n such a way that these events be mutually exclusve. 3. The results of the choce of any natural states must be evaluated. These evaluatons or results represent benefts or costs and may be presented as tables or matrxes. Such a table s also called the payment matrx (table). If the varants are V, =, m and the natural states N, =, n, then the result of the varant V n the natural state N s noted R and represents a beneft or a cost. Accordng to these remarks the way of makng the best decson wll be determned f the varants and the natural states wll be a fnte number. The degree of knowng the natural states s of hghest mportance for the decson maker. Accordng to the degree of knowng ther appearance the followng classfcaton ca be made. I. Makng the decson under certan crcumstances. Under the crcumstances, the decson maker s aware of the varants, he knows for sure what natural state wll appear and knows the results of choosng varants n all natural states. II. Makng the decson under uncertan crcumstances. The decson maker knows the alternatves, has no nformaton upon the probabltes of appearance of none of the natural states, but he can asses the results of choosng each alternatve n all natural states. III. Makng decsons under rsky condtons. The decson maker knows the alternatves, has enough nformaton to determne the probablty of appearance of each of the natural states and can asses the results. The problems of makng decsons under rsky condtons are also called stochastc problems. 85
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons 2. Decsons under uncertan condtons When nformaton regardng factors or events whch can nfluence the results of choosng the varants s mssng, a very mportant role belongs to the psychologcal factors. The decson wll depend mostly on subectve ratonalzatons of the decson maker, on the fact that he may be an optmstc or a pessmstc person. As t has already been mentoned, the decson maker can establsh all the varants or alternatves V, =, m and the results of choosng V n N state, noted R. He has no nformaton upon the probabltes of natural states appearance. The payment table looks lke ths: Table N N 2 N n V R R 2 R n V 2 R 2 R 22 R 2n V m R m R m2 R mn here: For makng the decson the decson crtera are beng used and we menton I. The max-max crteron: Ths crteron corresponds to an optmstc decson maker. He thnks that whchever varant he may choose, the nature wll act n such a way that he wll get the best result. He wll choose the maxmum value of the result of each varant (on lne) and then the maxmum value of these maxmums: V = max R, =, m max (max R ) = V or V = maxv where V represents the beneft (the result) that he hopes to obtan, and the lne whose corresponds ths V, determnes the varant whch have to be chose. Ths decson s rsky, especally n long term. II. The max-mn crteron Ths crteron corresponds to a pessmstc decson maker, who thnks that the nature acts aganst hm: any varant he would choose, he would get the worst possble cashng, so the worst result. 86
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons The decson maker wll choose, on lne, the mnmal result for each varant and then, he wll select the maxmal value of these mnmal results: V = mn R, =, m max (mn R ) = V or V = maxv Applyng ths crteron obvously lmts the obtanng of better results, wshng to obtan full securty of the result that corresponds to the varant that was chosen. III. Hurwcz s crteron The crteron may be appled for optmstc, pessmstc or n-between decson makers. Optmsm s expressed by the so-called optmsm ndex or coeffcent, α [ 0,], n such a way that (- α ) s the pessmsm ndex. Choosng the α coeffcent depends on the decson maker, so t s subectve. In ths case, a shared value of the result of each alternatve wll be ntroduced. V p = α max R + ( α) mn R V = maxv p If α =, the optmsm crteron s obtaned and f α = 0, the pessmsm one s obtaned. For example, for α = 0,3 the decson maker s nclned to pessmsm. IV. Laplace s crteron Ths crteron s also called equal chance crteron or the crteron of mathematcal hope. Equal chances of appearance are gven to each natural state, so to n states, the appearance probablty s /n. the natural states are equally probable. The expected value of the result, whch s the mathematcal expectaton, s: n V = R n = V = maxv 87
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons so the arthmetc mean of all elements on each lne s calculated and then we consder the maxmum value of these results. V. The mn-max crteron of regrets Ths one s also called Savage s crteron, the one who ntroduced the noton of regret, a measure of loss due to the mss choce of the best varant. Frst a regrets table s drawn, r, and then the mn-max crteron s appled. The regret s measured through the dfference between the best result that we would have got f we had known the natural state that was gong to appear and the result obtaned by makng the decson. r = max R R, =, n ; =, m. Practcally, for drawng the table of regrets, r, we must follow the followng steps: from the hghest element n each column n the payment table or the elements n the respectve column are subtracted; thus the column of the regret matrx wll result. The mn-max crteron, the table of regrets s appled, so for each lne the maxmum element s chosen and then we select the mnmum regret. The three stages necessary for the applcaton of the crteron are: - drawng up the regrets table; - dentfyng the maxmum regret for each alternatve; - choosng the alternatve that mnmzes the maxmum values of regrets; 3. Case study A frm owns a sellng department for ts products and wants to ncrease ts sales. The choces t has are: V : open a new sellng department; V 2 : to extend the exstng one; V 3 : to ncrease the number of ours the shop s open for buyers; The owner s decson wll depend on the buyer s request for hs products. Notng the natural states N, these can be: N : bg request; N 2 : average request; N 3 : low request; The frm manager evaluates the benefts R that he can get wth each varant, V, dependng on the natural states, whch are the market request, N, shown n table 2, where n varant, f the request s low, N 3, he loses 50 m.u. because he has nvestments. Usng the decson crtera, let us fnd the varant that the manager should choose. Solvng: 88
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons I. The max-max crteron, of optmsm: Table 2 N N 2 N 3 V 600 300-50 V 2 700 300 00 V 3 300 300 50 V V V 2 = max R = max R = max R 2 3 = max(600,300, 50) = 600 = max(700,300,000) = 700 = max(300,300,50) = 300 V = max V = max (600, 700, 300) = 700 The decson maker wll choose the second varant, whch s extendng the exstng shop, hopng that the request wll be hgher and the beneft obtaned wll be 7oom.u. The best result, ndcated by each of the fve crtera, s gven n Table 3. Table 3 N N 2 N 3 I II III IV V V 600 300-50 600-50 80 283,3 200 V 2 700 300 00 700* 00 220* 366,7* 50* V 3 300 300 50 300 50* 80 250 400 II. The max-mn crteron of pessmsm 89
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons We choose the mnmum value R on lne and then the maxmum values of these lmts. The decson: the choce of V 3, expectng the request to be lower, wth a beneft of 50m.u. V p = 0,2 III. Hurwcz s crteron The decson maker s nclned towards pessmsm, thus α = 0,2. max R + 0,8 R = 0,2 600 + 0,8 (-50) = 80 V p2 = 0,2 700 + 0,8 00 = 220 V p3 = 0,2 300 + 0,8 50 = 80 V = max (80, 220, 80) = 220 mn = 0,2 max (600, 300, -50) + 0,8 mn (600, 300, -50) The decson: he chooses V 2 expectng a beneft of 220 m.u. V = 3 IV. The crteron of equal chances, Laplace s crteron 3 = R = 3 (600 + 300 50) = 283,3 V 2 = 3 (700 + 300 + 00) = 366,7 V 3 = 3 (300 + 300 + 50) = 250 V = max (283,3; 366,7; 250) = 366,7. The decson: he chooses V 2 expectng the beneft of 366,7m.u. V. The mn-max crteron, the regrets crteron We are gong to buld the regrets table 90
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons = : r = max R R = max (R, R 2, R 3 ) R so: r = max (600, 700, 300) R whch means that the frst column s made of: r = 700 r r, r r 2 3 = 700 600 = 00 = 700 700 = 0 = 700 300 = 400 From the maxmum value of the results on the frst column, the results on the frst column were subtracted and the frst column from the table of regrets has resulted. The rest of the columns are smlarly obtaned. For the table of regrets, n table four the mn-max crteron s appled, the maxmum on lne value and then the mnmum of these maxmums. Table 4 Regrets Mn-max 00 0 200 200 0 0 50 50 400 0 0 400 The crteron ndcates the 2 nd varant, because t has the lowest regret. The fnal decson s V 2 : extendng the exstng space. References. Ackoff, R. L., Sasen, M. V. Bazele cercet[r operatonale, Bucurest, Edtura Tehnca, 975; 2. Budnck, F. S. Appled Mathematcs for Busness Economcs and the Socal Scences, Thrd Ed. Mc. Graw Hll Book Company, 988; 3. Cucu, G., Crau, V., Stefanescu, A. Statstca matematca s cercetar operatonale, Bucurest, Edtura Ddactca s Pedagogca, 978; 4. Mhoc, Gh., Cucu, G. Introducere n teora asteptar, Bucurest, Edtura Tehnca, 967; 9
Luca Căbulea, Mhaela Aldea-Makng a decson when dealng wth uncertan condtons 5. Rusu, E. Decz optme n management prn metode ale cercetar operatonale. Probleme s stud de caz, Edtura Economca, Bucurest, 200; 6. Rusu, E. Fundamenatrea deczlor n management prn metode ale cercetar operatonale, Edtura Junmea, Ias, 977; 7. Wagner, H. M. Prncples of Operatons Research wth Applcatons to Manageral Decson, Prentce-Hall Inc., New-Jersey, 969. Authors: Luca Cabulea and Mhaela Aldea, Unversty Decembre 98, Alba Iula., lcabulea@uab.ro, maldea@uab.ro 92