A decision rule based on goal programming and one-stage models for uncertain multi-criteria mixed decision making and games against nature

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1 Croatia Operatioal Research Review 6 CRORR 8(207), 6 76 A decisio rule based o goal programmig ad oe-stage models for ucertai multi-criteria mixed decisio maig ad games agaist ature Helea Gaspars-Wieloch, Departmet of Operatios Research/Poza Uiversity of Ecoomics ad Busiess Al. Niepodleglosci 0, Poza, Polad helea.gaspars@ue.poza.pl Abstract. This paper is cocered with games agaist ature ad multi-criteria decisio maig uder ucertaity alog with sceario plaig. We focus o decisio problems where a determiistic evaluatio of criteria is ot possible. The procedure we propose is based o weighted goal programmig ad may be applied whe seeig a mixed strategy. A mixed strategy allows the decisio maer to select ad perform a weighted combiatio of several accessible alteratives. The ew method taes ito cosideratio the decisio maer s preferece structure (importace of particular goals) ad ature (pessimistic, moderate or optimistic attitude towards a give problem). It is desiged for oe-shot decisios made uder ucertaity with uow probabilities (frequecies), i.e for decisio maig uder complete ucertaity or decisio maig uder strategic ucertaity. The procedure refers to oe-stage models, i.e. models cosiderig combiatios of scearios ad criteria (sceario-criterio pairs) as distict meta-attributes, which meas that the ovel approach ca be used i the case of totally idepedet payoff matrices for particular targets. The algorithm does ot require ay iformatio about frequecies, which is especially desirable for ew decisio problems. It ca be successfully applied by passive decisio maers, as oly criteria weights ad the coefficiet of optimism have to be declared. Keywords: ucertaity, multi-criteria decisio maig, goal programmig, games agaist ature, mixed strategies, oe-stage models, oe-shot decisios Received: October 3, 206; accepted: February 7, 207; available olie: March 3, 207 DOI: /crorr Itroductio This cotributio relates to multiple-criteria decisio maig for cases where criteria evaluatios are ucertai. This topic has bee ivestigated by may researchers because usually real decisio problems cotai umerous coflictig criteria ad a determiistic evaluatio of criteria is ofte impossible. Possible models, Correspodig author Croatia Operatioal Research Society

2 62 Helea Gaspars-Wieloch methods ad tools used to support ucertai multi-criteria decisio maig are described for istace i [2] (e.g. models with scearios, models usig probabilities or probability-lie quatities, models with explicit ris measures, models with fuzzy umbers). The method proposed i the article is desiged for oe-shot decisio problems ad multi-criteria decisio maig with sceario plaig. The procedure may be applied to totally ew decisio problems where the frequecy of particular scearios is ot ow. We assume that criteria payoff matrices are idepedet, which etails the opportuity to aalyze the ucertai multi-criteria problem as a oe-stage model. The ew approach eables oe to select a optimal mixed strategy. The procedure taes ito accout decisio maers objective prefereces (criteria weights) ad their attitude towards ris (coefficiet of optimism). The algorithm icludes a stage where a set of evets with the biggest subjective chace of occurrece (separately for each payoff matrix) is suggested. The last step cosists i formulatig ad solvig the optimizatio problem. The paper is orgaized as follows. Sectio 2 deals with the mai features of multicriteria DMU (decisio maig uder ucertaity), sceario plaig ad -stage models. Sectio 3 presets a procedure that may be used as a tool i multi-criteria optimizatio uder ucertaity for mixed strategy searchig ad -stage models. Sectio 4 provides a case study o the basis of the bi-criteria sigle-period ewsvedor problem. Coclusios are gathered i the last Sectio. The paper is a cotiuatio of several articles, where ucertai oe-criterio procedures [6], [8], [22] ad multi-criteria decisio rules for 2-stage models [20], [24], [25] were ivestigated. 2. Ucertai multi-criteria decisio maig ad -stage models I coectio with the ecessity to solve decisio problems with ucertai parameters, may diverse theories have bee developed, e.g. probability theory [39], possibility theory [77], [9], ucertaity theory [43], [44]. Nevertheless, it is worth emphasizig that there is o uaimity i defiig the otio of ucertaity [26]. Accordig to the first approach, the decisio maer (DM) may choose the appropriate alterative (decisio, strategy, variat) uder certaity (DMC each parameter of the decisio problem is determiistic), uder ris (DMR), uder partial iformatio (DMPI), uder complete ucertaity (DMCU) or uder total igorace (DMTI). I the case of DMR, DMPI ad DMCU, possible scearios (states of ature, evets) are predicted by experts or by the decisio maer. DMCU occurs whe the probability of those states of ature is ot ow or whe the DM does ot wat to mae use of the estimated probability distributio. If the lielihood of particular scearios is ow ad sigificat for the DM, we the tur our attetio to DMR [3], [37], [38], [54], [59], [60], [64], [72]. DMPI is characterized by probability distributios ot ow completely [33], [73], which

3 Ucertai multi-criteria optimizatio for oe-stage models 63 meas that the DM ows oly a) the order of scearios or b) the itervals with possible probabilities for each sceario. DMTI deals with problems for which the DM is ot able to defie possible evets. Ucertaity ad ris were formally itegrated i ecoomic theory by [68]. Supporters of the secod approach declare that ucertaity ivolves all situatios with o-determiistic parameters (ow, uow or icompletely ow probability distributio, lac of iformatio about possible scearios), while ris is related to the possibility that some bad (or other tha predicted) circumstaces will happe [8], [0], [4], [32], [53], [72]. Scietists stress that the defiitio of ucertaity varies depedig o the scietific domai. I the theory of decisio, ucertaity meas a situatio where particular decisios may lead to differet cosequeces ad the probability of evets is ot ow (see the first approach). I ecoomics, ucertaity is defied as a situatio where alteratives may lead to differet effects ad the probability of scearios is ow or ot ow. However, i the latter case, some probabilitylie quatities are ofte estimated ad applied (see the secod approach). Apart from two above approaches, we also refer to the Austria Ecoomic School which treats ucertaity as do decisio theorists, i.e. a situatio where the lielihood is ot ow. Accordig to that approach, the mathematical probability of the occurrece of a give sceario is ot ow sice probabilities (uderstood as frequecies) oly cocer repetitive evets, meawhile for the majority of real problems, the DM deals with o-repetitive evets [67]. Ucertaity is ot caused by the radomess of evets (as held by mai-stream ecoomists) but is due to umerous factors, of which oly some are ow i the decisio-maig process. I this paper, we rather treat ucertaity accordig to the third approach, but we ame it ucertaity with uow probabilities/frequecies (or complete ucertaity, strategic ucertaity) to be more precise. Nevertheless, the theory of ecoomics is also partially applied i this research give that uow iitial probabilities will be replaced with secodary probability-lie quatities. I may situatios, computig the lielihood may be difficult due to may discrepat defiitios of probability [6], [5], [38], [39], [67], lac of historical data (for totally ew decisios ad evets) [23], [33], lac of sufficiet owledge cocerig particular states or the fact that the set of possible scearios forecasted by experts i the sceario plaig stage does ot satisfy probability axioms (the sum of state probabilities should be equal to, the whole sample space must be precisely defied), see [39]. People may eve be uable to declare subjective probabilities they implicitly set probabilities i actig [4]. Additioally, accordig to Vo Mises [67], the theory of probability ca ever lead to a defiite statemet cocerig a sigle evet (the probability of a sigle evet caot be preseted umerically). There are may classical ad exteded decisio rules desiged for multi-criteria decisio maig uder ucertaity, e.g. [], [2], [7], [8], [], [3], [20], [24], [25],

4 64 Helea Gaspars-Wieloch [28], [29], [34], [36], [40], [4], [42], [45], [48], [49], [50], [5], [56], [57], [58], [6], [62], [63], [65], [66], [7], [74], [75], [76], however the majority of the methods refer to probability calculus. Hece, those cotributios are ot directly related to the topic ivestigated i this paper, sice here we cocetrate o totally ew decisio problems for which frequecies are ot ow. May existig procedures allow us to search for a optimal pure strategy, others are desiged for optimal mixed strategies. I the case of pure strategies, the DM chooses ad completely executes oly oe decisio. A mixed strategy implies that the DM selects ad performs a weighted combiatio of several accessible alteratives, see e.g. bods portfolio costructio, cultivatio of differet plats [24], [52], [55], [60]. This paper will deal with the latter case. Some rules ca be applied whe the DM iteds to perform the selected strategy oly oce. Others are recommeded for people cosiderig multiple realizatios of the chose variat. I the first case, fial solutios are called oe-shot decisios; i the secod case multi-shot decisios. This paper focuses o oe-shot decisio problems. Accordig to [2], ucertaities become icreasigly so complex that the elicitatio of measures such as probabilities, belief fuctios or fuzzy membership fuctios becomes operatioally difficult for DMs to comprehed ad virtually impossible to validate. Therefore, i such cotexts it is useful to costruct scearios that describe possible ways i which the future might ufold ad to combie MDMU (multi-criteria decisio maig uder ucertaity) with SP (sceario plaig). The result of the choice made uder ucertaity with sceario plaig depeds o two factors: which decisio will be selected ad which sceario will occur [24]. Istead of usig probabilities, here we apply probabilitylie quatities, i.e. coefficiets of optimism (β) or pessimism (α), which allow us to tae ito accout the DM s ature (attitude towards a give problem) ad defie the set of evets with the biggest chace of occurrece. These parameters belog to iterval [0,] ad satisfy the coditio α+β=. α (β) teds to 0 () for extreme optimists (ris-proe behavior) ad is close to (0) for radical pessimists (ris-averse behavior). Coefficiets of pessimism ad optimism have bee already used i decisio rules described, for example, i [6], [7], [8], [9], [20], [2], [22], [23], [24], [25], [35], [54]. As metioed before, the decisio rule preseted i this paper eables the DM to fid a optimal mixed strategy, but it is worth emphasizig that the existig oecriterio ad multi-criteria procedures for mixed strategies are related more to game theory, i.e. games betwee players [5], [27], [30], [46], [47], [69], [70], tha to games agaist ature (which costitutes a eutral oppoet). Therefore, devisig a approach for ucertai multi-objective mixed decisio maig ad games agaist ature seems vital ad desirable [24], [63]. Accordig to [2], [48] MDMU+SP models ca be divided ito two classes. The first class (A) icludes 2-stage models i which evaluatios of particular altera-

5 Ucertai multi-criteria optimizatio for oe-stage models 65 tives are estimated i respect of scearios ad criteria i two separate stages. Class A cotais two subclasses: A-CS ad A-SC. Subclass A-CS deotes a set of approaches that cosider decisios separately i each sceario before, set a m table ( umber of decisios, m umber of scearios) ad provide the aggregated (over attributes/criteria) performace of a alterative D j uder sceario S i. These evaluatios are the aggregated over scearios. I subclass A-SC, the order of aggregatio is reversed performaces are geerated across scearios ad measures are the calculated over the criteria. The secod class (B) cosists of oe-stage procedures that cosider all combiatios of scearios ad attributes (sceario-criterio pairs) as distict meta-criteria ad use a chose multiplecriteria approach for the trasformed meta-matrix. There is curretly o cosesus o the best way to solve ucertai multi-goal problems [2], [20]. We should otice that subclass A-CS may oly be applied to depedet payoff matrices. Hece, the umber of scearios ought to be the same for each criterio cosidered i the decisio problem ad evaluatio a ca oly be coected with evaluatios a,, a -, a +,, a p- ad a p (these values describe the performace of each criterio based o decisio D j provided that sceario S i occurs) where p is the umber of criteria. O the other had, subclass A-SC ca merely be used for idepedet payoff matrices, which meas that this time there is o relatioship betwee criteria. The performace of particular targets may be aalyzed totally separately sice the umber of states of ature ca be differet for each goal (m, m 2,, m p ). I the secod case, evaluatio a might be coected with ay evaluatio a (i =,, m ), ay evaluatio a 2 (i =,, m 2 ), ad ay evaluatio a p (i =,, m p ). Those values describe the performace of each criterio based o decisio D j ad assumig that ay sceario occurs for criteria C,, C -, C +,, C p [20]. Now we ca easily otice that oe-stage models (i.e. class B) are also dedicated to idepedet payoff matrices. Oe-stage models i the cotext of ucertai multi-criteria mixed decisio maig ad games agaist ature have ot as yet bee aalyzed i the literature. Nevertheless, we would lie to ivestigate this topic, as it gives us the opportuity to elaborate a faster procedure tha the methods desiged for subclass A-SC. The discrete versio (i.e. a set of alteratives is explicitly defied ad discrete) of MDMU+SP with idepedet payoff matrices cosists of decisios (D,, D j,, D ), each evaluated o p criteria C,, C,, C p ad m mutually exclusive scearios (S,, S i,, S m) where =,, p. The problem ca be preseted by meas of p payoff matrices (oe for each criterio) ad (m + +m + + m p ) evaluatios. Each payoff matrix cotais m evaluatios, say a, which deote the performace of criterio C resultig from the choice of decisio D j ad the occurrece of sceario S i. We assume that the distributio of payoffs related to a give decisio is discrete.

6 66 Helea Gaspars-Wieloch 3. Procedure for MDMU+SP, -stage models ad optimal mixed strategies I this sectio, we will preset a decisio rule that supports multi-criteria decisio maig uder complete ucertaity whe searchig for a optimal mixed strategy ad o the assumptio that the problem is aalyzed as a -stage model. We assume that payoff matrices are idepedet ad that, withi each criterio, payoffs coected with a give decisio costitute sequeces of outcomes (ot sets of outcomes). Thus, the positio of a payoff i the colum is ot accidetal, but strictly depeds o the sceario. The problem associated with sequeces of outcomes, but based o pure strategies ad oe-criterio aalysis, has bee ivestigated, for example, by [55]. We will otice that the procedure requires us to reduce the iitial sets of potetial scearios to the sets of states of ature with the biggest subjective chace of occurrece. The suggested method cosists of the followig steps: ) Give a set of potetial decisios ad payoff matrices for each criterio, defie a appropriate value of the parameter [0, ] accordig to your level of optimism ad choose weights w for each attribute (=,,p): 2) If ecessary, ormalize the evaluatios (use Equatio (2) for maximized criteria ad Equatio (3) for miimized criteria) separately withi each payoff matrix: a( ) a max i,..., m j,..., mi a i,..., m j,..., a mi a i,..., m j,..., p w,..., p; i,..., m ; j,..., () (2) a( ) max i,..., m j,..., max i,..., m j,..., a mi a a a i,..., m j,...,,..., p; i,..., m ; j,..., (3)

7 Ucertai multi-criteria optimizatio for oe-stage models 67 3) Create a meta-matrix cotaiig colums for each decisio ad (m + +m + +m p =R) rows for scearios assiged to subsequet targets. Complete that matrix usig (m + +m + +m p ) ormalized evaluatios. 4) Fid M * (the maximum ormalized value computed accordig to the max-max rule) ad calculate y * which is the maximized miimum guarateed ormalized value computed o the basis of Wald s model (Equatios 4-7): j y max (4) a( ) x y, i,..., R (5) j j x (6) j x 0, j,..., (7) j where x j is the share of alterative D j i the mixed strategy ad stads for the umber of decisios. Due to the existece of more tha oe criterio, value M * is usually uattaiable. 5) Choose the set of scearios with the biggest chace of occurrece (SPS ) for each criterio separately. This ca be doe i diverse ways, e.g. o the basis of the domiace cases ad the coefficiet of optimism [6], [22], [24], [25] or ituitively. The higher the value of β, the fewer scearios should be cosidered. Let us deote the umber of scearios with the biggest chace of occurrece i each set SPS by m *. Reduce the iitial meta-matrix to the most subjectively probable meta-matrix cotaiig colums for each decisio, (m * + +m * + + m * p=r * ) rows for scearios ad (m * + +m * + + m * p)= R * ormalized evaluatios. 6) Solve the optimizatio problem cosistig of Equatios (6)-(7) ad (8)-(0): w m * isps p w w max{ gi,0}... max{,0}... * gi * m m isps p p isps max{ g p i,0} mi (8)

8 68 Helea Gaspars-Wieloch j a( ) x r g, j i * i,..., R (9) * * * r ( M y ) y (0) Where r β is the expected level of the outcome depedet o β (Equatio 0) ad g i deotes the deviatio from r β of the outcome achieved by the DM if sceario S i occurs. Both sides of coditio (9) preset the true criterio performace obtaied if the shares of a give mixed strategy equal x, x 2,, x ad sceario S i taes place. The aim of the optimizatio model (Equatio 8) is to miimize, withi the reduced sets of scearios, the sum of all positive deviatios of the true payoffs from the expected oe (similar to goal programmig). Note that oly positive deviatios are disadvatageous sice the expected outcome the exceeds the true result [8]. The optimal solutio represets the multi-criteria mixed strategy reflectig the DM s level of optimism ad cosiderig his/her objective prefereces. Let us call the described procedure β-mmdm/, i.e. β decisio rule for multi-criteria mixed decisio maig ad -stage models. 4. Case study The method suggested i this paper will be illustrated by meas of the followig example. We aalyze a bi-criteria sigle-period ewsvedor problem (the oecriterio problem is described e.g. i [23], [26]). Usually, this issue is treated as a stochastic problem (with a ow probability distributio) [3], [60], but i [26], [33] authors stress the ecessity to ivestigate the topic as a strategic problem (with uow probabilities). The ewsvedor has 20 similar retail outlets (located i differet places, but the distaces betwee particular stores ad the wholesaler busiess are early the same) where he iteds to sell a totally ew short-cycle product. He assumes that the quatity procured will be used solely to satisfy the demad durig the curret period. The demad for this product is ot ow i advace. He cosiders order (q) ad demad (D) quatities betwee ad 5 boxes. The uit productio/purchase cost of box (c ) equals 5, the sellig price (c 2 ) equals 9 ad the discout price (price of leftover items) c 3 =2, hece the uit profit from sellig the product at price c 2 : b=c 2 -c =4 ad the uit loss from sellig it at price c 3 : s=c -c 3 =3. The ewsvedor maximizes the total profit (e.g. i thousads of Euros) resultig from buyig ad sellig the ew product ( st criterio depedet o the demad) ad miimizes the cost of supply (2 d criterio depedet o the supplyig, storage, weather coditios). Note that the total profit does ot iclude the cost of supply ad is equal to b q (for q D)

9 Ucertai multi-criteria optimizatio for oe-stage models 69 or b D-s (q-d) whe q>d. Payoff matrices are give i Table (first values i each cell). The ewsvedor iteds to fid a optimal mixed strategy, hece he is willig to order differet quatities of the ew product for particular retail outlets. Now, let us apply procedure β-mmdm/ for the aforemetioed problem. Crit. A = A 2 = 2 A 3 = 3 A 4 = 4 A 5 = 5 S = 4/0.43 /0.32-2/0.2-5/0. -8/0.00 S 2 = 2 4/0.43 8/0.57 5/0.46 2/0.36 -/0.25 S 3 = 3 4/0.43 8/0.57 2/0.7 9/0.6 6/0.50 S 4 = 4 4/0.43 8/0.57 2/0.7 6/0.86 3/0.75 S 5 = 5 4/0.43 8/0.57 2/0.7 6/ /.00 Crit. 2 A = A 2 = 2 A 3 = 3 A 4 = 4 A 5 = 5 S 2 0.5/ / / / /0.83 S 2 2 /0.79./0.75.2/0.7.3/0.67.4/0.62 S 2 3 2/ / / / /0.00 Table : Criteria payoff matrices ad ormalized values (example), source prepared by the author. First (step ), we assume that the DM is a moderate optimist (β=0.7, α=0.3) ad that w =0.6, w 2 =0.4. We ormalize values (step 2) - they have the same uits, but they are expressed i differet scales, see Table (secod value i each cell). The meta-matrix (step 3) cotais 5 colums (5 decisios), 8 rows (5 scearios for the st criterio ad 3 scearios for the 2 d criterio) ad 40 ormalized values (we do ot preset it due to page limitatios, but values i the meta-matrix are equal to ormalized values from Table ). Parameters M * ad y * are equal to ad (step 4). I step 5 we use the procedure suggested i [24], but other approaches are also possible, ad we obtai SPS ={S 3,S 4,S 5}, SPS 2 ={S 2 }. Now, the reduced meta-matrix cotais 5 colums, oly 4 rows ad 20 ormalized values (uderlied, Table ). I step 6 we solve the followig model where r β =0.7(-0.375)+0.375=0.82, variables x j are o-egative ad their sum equals /3(max{ g 3,0} max{ g4,0} { g5,0}) 0.4 max{ g,0} mi 0.43x g ` 0.57x2 0.7x3 0.6x4 0.50x

10 70 Helea Gaspars-Wieloch 0.43x g ` 0.57x2 0.7x3 0.86x4 0.75x x g ` 0.57x2 0.7x3 0.86x4.00x x g ` 0.96x2 0.92x3 0.87x4 0.83x The optimal solutio is as follows: x =0; x 2 =0; x 3 =0.3; x 4 =0.69; x 5 =0 ad g 3=0.7; g 4=0; g 5=0; g 2 = Hece, if the optimal strategy is executed, for three scearios: S 4,S 5,S 2 it will be possible to gai at least the expected ormalized value (depedet o β). The obtaied variable values sigify that for 3% of retail outlets (approximately 6) the order quatity should be equal to 3 boxes ad for 69% ( 4) the order quatity should be equal to 4. Note that the little chage of optimal results (3% 30%, 69% 70%) is required due to the discrete umber of retail outlets, but it does ot seriously affect the deviatio values: g 3=0.7; g 4=0; g 5=0; g 2 = As was metioed above, the etire mixed strategy covers oly oe seaso. 5. Coclusios The paper cotais a descriptio of a decisio rule supportig multi-criteria decisio maig uder ucertaity with uow probabilities (frequecies). Its goal is to fid a optimal mixed strategy (combiatio of pure strategies) which costitutes a oe-shot decisio (it is executed oly oce). The method is desiged for games agaist ature. It is based o oe-stage models. The fial model formulated ad solved i the last step of the algorithm is characteristic of weighted goal programmig, but here oly positive values of deviatios are disadvatageous sice the expected outcome the exceeds the true result. Advatages of applyig that approach are as follows: ) It does ot require ay iformatio about probabilities, which is especially desirable i the case of ew decisio problems, 2) It taes ito cosideratio the decisio maer s preferece structure ad ature, but oly criteria weights ad the level of optimism are supposed to be declared hece, the procedure may be successfully applied by passive decisio maers, 3) It ca be used i the case of totally idepedet payoff matrices for particular targets, 4) It is less time-cosumig tha procedures based o 2-stage models. The ovel rule has bee demostrated o the basis of a illustrative example cocerig the sceario-based bi-criteria ewsvedor problem. I the future, it would be desirable to explore the ucertai multicriteria mixed decisio maig problem o the assumptio that payoffs coected with particular decisios are preseted as sets (ot sequeces) of outcomes, sice i some real problems payoffs coected with particular ivestmets deped o totally differet scearios (eve withi the framewor of a give criterio)

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