ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS


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1 ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga, Spain. () Deparmen of Foundaions of Economic Analysis, Universiy Compluense of Madrid, Spain. ABSTRACT: The aim of his sudy is o analyse he resoluion of Sochasic Programming Problems in which he objecive funcion depends on parameers which are coninuous random variables wih a known disribuion probabiliy. In he lieraure on hese quesions differen soluion conceps have been defined for problems of hese characerisics. These conceps are obained by applying a ransformaion crierion o he sochasic objecive which conains a saisical feaure of he objecive, implying ha for he same sochasic problem here are differen opimal soluions available which, in principle, are no comparable. Our sudy analyses and esablishes some relaions beween hese soluion conceps. KEY WORDS: Sochasic Programming. Opimal soluion conceps. We hank J.B. Readman for his linguisic revision of he ex.
2 . Inroducion When a real problem is modelled and solved by means of a mahemaical programming problem i may happen ha some of he parameers which figure in he problem are unknown, wheher i be in he objecive funcion or in he feasible se. If hese parameers of unknown value can be aken as random variables he resuling problem is a sochasic programming one. Obviously, he fac ha one of he consrain funcions or he objecive funcion is affeced by random parameers gives rise o he view ha ha funcion is also a random variable and, given ha, in general, a random variable does no admi a relaion of order, i is necessary o specify a soluion concep for hese problems. The characerisics of sochasic programming problems and he deermining of opimal soluions for hem have been widely sudied in he lieraure. Among oher sudies we should make special menion of he books of Prékopa (995), Kall and Wallace (994), Kibzun and Kan (996) or he aricles of Kall (98), Leclerq (98) and Zare and Daneshmand (995). In hese sudies we can observe how he exising soluion conceps for sochasic programming problems originae from he ransformaion of he problem ino a deerminisic one which is given he erm deerminisic equivalen. This ransformaion is achieved by aking saisical feaures of he objecive funcion or of he consrain funcions dependen on random parameers in he problem. In his sudy we focus our aenion on problems of sochasic programming in which he random parameers affec only he objecive funcion of i or in which, if hey affec he consrains, his laer has already been ransformed ino is deerminisic equivalen by one of he exising procedures in sochasic programming. In addiion, we assume ha hey are coninuous random variables,
3 so ha he sochasic programming problem which we are considering is as follows: Min %(,) xc% (.) x D z where x is he vecor of decision variables, D R n is a closed se, bounded, convex and no empy, c ~ is a random vecor defined on he se E R s, wih known probabiliy disribuion and whose componens are coninuous random variables. Furhermore, we assume ha he probabiliy disribuion of vecor c ~ is independen of he decision variables of he problem, x,..., x. n From he hypoheses esablished for he problem (.) i follows ha he objecive funcion of he problem depends on random parameers and, as previously indicaed, his implies ha he funcion is a random variable. In order o solve he problem (.), differen soluion conceps have been defined in he lieraure. Each of hese conceps proceeds from he applicaion of a crierion o ransform he sochasic objecive ino a deerminisic funcion. These crieria draw on some saisical feaure from he sochasic objecive, so ha he problem (.) is aribued various deerminisic equivalens and, in general, various possible opimal soluions. The diversiy of conceps and soluions can give rise in some way o cerain confusion when a problem is being solved and immediaely poses various quesions, such as he exising differences beween some conceps and ohers, wheher any of he conceps so far defined is beer han anoher, ec. In his sudy we consider five soluion conceps o he problem (.) which correspond o he applicaion of differen crieria o ransform he sochasic problem ino is deerminisic equivalen, all of which have been defined previously, and some relaions are esablished beween hem. We begin by defining in Secion hese five soluion conceps and esablishing some differences beween 3
4 hem. In Secion 3 some relaions beween hese conceps are esablished and in Secion 4 we pose and solve an example o illusrae he resuls obained.. Some Soluion Conceps for Sochasic Programming Problems Le us consider he sochasic programming problem (.). Le z (x) be he expeced value funcion of ~ z ( x, ~ c ) and σ ( x ) is variance, which we assume o be finie for all x D. Le us now look a five opimum conceps for his problem. As previously indicaed, each one of hese conceps derives from he applicaion of ransformaion crieria o he objecive funcion of he problem and he obaining of a deerminisic equivalen problem from he iniial problem. The ransformaion crieria which we consider are: expeced value, minimum variance, expeced value sandard deviaion efficiency, minimum risk and Kaaoka. Definiion : Expeced value soluion Poin x D is he expeced value soluion o (.) if i is he opimal soluion o he problem: Min x D z( x) (.) In oher words, he expeced value opimal soluion o he problem (.) is simply he opimal soluion resuling from he subsiuion of he sochasic objecive funcion by is expeced value, hus aking up his measuremen of cenral endency of he random variable for is opimisaion. This soluion concep has been frequenly used in he lieraure for he resoluion of sochasic programming problems, alhough his crierion canno always be considered appropriae given ha if we remain wih only he expeced value of he random variable, cerain feaures of he sochasic objecive funcion of he problem migh no be included in he deerminisic equivalen problem. In his sense, some auhors such as Kaaoka (963) or Prékopa (995) make some criical commens abou he 4
5 applicaion of he above crierion and esablish same condiions under which i can be considered appropriae. Definiion : Minimum variance soluion Poin x D is he minimum variance soluion o he problem (.) if i is he opimum of he problem: Min x D σ ( x) (.) In oher words, in order o solve he sochasic programming problem, he crierion considered is ha of minimising he variance of he sochasic objecive funcion ~ z ( x, ~ c ), choosing, herefore, he vecor x for which he random variable ~ z ( x, ~ c ) is more concenraed around is expeced value. This implies ha he opimisaion crierion is ha of he minimum, regardless of wheher he problem of opimisaion be ha of he minimum (a hypohesis which we susain in his sudy) or of he maximum, and in some ways, can be considered a crierion of low risk. Definiion 3: Expeced value sandard deviaion efficien soluion Poin x D is a expeced value sandard deviaion efficien soluion o he problem (.) if i is an efficien soluion in he Pareo sense o he following bicrieria problem: Min x D ( z ( x), σ ( x) ) (.3) which includes he expeced value and he sandard deviaion of he sochasic objecive funcion. We denoe (.3) and E Eσ as he se of efficien soluions o he problem p E Eσ as he se of properly efficien soluions o his problem. This concep of efficiency was inroduced by Markowiz (95) as a means of solving problems in he field of porfolio selecion in finance economics. Markowiz considers he bicrieria problem of expeced value minimum variance, ha is: 5
6 ( x σ x ) Min z( ), () (.4) x D In his sudy we have chosen o subsiue he variance of he sochasic objecive for is sandard deviaion given ha, in his way, he wo objecives of he problem (.) are expressed in he same unis of measuremen. In any case, since he square roo funcion is sricly increasing, he efficien ses of boh problems coincide. Finally, we define he conceps of opimal soluion minimum risk of aspiraion level u and he opimal soluion wih probabiliy β. Boh soluions are obained by applying he minimum risk and Kaaoka crieria, respecively, referred o in he lieraure as crieria of maximum probabiliy or saisfying crieria, due o he fac ha, as we shall see in he following definiion, in boh cases he crieria o be used provide, in one way or anoher, good soluions in erms of probabiliy. Definiion 4: Minimum risk soluion of aspiraion level u Poin x D is he minimum risk soluion of aspiraion level u for he problem (.) if i is he opimal soluion o he problem: Max x D P ( ~ z ( x, ~ c ) u) (.5) Tha is, o obain a minimum risk opimal soluion o he problem (.) we apply wha in he lieraure is referred o as he minimum risk crierion. This consiss of fixing a level for he sochasic objecive funcion u R, o which we apply he erm aspiraion level, and of maximising he probabiliy ha he objecive will be less han or equal o ha level: P( z ~ ( x, ~ c ) u). In his way, he fixed level u can be inerpreed as being he uppermos level ha he Decision Maker (DM) is capable of admiing for he sochasic objecive ha we wish o minimise. 6
7 The firs sudies in which his crierion, also ermed model P, is proposed as a means of solving he problem (.), are hose carried ou by Charnes and Cooper (963) and Bereanu (964). Definiion 5: Kaaoka soluion wih probabiliy β Poin x D is he Kaaoka soluion wih probabiliy β o he problem (.) if here exiss u R such ha (x, u) is he opimal soluion o he problem: s. Min u x, u P( ~ z( x, ~ c) u) = β (.6) x D Thus, he obaining of opimal soluions wih probabiliy β is derived from he applicaion of he βfracile or Kaaoka crierion o he problem (.), inroduced by Kaaoka (963). I consiss of fixing a probabiliy β, for he objecive and deermining he lowes level u, which canno exceed he objecive funcion wih ha probabiliy. The objecive funcion of he sochasic problem goes on o become par of he feasible se as a probabilisic or chance consrain and he deerminisic equivalen problem is a problem wih n+ decision variables: he n decision variables of he sochasic programming problem carried in vecor x and variable u. Once hese five soluion conceps for sochasic programming problems have been defined, each one associaed wih a crierion which is considered adequae for he resoluion of he iniial problem, a comparaive analysis of he abovemenioned conceps enables us o commen on he following differences which emerge beween hem:. In order o apply he maximum probabiliy crieria (minimum risk and Kaaoka) i is necessary o fix previously a parameer: level u, if we apply he minimum risk crierion, or probabiliy β, if we apply he Kaaoka crierion, 7
8 while in he firs hree crieria his is unnecessary. Moreover, he minimum risk and Kaaoka opimal soluions will depend, generally, on he values assigned o hose parameers.. Whils he firs hree soluion conceps (expeced value, minimum variance and expeced value sandard deviaion efficiency) can be obained provided he expeced value and variance of he objecive funcion are known, he wo crieria of maximum probabiliy depend on he disribuion funcion of he sochasic objecive funcion and, herefore, in order o obain hem i is essenial o know he laer. 3. The las hree soluion conceps (expeced value sandard deviaion efficiency and he wo relaed o maximum probabiliy) give rise, in general, o a se of soluions: he se of expeced value sandard deviaion efficien soluions for he firs of hese and a soluion for each level, u, or probabiliy β o be fixed for he las wo conceps which are no generally comparable. From wha has been saed above i can be concluded ha he diversiy of soluion conceps leads o he need o choose one or he oher o solve he sochasic programming problems and, in his sense, we can sae ha he resoluion of hese problems always implicily conains a decision process. Obviously, he choice of a soluion will have o be made bearing in mind he characerisics of he problem o be solved and he preferences of he DM. However, as we shall see below, if specific hypoheses are shown o hold rue, he soluions which we have jus defined are relaed, alhough a he beginning one may hink he opposie, due o he fac ha hese are soluions o deerminisic equivalen problems which draw on saisical characerisics differen from he sochasic objecive. 8
9 3. Relaions beween he Soluion Crieria for Sochasic Programming Problems In his secion we deal wih he analysis of he exisence of relaions beween he soluions o he problem (.) previously defined, which are obained by applying o he problem he crieria discussed in he previous secion: expeced value, minimum variance, expeced value sandard deviaion efficiency, minimum risk and Kaaoka. Specifically, relaions are esablished beween he minimum risk opimal soluions and he Kaaoka ones and beween hese and he expeced value sandard deviaion efficiency soluions for a sochasic programming problem. Prior o ha, we shall consider some of he resuls of Muliple Objecive Programming which will be applied hroughou his secion. In addiion, as an immediae consequence of hese resuls we shall esablish relaions beween he opimal soluions expeced value and minimum variance and he expeced value sandard deviaion efficien soluions. Le us consider he following muliple objecive programming problem: Min x D ( f ( x),..., f ( x) ) q (3.) where f is a vecorial funcion, f: H R n R q, and he problem resuling from he applicaion of he weighing mehod o i gives us: Min µ f ( x) µ f ( x) x D q q (3.) wih vecor of weighs m, generaliy, we assume o be normalised. µ k 0, for all k {,,, q}, which, wihou losing The following heorem relaes o he efficien soluions o he muliple objecive problem (3.) and he opimal soluions o he associaed weighed problem, (3.). 9
10 Theorem.( Sawaragi, Nakayama and Tanino (985). Le us assume ha he funcions f,..., f q are convex and ha D is a convex se. Thus: a) If x* is a properly efficien soluion for he muliple objecive problem (3.), here exiss a weigh vecor m wih sricly posiive componens such ha x* is he opimal soluion for he weighed problem (3.). b) If x* is he opimal soluion o he weighed problem (3.), for a vecor of weighs wih sricly posiive componens, x* is a properly efficien soluion for he muliple objecive problem (3.). c) If x* is he only soluion o he problem (3.), wih m 0, hen x* is an efficien soluion o he problem (3.). If i is no he only soluion, he soluions obained are weakly efficien for (3.). This heorem of muliple objecive programming allows us o relae he expeced value and minimum variance soluions o he problem (.) o he expeced value sandard deviaion efficien se. Therefore, le us consider he problems (.), (.) and (.3). The weighed problem associaed o he problem (.3), for weighs µ and  µ, µ [0, ], is: Min µ z( x) + ( µ ) σ( x) x D From Theorem we can sae ha:. If he expeced value opimal soluion o he problem (.) is unique hen i is an expeced value sandard deviaion efficien soluion. Where i is no unique i can only be assured ha he expeced value opimal soluions are expeced value sandard deviaion weakly efficien soluions, bu we canno sae ha hey are expeced value sandard deviaion efficien soluions.. If he variance of he sochasic objecive is a sricly convex funcion, he minimum variance problem has a unique soluion (since we assume ha se D 0
11 is closed, bounded, convex and no empy), and so is opimal soluion is an expeced value sandard deviaion efficien soluion o he sochasic programming problem. Should he minimum variance problem have more han one opimal soluion, hese soluions are expeced value sandard deviaion weakly efficien soluions and he only hing we can be cerain (as in poin above) is he weak efficiency of hese soluions. We shall now analyse he relaions beween he minimum risk and Kaaoka problems. Following his, we shall reurn o deal wih he concep expeced value sandard deviaion efficien soluion and we shall relae i o he Kaaoka and minimum risk soluions o he sochasic programming problem. 3.. Relaions beween he minimum risk problem and he Kaaoka problem Given he sochasic programming problem (.) le us consider he deerminisic equivalen problem (.5) corresponding o he crierion of minimum risk of aspiraion level u, and he problem (.6) corresponding o he applicaion of he Kaaoka crierion for a probabiliy β (0, ). The following heorems esablish a relaion beween he opimal soluions o hese wo problems. Theorem Le us assume ha he disribuion funcion of he random variable ~ z ( x, ~ c ) is sricly increasing. Then x* is he minimum risk soluion of aspiraion level u* if and only if ( x *, u *) is he Kaaoka soluion wih probabiliy β*, wih u* and β* so ha: P( ~ z ( x *, ~ c ) u*) = β *. Proof: We demonsrae he heorem by process of reducio ad absurdum. ) If x* is he soluion o he problem (.5) i is rue ha: P ( ~ z ( x, c~ ) u *) P( ~ z ( x*, ~ c ) u *) = β*, x D
12 Le us assume ha ( x *, u *) is no he soluion o he problem (.6). In his case here exiss a vecor ( x, u ), feasible in (.6) proving ha u < u*, ha is: ~ x ~ β and u < u*. x D, here exis u R such ha P( z(, c) u) = * However as he disribuion funcion of he random variable ~ z ( x, ~ c ) is sricly increasing, i is also rue ha: P ( ~ z ( x, ~ c ) u* ) = P( ~ z ( x, ~ c ) u) + P( u < ~ z ( x, ~ c ) u *) = β * + θ ~ ~ ) > where θ = P( u < z( x, c u *) 0 ~ ~ ) β x D for which P( z ( x, c u *) > *, from which we deduce ha here exiss a vecor, which conradics he hypohesis. ) By hypohesis ( x *, u *) is he opimal Kaaoka soluion of level β *, herefore i proves ha: ~ ~ ) β a) I is feasible in (.6): x* D and P( z ( x*, c u *) = * b) If ( x, ) ~ x ~ ) β, hen u* u. u proves ha x D and P( z (, c u) = * Le us assume ha x* is no he opimal soluion o he problem (.5). Then, here exiss a vecor x D for which: P ( ~ z ( x, ~ c ) u *) > β * As he disribuion funcion of he random variable ~ z ( x, ~ c ) is sricly increasing, i holds rue ha here exiss a u < u* for which: which conradics he hypohesis b). P ( ~ z ( x, ~ c ) u) = β * From Theorem we can affirm ha if he disribuion funcion of he sochasic objecive is a sricly increasing funcion here exiss a reciprociy beween he minimum risk and he Kaaoka soluions, such ha, if we have he minimum risk soluion (Kaaoka soluion) we can affirm ha i is he Kaaoka soluion (minimum risk soluion), ha is:
13 . For each fixed aspiraion level u, he minimum risk soluion is also he Kaaoka soluion wih a probabiliy β equal o he maximum probabiliy obained in he minimum risk problem. For each fixed β he Kaaoka soluion is also he minimum risk soluion, if, for his laer, we esablish an aspiraion level u equal o he opimal value of he Kaaoka problem. 3.. Relaions beween he opimal soluions of Kaaoka and he expeced value sandard deviaion efficien soluions We shall now analyse he exisence of some relaion beween hose soluions o he sochasic programming problem (.) which correspond o he applicaion of he Kaaoka crierion and he se of expeced value sandard deviaion efficien soluions o he same problem. Up ill now, he relaions ha we have esablished beween he soluion crieria for sochasic programming problems have been of a general naure, ha is, applicable o any problem of sochasic programming. However, in his secion he relaions o esablish are exclusively applicable o sochasic programming problems of specific characerisics, given ha for he Kaaoka crierion o be applied i is necessary o know he disribuion funcion of he sochasic objecive, somehing which is, on he whole, complicaed, even when he probabiliy disribuion of vecor ~ c is known. This fac leads us o he heme of our sudy, assuming he objecive funcion of he problem o be of he linear ype, so ha from his poin onwards, he problem we are dealing wih is: Min D x cx % (3.3) Furhermore, we esablish a hypohesis regarding he probabiliy disribuion of vecor ~ c, and, specifically, we consider wo cases which correspond o he hypohesis ha vecor ~ c is normally disribued (normal case) 3
14 and o he consideraion ha vecor c ~ depends linearly on a single random variable (simple randomizaion case). In boh cases i is possible o deermine he probabiliy disribuion funcion of he sochasic objecive raise and solve he problem (3.3) by applying he Kaaoka crierion. ~ c x and, herefore, o To carry ou his analysis we begin by raising he problem of expeced value sandard deviaion efficiency of he problem (3.3) and he problems corresponding o he applicaion of he Kaaoka crierion in he cases menioned, and, following his, we analyse he relaions beween hem. Le c be he expeced value vecor of c ~ and V be is variance and covariance marix, which we assume is posiive definie. Then he expression of he expeced value of ~ c x is c x and is variance is x Vx, and, herefore, he se of expeced value sandard deviaion efficien soluions for he problem (3.3) is ha of he problem: / ( cx xvx ) Min, ( ) (3.4) x D As we assume ha he feasible se D is convex, by hypohesis from he / origin, and he funcion ( x Vx) is convex (see SancuMinasian (984) pages 9495), he bicrieria problem expeced value sandard deviaion is convex, and we can sae, on he basis of Theorem, ha he se of expeced value sandard deviaion properly efficien soluions o he problem (3.4), p E Eσ, can be generaed from he resoluion of he weighed problem: Min µ c x + ( µ )( x x D Vx) / (3.5) where µ (0, ). Subsequenly we go on o obain he deerminisic equivalen problems for (3.3) corresponding o he applicaion of he Kaaoka crierion, in he cases previously cied. a) Normal linear case 4
15 Le us suppose ha 0 D and c ~ is a random vecor mulinormal wih expeced value c and posiive definie marix of variances and covariances V. In his case he random variable ~ c x is a normal variable wih expeced value c x and variance x Vx, which implies ha: (( ~ ) / ( ) / c x c x ( x Vx) u c x ( x Vx) ) = Φ( u c x) ( x ) / ) ~ P( c x u) = P Vx where Φ is he sandardised normal disribuion funcion. This implies ha he feasible se of he problem (.6) is, according o he esablished hypoheses: { ( x, u) D R / u = c x + Φ ( β )( x Vx ) / } and he Kaaoka problem, for a probabiliy β, is: s.a u = c Min u x, u x + Φ x D ( β)( x Vx) / equivalen o: If α Min c x + Φ x D ( β )( x Vx) / Φ ( β), we can express he preceding problem as follows: = Min x D c x + α( x Vx) / (3.6) b) Simple linear randomizaion case Le us assume ha vecor c ~ depends linearly on a single random variable: ~ c c ~ = + c, where ~ is a coninuous random variable of expeced value, sandard deviaion υ > 0, υ <, and disribuion funcion F, sricly increasing. Moreover, le us suppose ha c x > 0 for all x D. Wih hese hypoheses he expeced value of ~ c is c = c + c and is marix of variances and covariances is V = υ c c. Proceeding from he previous hypoheses i follows ha: P( c ~ x u) = P( ~ c x u c x) = P ( ~ ( u c x) c x) = F ( u c x) c x) 5
16 so ha he Kaaoka problem is equivalen o: Min x D c x + F ( β ) c x As in he normal case, making α = ( ( β ) ) υ F we can express he preceding problem by means of he problem (3.6). Therefore, in he wo cases analysed, he problem corresponding o he Kaaoka crierion is problem (3.6), where α = Φ ( β ) in he normal case and ( ( β ) υ α = ) in he simple randomizaion case. F In order o esablish he relaions beween he opimal soluion of he Kaaoka problem in hese wo cases and he se of expeced value sandard deviaion properly efficien soluions we have only o relae he opimal soluions of he problems (3.5) and (3.6). Proposiion If α >0 and µ (0, ), wih µ = ( + α), hen problems (3.5) and (3.6) are equivalen. Proof: Since µ (0, ) he soluion o problem (3.5) coincides wih ha of: Min x D c x + ( µ )( x Vx) / µ and simply doing α = ( µ ) µ > 0 or, wha is equivalen, µ = ( + α) we obain he equivalence beween problems (3.5) and (3.6). Thus, from he Proposiion, we esablish an equivalence beween problem (3.6) and he weighed problem (3.5). If we denoe x*(α) he opimal soluion o problem (3.6), we can be sure ha: U α> 0 x *( α) = Le us see wha inerpreaion his resul has in he cases analysed. In boh cases he parameer α muliplies he sandard deviaion of he sochasic objecive E p Eσ 6
17 and depends on he fixed probabiliy for he applicaion of he Kaaoka crierion. Specifically:. In he normal case α = Φ ( β ), and consequenly his parameer is posiive if and only if he fixed probabiliy, β, is greaer han In he simple randomizaion case α = ( F ( β ) ) σ and herefore α > 0 if he probabiliy o be fixed in he Kaaoka problem β > F ( ), ha is greaer han he probabiliy of he random variable ~ being less han or equal o he expeced value. Consequenly, in boh cases, he fac ha α is sricly posiive can be inerpreed as fixing a high probabiliy. From he resuls obained we can sae ha: If he probabiliy fixed in he Kaaoka problem is high (β > 0.5 in he normal case and β > F ( ) in he simple randomizaion case), we can affirm ha he opimal soluion o his problem is a expeced value sandard deviaion properly efficien soluion of he sochasic problem (.). Given a expeced value sandard deviaion properly efficien soluion o he problem (.), in he cases analysed here exiss a high probabiliy, β, such ha his soluion is a Kaaoka opimum. Obviously, based on hese resuls one has o ask he quesion of wha happens when, in eiher of hese wo cases under sudy, he fixed probabiliy is such ha α 0. Noe ha for α = 0 he problem ha is obained in boh cases is he expeced value problem, previously menioned in he sudy, and i corresponds o he fixing of a probabiliy β = 0.5 in he normal case and β = F () in he simple randomizaion case. 7
18 Wha ype of soluion is obained when he probabiliy fixed is low? When his is so he opimal soluion o he Kaaoka problem does no have o be a expeced value sandard deviaion properly efficien soluion o he sochasic problem. In his sense i mus be noed ha he expeced value sandard deviaion efficiency crierion is appropriae when he individual is risk averse. So, in he cases analysed, he hypohesis of risk aversion is ranslaed ino he fixing of a high probabiliy. In addiion o his, if we analyse he objecive funcion of he Kaaoka problem, we noe ha when he fixed probabiliy is low, he sandard deviaion of he sochasic objecive is muliplied by a negaive facor. This ells us ha, in some way, a leas in he cases analysed, he minimum variance crierion may no be appropriae if he DM wans a risky soluion o he sochasic problem. On he oher hand, i mus be emphasised ha, as for hese problems, he disribuion funcion of he sochasic objecive is sricly increasing, his deermines as rue he reciprociy analysed in he previous secion amongs he opimal Kaaoka and minimum risk soluions, so ha i can be saed ha ha he minimum risk opimal soluions mainain wih he se of expeced value sandard deviaion properly efficien soluions, he same relaions ha we have obained beween hese laer and he opimal Kaaoka soluions. Finally, we esablish a furher connecion beween he problems (3.6) and (3.5) in Proposiion. Specifically we relae he weighs assigned o he expeced value and he sandard deviaion in he weighed problem (3.5) wih he value of parameer α in problem (3.6), whose opimal soluion coincides wih ha of (3.5). Given ha he weigh assigned, µ, in problem (3.5) can be considered as a measuremen of he relaive imporance which is given o he expeced value of he objecive funcion as opposed o he sandard deviaion in he obaining of expeced value sandard deviaion efficien soluions, we can also pose he quesion of 8
19 wheher here exiss any relaion beween he value of he parameer µ and he value of he parameer α which will weigh he sandard deviaion in problem (3.6). Proposiion Le x ( µ ), x( µ ' ), x( α) and x( α' ) be opimal soluions o he problems (3.5) for µ, (3.5) for µ ', (3.6) for α and (3.6) for α ', respecively wih µ, µ ' (0, ), α ( µ ) µ and α ' = ( µ ') µ ' =. Thus, µ < µ ' if and only if α ' < α. Proof: ) Since µ <µ, i is eviden ha:  µ ' < µ and µ ' < µ, consequenly: ) Obvious from ( ). ( µ ') µ ' < ( µ ) µ ' < ( µ ) µ α α ' = = Given ha α = Φ ( β ) in he normal case and α = ( F ( β ) ) σ in he simple randomizaion case, he resul is ha a weigh, µ, higher for he expeced value in he weighed problem (3.5), corresponds o a lower probabiliy in he Kaaoka problem and, herefore, more risky in erms of probabiliy. To illusrae he resuls obained in his secion, we shall now propose and solve he following example. 4. Example Le us consider he following problem of sochasic programming: 9
20 Min x cx % = cx % + cx % s.. 3x + x x 3x 8 x + x 5 x, x 0 where ~ c c ~ = + c, wih c = ( 4, 4), c = (, ) (4.) and ~ a random exponenial variable of parameer λ=, so ha is disribuion funcion is F ( η) = exp( η ) and is expeced value and sandard deviaion are = 0.5 y υ = 0.5, respecively, and consequenly he expeced value and he sandard deviaion of he random variable ~ c x are c x =.5x + and 4 x σ ( x) = 0.5x + 6 x, respecively. Le us proceed o solve he above problem using he soluion crieria seen in he sudy and illusrae he resuls obained. The expeced value and minimum variance problems associaed wih he sochasic problem have a unique soluion. These soluions are x E =(0, ) and x V =(/3, 0), respecively, as indicaed in Figure. Figure 0
21 The se of expeced value sandard deviaion efficien soluions, E Eσ, can be obained by solving he bi crieria problem: Min x s. ( 4.5x + x,0.5x + 6x ) 3x x + x 3x x + x x, x (4.) and herefore he payoff marix for his problem is shown in Table. Crierion Soluion Expeced Value Sandard Deviaion Expeced value x E =(0, ) 6 Variance x V =(/3, 0) 3 /3 Table : PayOff Marix The se of expeced value sandard deviaion efficien soluions is: { (, ) /3,, 0 } E = E x x R x + x = σ x x This se is he segmen of he feasible se which joins he verices x E and x V in Figure. Noe ha hese wo soluions (which are unique opima for he expeced value and minimum variance problems respecively) are expeced value sandard deviaion efficien soluions. In addiion, given ha (4.) is a linear problem, i is held as rue ha he ses of efficien and expeced value sandard deviaion properly efficien soluions coincide, ha is: Eσ p E Eσ E =. Le us now consider he soluion of he problem using Kaaoka crierion. For his, he feasible poins of he problem mus verify c x = x + x > 0. I is easy o prove ha in his problem his condiion holds rue. The soluions o he deerminisic equivalen problem which correspond o he applicaion of he
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