European Journal of Operational Research

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1 European Journal of Operational Research 216 (2012) Contents lists available at ScienceDirect European Journal of Operational Research journal homepae: Innovative Applications of O.R. Cost/risk balanced manaement of scarce resources usin stochastic prorammin Alexei Gaivoronski a,, Giovanni M. Sechi b, Paola Zuddas b a Department of Industrial Economics and Technoloy Manaement, Norweian University of Science and Technoloy, Trondheim, Norway b Department of Land Enineerin, University of Caliari, Italy article info abstract Article history: Received 23 April 2010 Accepted 29 June 2011 Available online 8 July 2011 Keywords: OR in natural resources Risk manaement Stochastic prorammin Water resources manaement Risk/performance tradeoff We consider the situation when a scarce renewable resource should be periodically distributed between different users by a Resource Manaement Authority (RMA). The replenishment of this resource as well as users demand is subject to considerable uncertainty. We develop cost optimization and risk manaement models that can assist the RMA in its decision about strikin the balance between the level of taret delivery to the users and the level of risk that this delivery will not be met. These models are based on utilization and further development of the eneral methodoloy of stochastic prorammin for scenario optimization, takin into account appropriate risk manaement approaches. By a scenario optimization model we obtain a taret barycentric value with respect to selected decision variables. A successive reoptimization of deterministic model for the worst case scenarios allows the reduction of the risk of neative consequences derived from unmet resources demand. Our reference case study is the distribution of scarce water resources. We show results of some numerical experiments in real physical systems. Ó 2011 Elsevier B.V. All rihts reserved. 1. Introduction In this paper we deal with quantitative plannin and manaement models for distribution of scarce resources under uncertainty and conflictin demands of different users and activities. A recurrin reference which we make is a distribution system of water resources in a reion where such resources are scarce; see for example Loucks et al. (1981), Dembo (1995) and Sechi et al. (2005). However, the methods which we consider here are applicable to a wide rane of manaement problems in other, different areas, like distribution of manpower in a call center, distribution of transportation or transmission capacities, assinment of empty containers, etc. We address the case when the plannin horizon extends for a substantial amount of time periods. These periods are connected with each other by the flow balance equations which connect the amount of the resource inherited from the previous period with the amount of resource which will be available in the next period, takin into account consumption and replenishment of the resource at the current time period. This relation makes the future state of the system dependent on the current decision about the resource expenditure. Consequently, it is necessary to foresee the future replenishment and consumption of the resource in order to make the optimal manaement decision now. There is substantial uncertainty about these forecasts, especially about replenishment process. Indeed, in the case of water resources it is necessary to Correspondin author. Tel.: addresses: Alexei.Gaivoronski@iot.ntnu.no (A. Gaivoronski), sechi@unica. it (G.M. Sechi), zuddas@unica.it (P. Zuddas). predict the hihly chanin and uncertain water inflow patterns for the time in consideration, and such forecasts are inherently very imprecise. Additional sources of uncertainty come from projections of future demand, costs and prices. An adequate manaement policy cannot be defined without adequate consideration of this uncertainty; therefore deterministic mathematical prorammin models are of limited use here. One way to address this problem is to construct a set of scenarios about the future values of uncertain parameters, like the values of replenishment and demand processes for each future time period. Then scenario analysis is employed in order to construct the optimal decision policies for each scenario and to investiate the sensitivity of these decisions with respect to the chanes in uncertain parameters. This can be a useful technique, especially when the scenarios do not differ drastically between each other. In such a situation it may be possible to identify a decision policy that will be relatively stable with respect to scenarios. However, in the problems which we address here this is not the case because different replenishment scenarios can differ substantially between themselves. In such circumstances an optimal policy for one scenario may have little in common with the optimal or even acceptable policy for some other scenario. These difficulties can be overcome by usin the stochastic prorammin approach (Ermoliev and Wets, 1988; Kall and Wallace, 1994; Pflu, 1996; Bire and Louveaux, 1997 and Kleywet and Shapiro, 2001); for application to water resources manaement see Dupacová et al. (1991), Pallottino et al. (2004) and Escudero and Mone (2008). One way to utilize this methodoloy is to describe the uncertainty by oranizin scenarios in scenario trees. A probability /$ - see front matter Ó 2011 Elsevier B.V. All rihts reserved. doi: /j.ejor

2 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) is assined to each scenario. These probabilities can be recovered from statistical processin of historical data, but they can be also assined subjectively by experts. The objective function of stochastic prorammin problem is defined by weihtin the costs or profits associated with each scenario with these probabilities. The structure of scenario tree allows formulatin a deterministic equivalent of multistae stochastic prorammin problem by collectin all constraints that define the feasibility sets for each scenario and addin nonanticipativity constraints, which assure that the decision variables associated with different scenarios, will be the same until the time period where scenarios start to differ. Besides references mentioned above, detailed description of different aspects of such multistae stochastic prorams with uncertainty described by scenario trees can be found in Rockafellar and Wets (1991), Hile and Sen (1996), Ruszczynski (1997), Lucas et al. (2001) and Römisch and Schultz (2001). Application examples to investment plannin are iven in Mulvey and Vladimirou (1991), Cariño and Ziemba (1998), Consili and Dempster (1998) and Zenios (2008), to air traffic manaement in Glockner and Nemhauser (2000), to insurance in Gaivoronski and de Lane (2000) and Høyland and Wallace (2001), to telecommunications plannin in Gaivoronski (2006), to electric power plannin in Fleten and Kristoffersen (2006), to facility location in Stouie et al. (2008), to empty containers allocation in Di Francesco et al. (2009), see also Wallace and Ziemba (2005) for representative collection of applied problems. Detailed application of stochastic prorammin approach to water resource manaement was considered in Pallottino et al. (2004) and Manca et al. (2004), where followin Rockafellar and Wets (1991) it is also referred to as scenario optimization. The specifics of this application field have not allowed the use of statistical techniques for scenario eneration due to considerable climate non-stationarities verified durin last years. Therefore scenario eneration was based upon expert knowlede, hence the term Subjective Stochastic approach, introduced there. Our reference application field in this paper is plannin and manaement of water resources. This field has attracted considerable attention of international research community. Recent examples and comparisons of deterministic multicriteria plannin models are presented in Mariano-Romero et al. (2007) and Hajkowicz and Hiins (2008). Different methods of incorporatin uncertainty into water resources plannin models were also a subject of recent research. In addition to already cited Dupacová et al. (1991) and other papers where stochastic prorammin approach was considered, one can mention Huan (1998) who developed a sinle period model with hybrid stochastic and interval representation of uncertainty. This work was extended to the case of two periods by Maqsood et al. (2005). A sinle period nonlinear stochastic optimization model with stochastic inflows and water deficit was developed by Azaiez and Haria (2001). A sinle period oal prorammin formulation under uncertainty that balances between the needs to satisfy aricultural demand and limit the environmental impact was considered by Bravo and Gonzalez (2009). Wen et al. (2010) develop multicriteria model for scenario analysis that is capable of incorporatin uncertainty. Our paper distinuishes itself from this literature by incorporatin risk manaement approach into multistae stochastic optimization model for water resources manaement with the uncertainty bein described by scenario trees. More specifically, it presents the followin novel contributions Taret resource delivery policy This is a modelin solution which responds to the specifics of distribution of scarce resource. Often such distribution is performed by a Resource Manaement Authority (RMA) responsible for functionin of the storae and distribution system. This authority should find a distribution pattern which will in the best possible way satisfy conflictin needs of different users and overall system performance criterion in the situation when the combined requirements exceed the system capacity. The overall manaement policy in such situation will consist of two components. 1. The RMA makes decision about the taret resource delivery to the users. This taret delivery is communicated to the users toether with the (usually hih) level of reliability of this delivery. This phase is performed by a cost/risk balanced model calibrated with intensive use of simulation tools. 2. After receivin this information about the taret delivery the users develop their own resource manaement policy that takes into account a possible shortfall between the taret delivery and their needs. This stae can include the neotiation phase with establishment of a joint RMA-users committee. This committee can develop an emerency plan to face resource shortae. In this paper we develop models and tools that RMA can use to determine the taret resource delivery on the first stae of this decision process, but also durin the neotiations of the second stae. We show how this concept of taret resource delivery can be interated in the multistae stochastic prorammin model by inclusion of a special distance term in the objective function. This term measures the distance between deliveries which occur under different scenarios and taret delivery, therefore the resultin policy will tend to be barycentric with respect to scenario dependent deliveries Risk/performance trade-off The risk of resource shortae is an important feature in this application field and resource manaement models should take it explicitly into account. Therefore we assume that the purpose of the system manaement is not optimization of averae costs or profits, but rather achievement of acceptable trade-off between costs/profits and risks. We call it cost/risk balanced manaement of scarce resource. The models for such manaement we obtain by extension of the multistae stochastic prorammin models to include explicitly the risk part, usin the risk measures appropriate for the distribution of scarce resources. One such measure is the barycentric deviation from the taret resource delivery outlined above. The rest of the paper is oranized as follows. Section 2 sets the stae for further discussion by formulatin the multiperiod stochastic prorammin model for distribution of scarce resources takin as a reference point the water resources. We discuss the difference between deterministic solutions for each scenario and interated stochastic solution. This model is extended in Section 3 to include the computation of the barycentric taret resource delivery. It is shown that the risk manaement is closely related to the concept of taret delivery and how the risk/performance trade-off can be obtained from extended stochastic prorammin model. In Section 4 we describe an application to manaement of real water storae and distribution system in Sardinia. Section 5 concludes the paper with the summary of results and directions for future research. 2. Multiperiod stochastic prorammin model for water resource manaement under uncertainty Here we set the stae for the development of models for cost/ risk balanced manaement of scarce resources by formulatin multistae stochastic prorammin model for water resources

3 216 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) manaement under uncertainty. This can be done in three phases: eneration of scenarios for evolution of uncertain parameters and construction of a scenario tree, development of a deterministic optimization model for water resources manaement for each scenario and areation of these models into a multistae stochastic prorammin model (Bire and Louveaux, 1997; Kall and Wallace, 1994) Construction of scenario tree The objective of this phase is to produce simplified yet adequate picture of the temporal evolution of uncertain parameters like water inflows and demands durin the time horizon of plannin interest. This is the stae where the specifics of the application field plays important role. First, we enerate a set G of N scenarios for evolution of water inflows and demands durin the time horizon of several years with a fixed time step. In Manca et al. (2004) and Pallottino et al. (2004) some water systems of south Sardinia were considered durin this process. From interaction with water manaement professionals it became clear that historical data are of limited use due to considerable climate perturbations durin the last decade. For this reason substantial part of scenarios was derived from subjective opinions of water resources experts. Each scenario 2 G was iven a weiht p which can be looked upon as a the subjective probability. Then the initial portions of scenarios were areated and scenarios were oranized in a scenario tree as follows. The root node of the tree corresponds to the beinnin of time period t = 1. From this node, n 0 scenarios, of inflow and demand, start and continue in parallel for T 1 periods. At t = T 1 each of n 0 scenarios is split into n 1 scenarios all the obtained scenarios n 0 n 1 continue in parallel for a further T 1 periods until t =2T 1 when each of them is split into n 2 additional scenarios. The process continues with splittin each T 1 periods. Thus, scenario tree after K splittins spans the time horizon T = KT 1 and consists of N scenarios, N ¼ YK 1 i¼0 n i An example of such a tree with T 1 = 12, T = 36, K =3, n 0 =1, n i =3 and i = 1:2 is shown in Fi. 1. Any specific scenario 2 G is represented by a path in this tree that starts at the root at time t =0 and ends at one of the leaves of the tree at time t = T. The scenario set G is composed from all such paths. Usually one time period corresponds to one month, and we take T 1 = 12 which corresponds to one year. This means that splittins occur at the end of each year, which conforms to the seasonal patterns of inflows and demands. This scenario eneration and areation process which produces a scenario tree from individual scenarios is described in more detail in Manca et al. (2004) and Pallottino et al. (2004) Deterministic optimization model This is a linear cost minimization model which is formulated for each scenario 2 G and has the followin form: min c T x A x ¼ b l 6 x 6 u The vector x includes all decision variables which describe operational and plannin decisions durin each time period of the plannin horizon, like decisions how much water deliver to the users, how much keep in different storae components of the system, how much water to exchane between different system components, and so on. The vector of unit costs c describe the costs of different activities like delivery costs, opportunity costs related to unsatisfied demand, opportunity cost of spilled water, etc. The set of standardized equality constraints describes the relationships between storae, usae, spill and exchane of water at different reservoirs in subsequent time periods. The riht hand sides b are formed from scenario data of inflows and demands. The lower and upper bounds l and u are defined by structural and policy constraints on the functionin of the system. All decision variables and data are scenario dependent, hence the index. One specific and more detailed example of the problem (1) can be found in Section 5 dedicated to numerical experiments Areated multistae stochastic prorammin model for scenario optimization More precisely, this model can be referred to as the deterministic equivalent of multistae stochastic prorammin model, see Bire and Louveaux (1997) and Kall and Wallace (1994) for details. It is constructed from collection of deterministic models (1), as follows: min x ;2G p c T x ð1þ ð2þ A x ¼ b ; 8 2 G ð3þ l 6 x 6 u ; 8 2 G ð4þ x 2 S ð5þ The objective function (2) is defined as the averae of the cost objectives of all scenarios weihted with by the probabilities of the scenarios, p. All constraints (3), (4) are collected from all scenarios and put in the areated model. An additional set of nonanticipativity constraints (5) is added. These constraints ensure that the decision variables that describe decisions at time periods when scenarios are not yet split coincide. In other words, they ensure that decisions do not depend on the future. For example, the values of decision variables related to scenarios 1 and 2 from Fi. 1 must coincide durin time period [0,2T 1 ] and decision variables related to scenarios 1, 2 and 3 must coincide durin time period [0,T 1 ], so the set of constraints (5) for this scenario tree will contain correspondin equalities. Fi. 1. Scenario tree for K =3,T 1 = 12, n 0 =1,n 1 = n 2 =3. In Pallottino et al. (2004) authors provide some results obtained by solution of multiperiod stochastic prorammin model (2) (5)

4 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) with scenario optimization and its comparison with deterministic solution in the case of one real water resources system when scarcity occurs. These results were referred to a two-stae, two-scenario tree. The first scenario was obtained from historical hydroloical inflows while the second simulates a severe but possible shortfall in water inflows. It is derived from the first one assumin that a reduction of 50% will occur after the branchin time. Resources stored in the reservoir and resources transferred to a cluster of users for stochastic and deterministic solutions are compared. Comparison shows that the deterministic policy leads to earlier emptyin of reservoir for scarce scenario compared to the stochastic policy for the same scenario. The consequence of the deterministic policy is a drastic cut in water resources that can be delivered to civil and industrial communities. Stochastic policy for scarce scenario, resultin from scenario optimization, exhibits smoother resources distribution and a lower variance with respect to the deterministic policy. Thus, the manaement policy suested by scenario optimization of stochastic prorammin model is more realistic compared to the policy that results from the deterministic optimization in the case of scarce resources. An effective manaement policy must be able to establish a taret value for deliverin resources to the demand centre. The community suffers less from resource rationin if it has been forewarned of a possible shortae. This taret value should take into account the entire rane of possible scenarios of resource availability, neither too pessimistic in case of abundance, nor too optimistic in case of scarcity of resources. In other words, a taret value should be sufficiently balanced with respect to the different possible scenarios that could take place. In what follows we refer to this balanced taret, evaluated by a balancin process amon the whole set G of considered scenarios, as barycentric. The more formal definition of this notion is iven in Section 3, see problem (10) (13). Establishin the resource demand level at this taret value would permit notifyin the resource users (the community) in a timely fashion. As a consequence, preventive measures could be adopted in order to avoid, at least in part, damaes derived from an unexpected drastic cut in resources. In what follows we develop enhanced stochastic prorammin models that address this issue. 3. Taret resource delivery and cost/risk balanced manaement In this section we shall be concerned with further development of the approach described in Section 2 from the perspective of relation between the Resource Manaement Authority (RMA) and end users. We shall see that the purely cost minimization point of view developed in the previous section is not sufficient and should be enhanced by risk manaement considerations. Indeed, the party which bears the risk of shortaes is the end user. For this reason resource plannin models employed by RMA should include the balance between costs which authority bears and risks to which the end users are exposed. We take as a startin point the eneral form of resource manaement problem of the RMA from Sections 2 5. It is also possible to formulate this problem in the node form, where nonanticipativity constraints (5) will become unnecessary due to the absence of redundant decision variables, which correspond to coincidin parts of scenarios. Solution of this problem will yield the amount of resource which will be allocated to different users durin time periods of the plannin horizon under different scenarios. Assumin that there is a sinle user (possibly consumin resource at different locations) we shall denote the set of known user s demands, in all time periods, under scenario as D. This will be a subvector of the vector b in Eq. (3). We shall also denote the set of decision variables representin resources delivered to this user in all time periods, under scenario as ^x. This will be a subvector of the vector x of all decisions of the RMA under scenario. In the case when there is a number of users for which the RMA can have different policies, we shall index these users with index l =1:L. Then the vectors D of users demands and ^x of resource delivery will be further subdivided into subvectors D l ; : D ¼ D 1...; DL and ^x l : ^x ¼ ^x 1 ;...; ^xl. The vector of all demands of user l under all scenarios will be denoted by D l ; D l ¼ n o D l j 2 G and the vector of all demands of all users under all scenarios will be denoted by D. The vector of all deliveries to the n o user l under all scenarios will be denoted by ^x l ; ^x l ¼ ^x l j 2 G and the vector of all resource deliveries under all scenarios to all users will be denote by ^x; ^x ¼ ^x j 2 G. We assume that the resource in question is scarce and for this reason the demand will not be satisfied in many scenarios. In such situations users should develop and adopt an emerency policy to alleviate and manae the effect of shortaes on their activities. In order to do this a user l should know in advance the taret level of demand satisfaction x b l that the RMA is willin to deliver to him no matter what scenario will occur. Usually this taret level will be less than the user s demand D l, due to inherent scarcity of the resource. This difference between demand D l and delivery x b l will represent the planned shortae of the resource, which the user l is asked to accept. Besides this planned shortae there can also be unplanned shortaes when due to severe lack of resources under some scenarios the RMA will not be able to deliver even the reduced volume x b l. In order to manae this situation and develop an appropriate emerency policy, the user should be informed by the RMA about the reliability of the delivery of the taret level of resource or, in other words, the quantitative level of risk that the actual delivery will fall short of the promised one. This measure of risk for user l will be denoted by R ^x l ; x b l. In what follows we are oin to discuss different approaches which the RMA can adopt for definin this taret level of delivery and the risk of not meetin this taret. In order to simplify our notations, this discussion will consider the case of a sinle user who can actually be an areation of many users; the taret resource delivery in this case is denoted by x b and the user s demand by D. These are the vectors with the dimension that equals the total number of periods in the time horizon Expected delivery from the cost minimization The simplest way to define the taret delivery is to solve the plannin problem (2) (5) and take the expected delivery as the taret delivery: x b ¼ E ^x ¼ p ^x Then similarly to the classic risk manaement theory of finance (Markowitz, 1991) the risk can be measured as the variance r 2 ¼ r 2 ð^x; x b Þ of deliveries: Rð^x; x b Þ¼r 2 ð^x; x b Þ¼ p k^x x b k 2 Knowin the taret delivery x b and the variance between the taret and the actual delivery Rð^x l ; x b Þ, the user can now desin the policy for confrontin the possible resource shortaes.

5 218 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) Balance between costs and risk. Definin the taret resource delivery as the averae resource delivery under cost minimization has the followin shortcomin. It assumes that the objective of RMA is fully described by the cost minimization. The reality is more complex because the purpose of plannin of the distribution of a scarce resource is to achieve a balance between sustainable maintenance of the whole system and satisfaction of the demand of end users that is in some sense reasonable. Pure cost minimization may result in unacceptably hih risk of not achievin the taret delivery to the end user. For this reason the balance between the total costs and risks should be explicitly included in the optimization model of the type (2) (5). This can be done similarly to the way in which the balance between risk and performance is modeled in financial risk manaement (Markowitz, 1991). Namely, the risk taret q is fixed and the additional risk constraint is included in the resource plannin problem (2) (5) which takes the form: min p c ðx Þ x ;x b p dð^x ; x b Þ 6 q x b ¼ p ^x with additional constraints (3) (5). Here c (x ) is the cost associated with decision x under scenario and dð^x ; x b Þ is the measure of distance between the taret delivery x b and actual delivery ^x. In the case of Euclidean distance dð^x ; x b Þ¼k^x x b k 2 The constraint (7) transforms into the bound on the variance of delivered quantity. The value of q defines the relative importance of cost and risk. For lare q constraint (7) becomes nonbindin and (6) (8) with (3) (5) reduces to the cost minimization as in (2) (5). The smaller q the hiher importance of risk in cost/risk balance and for the smallest q for which the problem remains feasible (6) (8) is equivalent to minimization of risk. Equivalently, the cost/risk balancin problem (6) (8) can be formulated with the objective function containin both the risk and the cost terms as follows:! min x ;x b ð1 kþ p c ðx Þþk p dð^x ; x b Þ subject to constraints (8) and (3) (5). The parameter k can vary between 0 and 1; k = 0 corresponds to the pure cost minimization, while for k = 1 the problem becomes one of minimization of risk. Intermediate values of k provide different tradeoffs between costs and risk. Alternatively, constraint (8) can be dropped. Then in the case of Euclidean distance we call it the barycentric problem: min ð1 kþ p c ðx Þþk! p k^x x b k 2 x ;x b ð10þ A x ¼ b ; 8 2 G ð11þ l 6 x 6 u ; 8 2 G ð12þ x 2 S ð13þ We remind that in (10) the vector ^x of deliveries under scenario is a subvector of the full decision vector x. Similar to the problem (2) (5) the vector D of user s demands under scenario enters this problem as a subvector of the vector b of riht hand sides in (11). How can the appropriate value of the weiht k which balances the risk and costs be selected? ð6þ ð7þ ð8þ ð9þ Since the RMA and users are jointly responsible for the functionin of the whole system, this should be a matter of compromise between these actors. Here also we can draw from risk manaement approaches developed in finance. Let us look at resource distribution decisions ^x as a portfolio of resource distribution. Then followin the approach of portfolio theory (Markowitz, 1991) we shall construct the efficient frontier in the space of risk/ cost by solvin the problem (9) for different values of k 2 [0,1]. This frontier will have the shape shown in Fi. 2. Now the acceptable level of risk R will be neotiated between RMA and resource user. For the user it will include unplanned deviation around the taret delivery. The weiht k will correspond to this value of risk. The planned shortae is not included in this risk formulation, but it is included in the cost part instead. Similar cost/ risk balancin methodoloy was also utilized outside finance; see Gaivoronski and Zoric (2008) for application to collaborative delivery of advanced mobile data services. Formally speakin, the symmetric risk measure utilized in (10) considers as risk both shortae of resource delivery and excess of resource delivery. The reasons for this are twofold. From the point of view of RMA both shortae and excess carry undesirable effects and thus represent risks. Besides, the properties of the problem with a symmetric measure make it easier to solve in many cases. The part of the risk that is relevant for the users, namely the risk of shortae, can be recovered from the total risk by utilizin a multiplier that depends on the distribution of random parameters. Alternatively, some asymmetric risk measure can be used, that takes into account only the part of risk that is relevant for the end user, like expected shortfall with iven level of reliability (see the problem (17) (21) and discussion below) Reoptimization Thus, as the result of this cost/risk balancin process the user will obtain the value x b of planned delivery of the scarce resource which corresponds to the cost/risk weiht k neotiated as described above and the level of risk R that the actual delivery will differ from the planned value. The value of risk can be utilized by the user to estimate the confidence intervals within which the actual delivery will be contained under assumption of approximate normality. This information can be utilized by the user to construct emerency policies for the case when the actual delivery falls short of the planned one. The RMA can also ive the user its estimate of the worst shortae which could happen with respect to the planned delivery. This is done by reoptimization of the plannin problem (10) (13) for the worst possible scenario 2 G in the case Fi. 2. Tradeoff between costs and risk.

6 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) when such a scenario is clearly identifiable. User demand in such a reoptimization problem will be taken to be equal to the planned delivery x b, this demand will form part of the riht hand sides of (11). The risk term will be absent from the objective function in (9). Therefore the reoptimization problem will take the form: min c ðx Þ ð14þ A x ¼ ^b ð15þ l 6 x 6 u ð16þ where ^b is constructed with x b in place of the oriinal user s demand D. The solution of this problem, x, will be communicated to the user as the worst that can happen with reard to the delivery of scarce resource. In the case when the worst case scenario is not clearly identifiable then the problem (10) (13) can be reoptimized for some subset G of the oriinal scenario set G which is associated with the worst scenarios, aain takin the planned delivery x b as the user demand. Then the worst delivery will be the smallest delivery with respect to all scenarios in G. In the limit, the reoptimization may be performed on the whole set G of oriinal scenarios with the planned delivery x b as the user s demand and without the risk term in (10) Guaranteed delivery with constraints on expected shortfall Up till now we have defined the planned delivery x b of scarce resource as the point which in some sense will be located in the middle of deliveries x which will occur under different scenarios 2 G, hence the name barycentric. Let us now adopt a different philosophy and suppose that the RMA wants to set the deliveries to some level that it will uarantee with some specified reliability. In this case the RMA will ive the user the followin information about the planned deliveries: The minimal planned delivery x b which the user will obtain in (1 a) % of scenarios (or with probability 1 a). Here a is the level of reliability; it will be a small positive number defined durin the neotiation process with the user. The averae shortfall to the minimal delivery which will occur in the remainin small percentae a of bad scenarios. This can be supplemented with reoptimization in the case of the worst scenario(s), takin the uaranteed level x b as the new user s demand, as was discussed above. Here we develop the cost/risk optimization model which will allow settin deliveries accordin to these principles. For the moment, let us assume that the level of reliability a is fixed and denote by Z a (x b ) the averae shortfall from the uaranteed level x b in the a% of scenarios. We are oin to set the level x b by minimizin the averae shortfall from this level under structural and cost constraints. This leads to the followin optimization problem: min Z a ðx b Þ ð17þ x ;x b p c ðx Þ 6 C ð18þ A x ¼ b ; 8 2 G ð19þ l < x < u ; 8 2 G ð20þ x 2 S ð21þ This problem implements the same cost/risk balance paradim shown on the Fi. 2 as the problem 10, 11, 13. The objective function Z a (x b ) measures the risk which in this case is the shortfall from the uaranteed delivery x b. The upper bound C on the averae costs in (18) plays the same role as the weihtin parameter k in (19). For lare C the constraint (18) will become nonbindin and the problem (17) (21) reduces to the minimization of risk. The smallest C for which the admissible set (18) (21) remains feasible ives the solution of the problem of the cost minimization without the risk considerations. Intermediate values of C yield different compromises between minimization of costs and minimization of risk. By solvin the problem (25) (29) for different values of C we shall et the cost/ risk efficient frontier shown in Fi. 2. The delivery policy will be chosen on this efficient frontier throuh neotiations between the RMA (which bears costs) and the user (which bears the risks). In the rest of this subsection we show that the problem (17) (21) is computationally feasible. This is not obvious because the objective (17) is expressed throuh function Z a (x b ) with as yet non clarified properties. However, we shall show now that the problem (17) (21) is equivalent to a linear prorammin problem. The first step is to define the shortfall Z a ðxb Þ for a fixed scenario 2 G. This can be done in different ways, for example: Z a ðxb Þ¼max t ¼ T t¼1 x bt ^x t ; Z a ðxb Þ¼ T max 0; x bt ^x t t¼1 x bt ^x t ; Z a ðxb Þ ð22þ where x bt is the t-th component of the vector x b and ^x t is the t-th component of the vector ^x Now (17) (21) can be transformed as follows: min f þ 1 p x ;x b ;f;y þ 1 a y þ ð23þ y þ P Z a ðxb Þ f; 8 2 G ð24þ y þ P 0; 8 2 G ð25þ with additional constraints (18) (21). This transformation is similar to the representation of the problem of CVAR minimization as LP, see Uryasev and Rockafellar (1999); Zenios (2008). Now it is necessary to select a specific expression for the shortfall. Selectin the first definition from (22) we obtain the followin LP: min f þ 1 p x ;x b ;f;y þ 1 a y þ ð26þ y þ xbt þ ^x t þ f P 0; 8 2 G; t ¼ 1 : T ð27þ y þ P 0; 8 2 G ð28þ p c ðx Þ 6 C ð29þ A x ¼ b ; 8 2 G ð30þ l < x < u ; 8 2 G ð31þ x 2 S ð32þ It is possible to combine this uaranteed approach to resource delivery with the barycentric approach (10) (13). This is done by addin the followin constraint f þ 1 1 a p y þ 6 r and constraints (27), (28) to the problem (10) (13). Here r is the larest admissible averae shortfall from uaranteed delivery in a fraction of scenarios when shortfall occurs. What is the most appropriate risk measure in the context of manaement of scarce resources? This is an interestin question that we are oin to explore in our subsequent research. Interestin perspective on this issue is provided in Artzner et al. (1999), amon others.

7 220 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) Examples and numerical experiments: a water resource system The approach described in the previous sections is implemented in the prototype of decision support system for the manaement of scarce water resources in the south of Sardinia, Italy. It consists of the followin components: Graphical interface for modelin a water resource system. It allows a decision maker to describe in intuitive and raphical form a water distribution network consistin of several reservoirs, several types of demand, subject to different hydroloical phenomena like evaporation and infiltration. It contains the means to represent scenarios formulated by experts. Input enerator that transforms the raphical model of the water resource system into the input file for the optimization module. Optimization module with solver. Here we utilize the commercial software available on the market, in particular CPLE. A part of experiments was conducted with Matlab. We have conducted a series of experiments with water systems of different confiurations. Here we describe an example of a simple but important reservoir-demand water system. In the scarcity conditions, the reservoir can deliver the resource (if available) in the current period or store it to deliver in a successive time-period. We assume that the dimensions of the reservoir and the demand centres are known. We want to determine the resource manaement policy in terms of stored water in the reservoir and amount of resource delivered over a time horizon under consideration. The purpose of this policy is to satisfy as much as possible the known resource demand and minimize the costs associated with unsatisfied demands. We consider here a water system in south Sardinia, named Flumendosa-Campidano. Here we report some results selected amon a wide set of instances. We adopt a time-horizon of 4 years, comprisin 48 monthly time-periods, from October 70 to September 74. To reach the simple reservoir-demand confiuration, we consider that the main reservoirs are rouped in one with a reulation capacity obtained by addin the capacities of all reservoirs included in the system (584, meter 3 ). The sinle demand center includes all the demand centers supplied by the reservoir. The total demand equals 235, meter 3 /year. For a detailed description of the Flumendosa-Campidano system we can refer to (RAS, 2005). This water system was also used as a test case in the SÉcheresse and DÉsertification dans le bassin MÉDiterranéen (SEDEMED, 2003) European Project. To validate the approach we consider both the case of fixed resource demands (the same in all time-periods) and variable resource demands (different demands in different periods). Fixed demand centres correspond, e.., to civil, industrial and hydro electrical demand centres, and variable demand centres correspond, e.., to aricultural demand centres. In order to provide a more transparent comparison of different solutions we report here the case of simplified structure of the scenario tree containin two scenarios 1 and 2 which differ in inflow data, maintainin the data related to the water demands equal. This means that the riht hand sides b 1 and b 2 in Eq. (11), differ only in part. The branchin of scenarios occurs in the 12th period. We assume that scenario 1 represents the pessimistic evolution of hydroloical events, when available resources are scarce, while scenario 2 represents the optimistic evolution when sufficient resources are available. In particular, the hydroloical inflows data in the scenario 2 are obtained from the historical data while those of scenario 1 are obtained assumin that a reduction of 50% in hydroloical inflows will occur after the branchin time. In this way, the scenario 1 simulates a severe but possible shortfall in water inflows. We compare the behavior of stored volumes in the reservoir and resources transferred to the demand center obtained by the deterministic optimization (1) for each scenario (deterministic policy), scenario optimization usin stochastic prorammin approach (2) (5) (stochastic policy), and a the reoptimization phase of cost/ risk manaement policy (barycentric policy) puttin special emphasis on the results obtained in the conditions of scarce resources. In order to et the barycentic policy we compute first the barycentric taret value x b by solvin the problem (10) (13) for k = 0.5. Then we solve the problem (14) (16) for the critical scenario = 1 settin user demands at this taret value x b in place of the oriinal user s demand. The solution of this problem yields the new policy to be adopted when scarce critical scenario 1 occurs, which we call barycentric policy. Fis. 3 and 4 represent the behavior of stored volumes in the reservoir and resources delivered to the demand centre in the case of fixed demands, i.e. when water demands in each scenario are the same in all periods of the time horizon. This case represents reasonably well industrial consumption. We denote it as the oriinal demand D. Fi. 3 shows the stored volumes (measured in 10 6 cubic meters, Mm 3 ) obtained by stochastic and deterministic policies for both scenarios, and those achieved by the barycentric policy in the case of the critical scenario 1, i.e. in the case of the critical shortae of resources. When scenario 2 occurs, deterministic and stochastic policies exhibit the same distribution of stored water because in this scenario the amount of resources is enouh to fully satisfy the demand. On the fiure they are both represented by red line 1 (the two policies are overlapped). When scarce scenario 1 occurs, the deterministic policy leads to earlier emptyin of the reservoir (reen line) while the stochastic policy is able to store water until the end of the time horizon (blue line). Besides, Fi. 3 shows that the barycentric policy (pink line) enhances the resource distribution further. As a matter of fact this last policy allows to reach the end of the time horizon while maintainin a hiher level of the stored water with respect to other policies when the critical scenario occurs. As a consequence, the decisions on stored volumes followin barycentric policy better preserve the resource available in the reservoir in each time-period even with respect to the previous stochastic policy under critical conditions. Fi. 4 presents the comparison amon the different policies with respect to the resources delivered to user. As before, when scenario 2 with abundant resource supply occurs, deterministic and stochastic policies produce the same behavior. On this fiure both policies are represented by red line (the two policies are overlapped). In the case of this scenario the resource demand is fulfilled until the end of the time horizon. When scenario 1 with scarce resource supply occurs, results of deterministic policy are represented by the reen line that coincides with the red line until the 42nd period and then drops to zero. This means that an abrupt cut of resource will occur, causin an unexpectedly heavy disruption in the user activities. The stochastic policy in the case of poor scenario 1 allows the full satisfaction of the resource demand D until the 12th time period while an increasin resources shortae can be observed from the next period to the end of the time horizon (blue line in the fiure). Also this policy will cause a shortae for the user, but it is less dramatic than the shortae resultin from the deterministic policy because it avoids an unexpected full cut of resource. 1 For interpretation of color in Fi. 3, the reader referred to the web version of this article.

8 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) Fi. 3. Stored water in the reservoir-fixed demand. Fi. 4. Transferred resource to the demand center-fixed demand. In the reoptimization phase, the demand level is set at the taret value x b obtained by adoptin the poor scenario 1. The fiure illustrates how the barycentric policy (pink line) enhances the performance of the system represented by the response to the demand. In the case of scarce resources, this policy allows identifyin a lower demand level x b ¼ D b with respect to the oriinal demand D that is fulfilled until the 24th period. In the followin periods the delivered resource radually decreases while maintainin a hiher level of deliveries with respect to other policies. This can be considered as the preferable policy because it yields the smoother resources distribution that facilitates the arranement of the preventive measures by the user in the face of the shortae. Fis. 5 and 6 show the behavior of stored volumes in the reservoir and resources delivered to the demand center in the case of variable demands. We have adopted the typical pattern of demand enerated by aricultural production with maximums occurrin durin the summer months and the minimums durin the winter months. These fiures exhibit the properties of the polices that are similar to the case of the fixed demand. They show better performance of the stochastic approach with respect to the deterministic one also in this case. Moreover the reoptimization phase allows reachin a better resources distribution also in the case of variable demand. Summarizin, the manaement policy suested by the stochastic scenario optimization model results more risk-adapted and more implementable with respect to that obtained by the deterministic optimization in the case of scarce resources, confirmin the trend observed in Pallottino et al. (2004). Besides, the reoptimization phase leads to an effective manaement policy that is able to establish a reduced taret value for deliverin resources to the demand centre under water scarcity. With respect to both cases of constant and variable demand, we observe that the barycentric taret value takes into account the entire rane of possible scenarios of resource availability. In the case of abundance it does not result in excessively restrictive use of resources, while in the case of scarcity it does not lead to premature

9 222 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) Fi. 5. Stored water in reservoir-variable demand. Fi. 6. Transferred resource to the demand center-variable demand. exhaustion of resources supply. In other words, the taret value is sufficiently barycentric in respect to the different possible scenarios that could take place. Reducin the resource demand level to this taret value can be considered as a preemptive action to mitiate the consequences of water scarcity durin drouhts. It would also permit the resource users (the community) to be notified in a timely fashion. This risk-adapted network can be adopted in order to avoid, at least in part, damaes derived from an unexpected drastic cut in resource deliveries. The community suffers less from resource rationin if it has been forewarned of a possible shortae. Finally, Fis. 7 and 8 show the shape of the cost/risk efficient frontier obtained from solutions of the barycentric cost/risk balancin problem (10) (13) for different values of parameter k. Fi. 7 shows the case of Flumendosa-Campidano water system with constant demand and two scenarios discussed above. Example reported Fi. 8 refers to another example with scenario tree that spans five years. Splittin of scenarios occurs at the end of each year and each node is a predecessor of three nodes. Thus, this Fi. 7. Cost/risk frontier, barycentric risk, two scenarios. scenario tree has 81 leaves, each leaf correspondin to one scenario. The initial part of this scenario tree correspondin to the first

10 A. Gaivoronski et al. / European Journal of Operational Research 216 (2012) three splittin periods is shown Fi. 1. The barycentric problem (10) (13) has in this case 5820 variables and 2904 equality constraints, not countin nonanticipativity constraints. Computation of cost/risk frontier from Fi. 8 was performed by solvin 201 instances of this problem for different values of k between 0 and 1. It required 533 seconds on HP laptop with dual core processor runnin at 2.33 GHz and havin 2 Gb of RAM, 36 bit architecture. Both fiures show the concave dependence of costs on risk presented durin the eneral discussion Fi. 2. The costs as expressed by linear term in (10) and risk is expressed by the square root of quadratic term in (10). Now the estimate of the size of unplanned lack of delivery can be made from the cost/risk profile shown on this fiure and the acceptable cost/risk balance can be achieved by neotiations between the end users and RMA. Two observations can be made lookin at Fis. 7 and 8. It is possible to reduce substantially the risk of unplanned lack of delivery to end user(s) by increasin the costs for RMA. The (almost) risk free policy is costly and at the low risk levels the costs can be reduced considerably by moderate increase in risk. For example, in the case shown on Fi. 8 the reduction of costs by 1/3 of the total maximal risk manaement budet leads to acceptance of only about 1/6 of the maximal risk value. Even if the specific application is referred to a water resources system, the presented approach can be adopted in a wide rane of manaement problems when scarce resources occur. The study of different alternative risk/performance trade-off models is in proress providin different interated plannin and risk manaement models. The natural next step is to utilize developed here methodoloy for desin of such systems. This will be the subject of our future research. 5. Conclusions Fi. 8. Cost/risk frontier, barycentric risk, 81 scenarios. In this paper we have developed a quantitative approach for cost/risk balanced plannin of manaement of scarce resources under uncertainty and conflictin demands of different users and activities. This approach is based on the eneral methodoloy of stochastic prorammin/scenario optimization and utilizes the risk manaement tools from modern financial theory and investment science. We have been concerned here with the manaement of scarce resources in a iven system for their storae and delivery. Adoptin a risk manaement perspective, we defined a taret barycentric resource delivery by solvin an appropriate multistae stochastic prorammin problem. This delivery level is communicated to the resource users such that they can decide their local resource manaement policies adapted to barycentric delivery in a timely fashion. For the worst case scenarios additional reoptimization is performed takin the barycentric level for the users demand. The reoptimization step allows to obtain the best resource manaement policy when even barycentric delivery level can not be met. As a consequence, preventive measures can be adopted in order to avoid, at least in part, damaes derived from an unexpected drastic cut in resources. Numerical experiments confirm that this approach possess certain desirable properties from the application viewpoint, like smoother delivery patterns of scarce resources compared to the deterministic scenario optimization. Althouh the specific application is described with reference to a water resources system, the presented approach can be adopted in a wide rane of manaement problems when scarce resources occur. The study of different alternative risk/performance trade-off models is in proress providin different interated plannin and risk manaement models. The natural next step is to utilize developed here methodoloy for desin of such systems. This will be the subject of our future research. Another promisin direction can be the utilization of stochastic dynamic prorammin (SDP) methodoloy in the context of definin the performance/risk tradeoff pursued in this paper, extendin SDP approach to water reservoir manaement considered in Cervellera et al. (2006). Acknowledements The authors are rateful to the anonymous reviewers for their comments that helped to improve the exposition of the paper. References Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Coherent measures of risk. Mathematical Finance 9, Azaiez, M.N., Haria, M., A sinle-period model for conjunctive use of round and surface water under severe overdrafts and water deficit. European Journal of Operational Research 133, Bire, J.R., Louveaux, F., Introduction to Stochastic Prorammin. Spriner, New York. Bravo, M., Gonzalez, I., Applyin stochastic oal prorammin: A case study on water use plannin. 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