Management of Pricing Policies and Financial Risk as a Key Element for Short Term Scheduling Optimization

Transcription

1 Ind. Eng. Chem. Res. 2005, 44, Management of Pricing Policies and Financial Risk as a Key Element for Short Term Scheduling Otimization Gonzalo Guillén, Miguel Bagajewicz, Sebastián Eloy Sequeira, Antonio Esuña, and Luis Puigjaner*, Deartment of Chemical Engineering - ETSEIB, Universitat Politècnica de Catalunya, Avda. Diagonal, 647, G2, E-08028, Barcelona, Sain, and School of Chemical Engineering and Materials Science, University of Oklahoma, 100 East Boyd Street, T-335, Norman, Oklahoma In this article the scheduling of batch lants is integrated with ricing decisions. The roosed integrated model simultaneously rovides the otimal rices and schedule as oosed to earlier models where rices are usually considered as inut data. The main advantages of such formulation are highlighted through a case study where comarison with the traditional aroach is carried out. A two-stage stochastic mathematical model is also develoed in order to address the uncertainty associated to the demand curve. Finally, financial risk management is discussed. 1. Introduction With the recent trend of building small and flexible lants that follow the market dynamics closer, there has been renewed interest in batch rocesses. 1 Thus, a high effort to develo different strategies and tools for modeling, simulation, and otimization of these rocesses is underway. 2 Nevertheless, in all these develoments, significant asects related to the influence of the currently dynamic environment into the batch lant activity have not been roerly studied. One such asect is related to the way ricing decisions are made in these batch-chemical comanies. Industry managers and academia are realizing the strategic imortance of the rice variable. Pricing strategies consist of selecting the most aroriate rice for a articular economic environment or market situation. Traditionally, ricing decisions were the resonsibility of the marketing manager who sets a rice within the context of his overall market strategy, 3-5 but an effort to integrate ricing strategies with different decision making rocess is now underway. For instance, Chen et al. 3 analyze a finite horizon, single roduct, eriodic review model in which ricing and inventory decisions are made simultaneously. They also reort some articles where rice, inventory control, and quality of service (retail and service industries) are integrated. Moreover, issues so far related to the integration of decisions at different levels (scheduling, rice determination, etc.) and the associated uncertainty have not been considered. Therefore, while many models have been roosed for scheduling 2,6-8 few are devoted to include uncertainty Sources of uncertainty in scheduling can be divided into short-term (rocessing time variations, equiment breakdowns, etc.) and long-term (market trends, technology changes, etc.). For the shortterm, reactive scheduling has been used, while some form of stochastic rogramming has been considered for the long-term. 15 Furthermore, although stochastic models otimize the total exected erformance measure, they usually do not * To whom corresondence should be addressed. Tel.: Fax: luis.uigjaner@ uc.es. Universitat Politècnica de Catalunya. University of Oklahoma. rovide any control on its variability over the different scenarios; i.e., they assume that the decision maker is risk neutral. However, different attitudes toward risk may be encountered. In general, most decision makers are risk averse imlying a major reference for lower variability for a given level of return. In relation to this, Bonfill et al. 16 resented some techniques to manage financial risk in scheduling roblems similarly to the way it was done by Barbaro et al. 17 for lanning roblems. Some of these techniques were also used by Guillén et al. 18 for maniulating the financial risk associated to a given suly chain configuration under demand uncertainty. In this work a new strategy for integrating ricing decisions with the scheduling of batch lants and managing the financial risk associated with the consideration of the uncertainty associated with the demand curve is introduced. The starting oint is the modeling and forecasting of the relationshi between roduct rices and demand aiming at the incororation of ricing as a decision variable instead of treating it as inut data. Once this relation is obtained, it is integrated into a scheduling mathematical model in order to simultaneously determine the rices and the associated otimal schedule which maximizes the resulting rofit. The caabilities of such model are highlighted through a case study where comarison with the traditional aroach alied to fix rices is carried out. A two-stage stochastic formulation is then develoed to deal with the uncertainty associated with the demand curve, and the main advantages of the roosed formulation are highlighted through comarison with the deterministic aroach. A methodology to managing financial risk which relies on the samle average aroximation (SAA) method as a way of generating solutions which erform in dissimilar ways under the uncertain environment is next described and alied to our roblem. Some risk erformance indicators are finally used for assessing the obtained solutions and guiding the decision maker s choice. 2. Pricing Background Prices are marketing variables often characterized by quick market resonses; that is, any decision on ricing /ie049423q CCC: $ American Chemical Society Published on Web 01/08/2005

2 558 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 has a fast resonse of the market demand and must be therefore considered at the tactical (short term) level. The decisions related to the determination of rices begin with the available information on the fixed and variable costs, which are easily obtained from accounting and roduction registers. The relationshi of demand and rices is also required as inut data, and it is usually obtained by using historical data and, what is less usual, by direct exerimentation over the consumer s resonse to different rice levels in several markets. The objectives ursued by ricing need to be consistent with the enterrise goals. The most widely used objective function when determining rices is the maximization of rofit. We first start by discussing single roduct ricing followed by mutliroduct ricing, highlighting some of the shortcomings of these models Single Product Pricing Model. Consider the benefit maximization for a single roduct. Starting from the benefit equation: B ) I - C (1) where B reresents the benefits, I, the incomes, and C, the costs. A tyical model for I is I ) Sales Price (2) Sales e Q (3) Sales e Demand (4) In these equations Price is the roduct rice and Sales are the roduct sales which are constrained to be smaller than the volume of roduction Q and the demand Demand (eqs 3 and 4). In fact, when no caacity constraints are considered, the sales take the same value as the demand and the roduction volumes. There are several models that reflect the relationshi between demand and rices. 4 The most simle one involving one roduct and assuming that the roducts of the cometition have fixed demands and rocess is given by the exression: Demand ) k Price -e (5) where, k and e (the demand elasticity) are determined according to the historical information over a commonly bounded range of rices. The elasticity e can be a function of rice, but in ractice, given the narrow ranges of rices one is able to maniulate and, for the sake of simlicity, one can assume a linear relationshi between rice and demand: Demand ) Demand 0 - m Price (6) Price lo e Price e Price u (7) In addition, relacing the demand by its value in the classical model for the roduction cost C without economy of scale considerations, we obtain: C ) C F + C v Demand ) C F + C v Demand 0 - C v m Price (8) where C F reresents fixed costs and C v reresents the unitary variable costs. Therefore, to determine the otimal rice (denoted by Price*), the following equation assuming an unlimited roduction caacity should be solved: db d ) d dprice { I - C ) Price (Demando 0 - m Price) - -(C F + C v + Demand 0 - C v m Price) } ) 0 (9) which leads to Demand 0 + C m v Price* ) (10) 2 if Price lo e Price* e Price u and m > 0 (11) Otherwise Price* ) Price u (12) Thus, as long as C v is constant (Demand 0 and m can be considered constants), the ricing and the associated scheduling roblem are trivial Multile Product Pricing Model. For the case of several roducts, one may determine searately the rices for each one using single-roduct techniques and obtain the desired quantities to roduce and sell by adding all the individual terms. However, the shared resources are finite (R i ), and therefore they have to be included exlicitly as constraints. Thus, the ricing roblem becomes SIMPLE PRICING MODEL max B T ) s.t. B ) (I - C ) ) (Sales Price - Q C v ) (13) Sales e Q (14) Sales e Demand (15) Demand ) Demand 0 - m Price (16) Price lo e Price e Price u (17) r i Q e R i i (18) where, in this case, Sales reresent the total sales of roduct. Although taxes may lay an imortant role, for the sake of simlicity, they are ignored. Constraint (18) is a summation over all roducts of the resources used to roduce them. In this equation r i is the amount of resource i needed for the roduction of one unit of roduct. Its summation over is limited by the availability of i (R i ). In general, R i refers to any kind of resources which are generally classified as renewable (caacity, human ower, etc.) and nonrenewable (raw materials, intermediate roducts, etc.). At this oint, additional considerations can be introduced in order to obtain a more realistic formulation of the roblem. Such formulation is shown below (model DETPRICE):

3 Ind. Eng. Chem. Res., Vol. 44, No. 3, MODEL DETPRICE max B T ) s.t. ) B ) t (I - C ) ) (Sales t Z r dprice r - Q t C v - r ESales t EPrice ) (19) Sales t e Demand t, t (20) ddemand tr ) ddemand t0 - m t dprice r, t, r (21) Demand t ) Z r ddemand tr, t (22) r Price ) Z r dprice r (23) r Z r ) 1 (24) r Sales t ) ESales t + ISales t, t (25) t )t-1 ISales t e Q t + (Q t - Sales t ) + Inv 0, t t )1 (26) t r i Q t e R i i (27) In first lace, discrete variables instead of continuous ones can be used to reresent rices. This modification tries to reflect industrial ractices in which rices are commonly selected from a set of allowable values given by marketing surveys rather than icked from continuous ranges. Such discrete values are commonly obtained by discretizing the demand curve of each roduct into r intervals as stated by eqs The continuous variable used for reresenting ricing decisions Price is therefore substituted by a binary variable Z r which takes the value of one if the discrete rice r(dprice r )is selected for roduct or zero otherwise (eq 24). Additionally this use of discrete variables avoids the nonlinearities which aeared in the revious formulation and were due to roducts of continuous variables reresenting the sales and rices of roducts. The ossibility of roduction outsourcing (25), a common ractice in industry, is introduced next into the formulation. Secifically, the satisfaction of the demand is obtained either by in house roduction, i.e., sales of roducts manufactured at the lant (ISales t ), or by aying a remium (EPrice ) to other external roducers to deliver the roducts (ESales t ) to the consumer. Therefore, the total sales (Sales t ) for each roduct and time interval comrises the internal and external sales terms. The mathematical formulation also includes more than one eriod of time. A subscrit t is therefore introduced in those decision variables related to roduction and sales of materials. The ossibility of selling in a certain time interval t stock materials which have been roduced in earlier eriods is therefore taken into account in eq 26, where Inv 0 reresents the initial inventory of roduct, Q t, the total amount of roduct manufactured during eriod t, and ISales t, the internal sales, i.e., the amount of roduct sold to customers during time interval t coming from internal roduction carried out by the lant. In general, it is desirable that the internal sales equals the roduction rate. However, chemical lants are often forced to store materials in order to deal with the variability and uncertainty associated with the demand. For this reason, both terms are different; Q t is the one that we consider aroriate to be included in the objective function because it reresents the total roduction rate of the lant that originates the variable roduction costs. This simle aroach, however, does not reflect correct manufacturing costs. Indeed, in discrete or batch manufacturing scenarios, the coefficients C v and r i deend on the way the roduction is erformed, esecially the schedule. Therefore, the first one (C v ) should include inventory costs, the cost of the utilities consumed by labor intensive tasks, and the costs of the wastes generated when changes of batches of different roducts take lace in the roduction lines. On the other hand, the resource utilization factor (r i ) must constrain the total roduction rate of each of the items, which is indeed a function of the sequence of batches. Thus, the solution of the roblem requires a scheduling model (DETSCHED) in order to check the feasibility of the roosed lan (amount of materials to be roduced and their corresonding rices) and comute realistic values of C v and r ij. Such model is described next. 3. Multile Product Pricing Model with Scheduling Considerations As it has been mentioned before, the simlified ricing model (DETPRICE) determines the amount of materials (Q t ) to be roduced and their corresonding rices (Z r ). We now resent a scheduling model which will allow the determination of the arameters r ij and C v. The roblem is formally stated as follows: Given (a) The amount of roduct to be roduced and sold in time interval t (Q t and ISales t ). (b) The cost of the utilities consumed when fabricating roduct (C u ) (c) The cost of the wastes originated when changing roducts in the manufacturing lines (PCW ). (d) The nominal batch size and inventory costs of roduct to be fabricated (bs and R ) (e) The rocessing time of stage j involved in the fabrication of roduct (to j ) (f) The time horizon (H) and number and length of time intervals (T t ). Determine The schedule that satisfies the requirements given by the lan (amount of materials to be manufactured and their corresonding rices) comuted by means of DETPRICE and that minimizes the total costs (inventory, utilities and changeover waste cost). The aforementioned model comrises three major sets of constraints which are described in detail next. Scheduling Constraints: These constraints, which are required in order to introduce scheduling considerations into the formulation, should enable the comutation of the initial and finishing times of all the tasks involved in the roduction of the batches and ensure the feasibility of the resulting schedule. In this work, a very simle scheduling model, which is suitable for a multile stage and multiroduct batch lant with one single unit er stage, is alied to

4 560 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 illustrate the way scheduling and ricing decisions interact in manufacturing environments. However, such a model can be extended to more comlex structures. The roosed formulation is derived based on a batch slot concet. With this formulation the time horizon is viewed as a sequence of L batches that can be assigned to only one articular roduct. Concerning the maximum number of batches to be roduced it should be ointed out that such value can be either estimated based on caacity limitations or given by the decision maker. Sequence decisions are linked to a binary variable X l which reresents the existence of a batch l of roduct and takes the value of 1 in case l belongs to and 0 otherwise. 1 if batch l is of roduct X l ) { 0 otherwise X l e 1 l (28) X l e X l+1 l < L (29) Constraint 28 states that a batch l can belong at most to only one roduct, while equation 29 enforces the condition for which the nonroduced batches are located at the beginning of the schedule. Although constraint 29 is not necessary, it hels comutations. Indeed, by fixing the osition of the nonroduced batches we get smaller branch-and-bound trees and shorter comutational times. The secific osition of these nonroduced batches (at the beginning of the schedule) is absolutely arbitrary and is chosen for simlicity. To reduce the comlexity of the formulation, only the case of a zero wait olicy (ZW) is here considered. Neither intermediate storage nor waiting times in the rocessing units are available. Under ZW olicy, there can be no delay between the time a roduct batch finishes rocessing on stage j (TF lj ) and the time it commences rocessing on stage j + 1(TI lj+1 ) as stated by constraint 30: TI lj+1 ) TF lj l, j < J (30) The finishing time of stage j (TF lj ) involved in the fabrication of batch l is comuted from the initial time of l in j (TI lj ) and the duration of stage j which is given by the recie of the roduct the batch belongs to (to j ) as exressed by constraint 31. TF lj ) TI lj + to j X l l, j (31) The ZW assumtion resulting in the above commented constraints (eqs 30 and 31) can be easily modified in order to consider other transfer olicies such as the unlimited intermediate storage and nonintermediate storage ones (UIS and NIS). For the sake of simlicity it has been also assumed that only one roduction line with one assigned equiment er stage is available. Constraint 32 reflects the aforementioned olicy by forcing stage j involved in the fabrication of batch l to start after the end of the same stage j erformed in any revious batch l. Such olicy could be easily modified in order to reflect other equiment-task allocations ossibilities. TI l j g TF lj j, l > l (32) Inventory Constraints: These constraints are introduced in order to comute the inventory costs. We follow here the work develoed by Tsiakis et al. 19 The comutation of the total amount of roduct manufactured in time interval t (Q t ) is carried out by adding all the batch sizes b s of the l batches of roduced during eriod t as exressed by eq 33. Q t ) l bs X l Y lt, t (33) In this equation, Y lt is a binary variable used to allocate batches to eriods of time and takes the value of one if the last stage of batch l is finished within time interval t and zero otherwise. If the limits of interval t [1, NT], where NT reresent the total number of eriods in which the time horizon is divided, are denoted as T t-1 and Th t, the above definition can be reresented by the following linear constraints: Y lt ) { 1ifTF lj [T t-1 and Th t ] 0 otherwise Y lt ) 1 l (34) t Y lt T t-1 e TF ljt e Y lt T t l, t (35) TF ljt ) TF lj l (36) t Constraint 34 ensures that each batch l is finished within only one eriod of time; i.e., only one of the variables Y lt (say, for t ) t*) takes a value of 1, with all others being zero. Constraint 35 allocates each of the batches to its corresonding time interval using the defined binary variable Y lt. Such an equation forces the auxiliary continuous variable TF ljt to 0 for all t * t*, while also bounding TF ljt* in the range [T t*-1, T t* ]. Finally, constraint 36 exresses the condition for which the summation of the auxiliary variable TF ljt over t must be equal to the time in which batch l finishes its last stage J, what imlies that TF lj ) TF ljt* and, therefore, TF lj [T t*-1, T t* ], as desired. Furthermore, the in-house sales (ISales t ) must be higher than the sales requirements (ISales t ) and lower than the demand (Demand t ) comuted by model DET- PRICE as exressed by constraints 37 and 38. In fact, these two constraints reresent the link between the lanning and the scheduling model, i.e., the scheduling model (DETSCHED) must satisfy the sales requirements determined by the lanning one (model DET- PRICE). ISales t g ISales t, t (37) ISales t e Demand t, t (38) In addition, the sales derived from in-house manufacturing (ISales t ) must be also smaller than the amount of roducts available in each time interval t as stated by constraint 39: ISales t e Q t + Inv t, t (39) Finally, the average inventory of roduct in eriod t (Inv t ) is comuted with the initial inventory of

5 Ind. Eng. Chem. Res., Vol. 44, No. 3, roduct ket at the beginning of eriod t (Inv t ) and the roduction rate (Q t ) (constraint 40). In this equation it has been assumed that sales are executed at the end of each time interval and that the real rofile of materials can be aroximated by a linear equation. The error incurred by the use of such aroximation is believed to be negligible for a high number of batches and time intervals. Inv t ) Inv t + Q t, t 2 (40) Inv t ) Inv t-1 + Q t-1 - ISales t-1, t (41) Integer Cuts: A way to reduce the comutational effort associated to the resolution of the mathematical formulation consists of generating integer cuts that exlicitly forbid some infeasible solutions. In our case, some integer cuts can be inferred from the sequence constraints. Equation 42 states that nonroduced batches must be located in the first eriod, while constraint 43 exresses the condition for which the classification of batches into time intervals must be erformed following the sequence of batches in the roduction lines. Y lt g 1 - X l l, t ) 1 (42) Y l t + t>t Y lt e 1 l > l (43) Objective Function: The resented model accounts for the minimization of the total cost (C ) associated to the schedule that satisfies the requirements comuted by DETPRICE. It includes the cost of the utilities, the cost of the changeover wastes, and the inventory cost as stated by eq 44. [ C ) t t + Q t C u + R Inv t ] l X l X l+1 PCW + (44) Finally, the overall roblem can be exressed as follows: MODEL DETSCHED minimize C subject to eqs Hierarchical Scheduling-Pricing Procedure In this section, one way by which the ricing and scheduling models can be used to reach an imroved solution is discussed. Pricing olicies and scheduling decisions are nowadays taken in many comanies through a hierarchical decision-making rocess involving the financial and the roduction deartments of the comanies reresented by the chief financial officer (CFO) and the roduction manager, resectively. The CFO is resonsible for the financial area of the enterrise and has therefore to take care of financial-lanning decisions with the hel of the available decision-suort tools (lanning models such as DETPRICE). On the other hand, once the financial decisions are taken, the feasibility and cost of the roosed lans are evaluated by the roduction manager using scheduling models such as DETSCHED. This information is finally reorted to the financial deartment (CFO) and the overall rocess is sometimes Table 1. High Level Data (I) roduct Erice (m.u./kg) Dem t0 (kg) m t (kg/m.u.) P P P Table 2. High Level Data (II) roduct r i (u.r./kg) r i (u.r./kg) C v (m.u./kg) P P P Table 3. Low Level Data (I) roduct bs (kg) R 10 2 (m.u./kg h) C v (m.u./kg) P P P reeated until a final solution which satisfies the involved managers is reached, but in many others these iterations do not even take lace. An algorithm that tries to reflect this tye of traditional enterrise ractices is described below. Such an algorithm is alied with the aim to highlight the drawbacks of such methodologies, and it has not been develoed as a way of decomosing the roblem in order to overcome numerical difficulties. Moreover, the changeover costs term, which imlies the comutation of scheduling variables and aears in the objective function of the overall roblem, would make it difficult to aly decomosition strategies available in the literature 10,20 which rely on either combining mixedinteger rogramming (MILP) to model the assignment art and constraint rogramming (CP) for modeling the sequencing art, or else combining MILP models for both arts. We now resent the iterative model. TRADITIONAL PROCEDURE: Do Solve the simlified ricing model (DETPRICE) assuming initial values of costs (C v ) and use of resources (r i ). Use the results (ISales t ) as data for the roduction lanning detailed model (DETSCHED). If DETSCHED is feasible Then Udate C v and r i in the ricing model (DETPRICE) using the results from DETSCHED. Else Increase r i if the roduction of roduct is not zero. Kee it constant otherwise. End if Until a finalization criterion is met. The way of udating these coefficients during the iterative search and the starting oint of the iterative rocedure may have an imact in the final solution. To illustrate the caabilities of the roosed aroach, the algorithm shown above is alied to a case study of a multiroduct batch lant manufacturing three different roducts in three stages. It is assumed a time horizon of 12 days divided into two eriods of equal length. Moreover, sales of roducts are suosed to take lace each 6 days, and only one resource Ri is considered. The data of the roblem are listed in Tables 1-5. The set of discrete rices and their associated demands are generated discretizing the demand curve into 10 intervals of equal length. The overall roblem consists of finding the

6 562 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 Table 4. Low Level Data (II): Processing Times, to j (h) stage roduct j1 j2 j3 P P P Table 5. Low Level Data (III): Changeover Waste Costs, PCW roduct roduct P1 P2 P3 P P P roduct rices and lant schedule that maximizes the total enterrise rofit. Regarding the details of the sequential aroach, such a rocedure is carried out alying eqs 45 to 47 to udate C v and r i in DET- PRICE and considering constant values of r i. Feasible DETSCHED: C v ) C u + Σ l X l X l+1 PCW + Σ t R Inv t (45) Q t t r i ) r i - r i if Q t * 0 i, (46) t Infeasible DETSCHED: r i ) r i + r i if Q t * 0 i, (47) t Constraint 45 reflects the heuristic consisting of estimating the roduction cost of a roduct by assigning a art of the overall costs generated by the schedule to it. Table 6. Iterative Method initial solution final solution roduct rice (m.u.) Q (kg) rice (m.u.) Q (kg) P P P B T (m.u.) The iterative rocedure is therefore imlemented assuming initially a low lant caacity (R i equal to 100 caacity units) and high variable costs (elevated C v ). The caacity is then rogressively increased (lower values of r i ) until infeasibility in DETSCHED is detected. Figure 1 shows the evolution of C v and r i, while Figure 2 and 3 deict the decision variables (Price and Q ) and the rofits (the original one given by DETPRICE and the corrected one udated with the results rovided by the detailed model DETSCHED) during the iterative rocedure. The Gantt charts for the initial and final solutions are given in Figure 4, while Table 6 shows the values of the decision variables and the rofits obtained for the initial and final solutions. The first infeasibility is reached in iteration 108. At this oint the lant is not caable of manufacturing the amount of materials rovided by model DETPRICE. This makes the corrected benefit dro to zero as deicted by Figure 3. The coefficients are next udated, and a second infeasibility is then reached at iteration 110. The iterative arameters are then udated again, and the iterative scheme finally converges at iteration 115. Figure 2 indicates low oscillations of roduction volumes during the iterative rocess while rices remain more or less constant. Moreover, Figure 3 shows the difference between the original benefit given by DETPRICE and the corrected one comuted once DETSCHED is solved varies from 0.4% to 8% for the feasible lans. Results indicate that the earliest lans involve the fabrication of small amounts of roducts thus leading to lower rofits. On the other hand, higher amounts of Figure 1. Evolution of C v and r i.

7 Ind. Eng. Chem. Res., Vol. 44, No. 3, Figure 2. Evolution of rice and Q. Figure 3. Evolution of B T. materials and hence bigger benefits are associated to the last iterations. There are four drawbacks of this rocedure: (a) The number of iterations. For instance, in the roosed case study the first solution reached after 50 iterations exhibits a rofit which is nearly 21.4 lower than the final one. The initial solution also leads to 26.6% less benefit than the last one. (b) The way of udating C v and r i in DETPRICE. Different starting oints and udating rocesses may lead to dissimilar solutions. Therefore, good solutions may not be found, although they may exist. (c) The infeasibilities which arise when solving model DETSCHED. Such infeasibilities aear when DET- PRICE generates rather otimistic lans that assume higher caacities than the available ones.

8 564 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 (d) Even if these two issues are resolved, the otimality of the final solution cannot be guaranteed. To ameliorate all the roblems derived from this aroach an integrated model is suggested. Such a model is resented in the next section. 5. Pricing Policies Model In this section a mathematical model is roosed in order to simultaneously determine the amount of roducts to be manufactured, their corresonding rices, and the associated scheduling. The roosed model can be derived from the scheduling mathematical formulation and, in general, from any of the existing scheduling formulations, 6-8 by eliminating the constraints which link DETSCHED with DETPRICE (eqs 37 and 38) and adding eqs 48 to 53, which are in fact taken from DETPRICE: A stochastic rogramming aroach based on a recourse model with two stages is roosed to incororate the uncertainty associated with the relationshi between demand and rice. In a two-stage stochastic otimization aroach, the uncertain model arameters are considered random variables with an associated robability distribution and the decision variables are classified into two stages. The first-stage variables corresond to those decisions which need to be made here-and-now, rior to the realization of the uncertainty. The second-stage or recourse variables corresond to those decisions made after the uncertainty is unveiled and are usually referred to as wait-and-see decisions. After the first-stage decisions are taken and the random events realized, the second-stage decisions are made subject to the restrictions imosed by the second-stage roblem. Due to the stochastic nature of the erformance associated with the second-stage decisions, the objective function consists of the sum of the first-stage erformance measure and the exected second-stage erformance. Birge et al. 22 rovides an overview of this kind of stochastic techniques. In our roblem, roduct demands are reresented by a set of scenarios with a given robability of occurrence, which are obtained by alying a Monte Carlo samling. Decision variables related to the roduction schedule (X l ) and the ricing olicies (Z r ) are considered as firststage decisions, since it is assumed that they have to be taken at the scheduling stage before the uncertainty is unveiled. On the other hand, the sales (Sales ts, ISales ts, and ESales ts ) and thus the inventory rofiles (Inv ts and Inv ts ) are second-stage variables. Therefore, at the end of the scheduling horizon, a different rofit value is obtained for each articular realization of demand uncertainty (B Ts ). The roosed model accounts for the maximization of the exected value of this rofit distribution (E[B T ]). The overall stochastic forddemand tr ) ddemand t0 - m t dprice r, t, r (48) Demand t ) Z r ddemand tr, t (49) r Price ) Z r dprice r (50) r Z r ) 1 (51) r Sales t ) ESales t + ISales t, t (52) Sales t e Demand t, t (53) The new objective function is MaxB T ) [ t t t Sales t Price - Q t C u + X l X l+1 PCW + l (54) R Inv t + ESales t EPrice And finally the overall roblem can be exressed as follows: MODEL DETINT maximize B T subject to eqs 28-36, 39-43, and To illustrate the caabilities of this aroach, the roosed case study is solved with the integrated model and the results are comared with those obtained by means of the iterative rocedure. For the roblem discussed above, the resulting formulation has 1169 constraints, 474 continuous variables, and 241 binary variables. It was imlemented in GAMS 21 and solved using the MIP solver of CPLEX (7.0). It takes about 26 s to reach a solution with 0% integrality ga on an AMD Athlon 3000 comuter. Table 7 shows the obtained results, while Figure 5 deicts the Gantt chart associated with the resulting schedule. The rofit achieved by the integrated model is nearly 14.6% higher than the best solution rovided by the iterative scheme ( m.u. for the integrated vs 9544 m.u. for the hierarchicaliterative method). Moreover, this ercentage increases u to 37.2% when comaring the integrated solution ] Table 7. Integrated Model roduct rice (m.u.) Q (kg) P P P B T (m.u.) with the first solution given by the iterative method ( m.u. for the integrated vs 7010 m.u. for the iterative aroach). There are two main reasons for why the integrated model yields higher rofit than the sequential one. In first lace, the DETPRICE model used in the sequential aroach does not take into account that roduction is carried out in a discontinuous way; i.e., a continuous variable is used for reresenting the roduction rate instead of taking into account the batch sizes associated to the existing equiments. This issue makes the DETSCHED model comute schedules which either are infeasible or require higher amounts of roducts than the ones requested by DETPRICE, which results in higher storage costs. Second, the DETPRICE model does not consider the changeover costs, which are sequence deendent, and therefore cannot roerly assess the rofitability of the different roducts thus leading to subotimal solutions. 6. Integrated Model under Uncertainty

9 Ind. Eng. Chem. Res., Vol. 44, No. 3, Figure 4. Gantt charts of the iterative aroach. Figure 5. Gantt chart of the integrated solution. mulation can be derived from the deterministic one by modifying those constraints which are scenario-deendent; that is, deterministic constraints which include second-stage variables 39-41, 48, 49, 52, and 53 are

10 566 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 Table 8. Uncertainty roduct MeanDem 0 (kg) SDevDem 0 (%) K (m.u.) P P P relaced by the following equations which must be satisfied for each scenario s: ISales ts e Q t + Inv ts, t, s (55) Inv ts ) Inv ts + Q t, t, s (56) 2 Inv ts ) Inv t-1s + Q t-1 - ISales t-1s, t, s (57) ddemand trs ) Demand t0s - m ts dprice r, t, r, s (58) Demand ts ) Z r ddemand trs, t, s (59) r Sales ts ) ESales ts + ISales ts, t, s (60) Sales ts e Demand ts, t, s (61) The resulting formulation accounts for the maximization of the exected rofit as stated by eq 62: MaxE[B T ] ) s ) s [ t rob s - l rob s B Ts ) Sales ts Price - Q t C u - t R AvInv ts ] + t X l X l+1 PCW - ESales ts EPrice (62) The overall roblem can be therefore formulated as follows: MODEL STOCINT maximize E[B T ] subject to eqs 28-36, 42, 43, 50, 51, and To highlight the convenience of using the stochastic aroach, the roosed formulation is alied to the case study described before considering the uncertainty associated to the arameters of the demand curve. Uncertainty is reresented by 250 scenarios, each of them comrising a certain demand value Demand trs for the same discrete rice dprice r. These scenarios are comuted assuming that the arameter Demand t0 follows a normal robability distribution with mean and standard deviation given in Table 8 and that there exists a constant rate between such coefficient and m ts for each scenario s (eq 63). m ts ) Demand t0s, t, s (63) K This stochastic formulation involves constraints, continuous variables, and 241 binary variables, and it is also imlemented in GAMS 21 and solved using the MIP solver of CPLEX (7.0). It takes about s to reach a solution with 0% integrality ga on an AMD Athlon 3000 comuter. All the scenarios where considered simultaneously in this model. Table 9. Deterministic vs Stochastic deterministic stochastic roduct rice (m.u.) Q (kg) rice (m.u.) Q (kg) P P P deterministic stochastic E[B T] (m.u.) VaR (m.u.) OV (m.u.) RAR (adim.) 0.31 For comarison uroses, the solution which was originally obtained by alying the deterministic model for the mean demand scenario is evaluated against the same 250 scenarios by fixing the first-stage variables (rices and schedule) and comuting the second-stage ones with the stochastic formulation. Table 9 shows the decision variables and some risk erformance indexes, which are described in the next section, associated to both solutions. The stochastic Gantt chart is given in Figure 6, while the deterministic and stochastic risk curves are also shown in Figure 7. Such curves are none other than cumulative rofit robability curves, which can be used to assess risk. 1 As it can be observed, the stochastic solution roduces a less amount of items with higher demand uncertainty (P1) and higher rofitability and also decreases the roduction of materials characterized by demands with low standard deviation (P3) but which are on the other hand less rofitable. The stochastic aroach suggests also variations in rices in order to adjust the demand to the new roduction volumes. For instance, the rice of P3 is reduced in order to increase its demand thus allowing higher sales. The exected rofit associated to the stochastic solution is nearly 3% higher than the deterministic one ( m.u. for the deterministic solution and m.u. for the stochastic one). Moreover, both aroaches lead to rather different solutions from the risk management oint of view. In addition to roviding higher rofits, the stochastic solution yields lower robabilities of low rofits, making the roduction also less risky in economic terms. For instance, a 10% robability of scenarios with earnings below m.u. is achieved in the stochastic formulation, while this robability increases u to 35% in the deterministic aroach (see Figure 7). On the other hand, the deterministic solution yields higher robabilities of larger benefits. For instance, a 40% robability of earnings above m.u. is reorted by this solution, while the stochastic one rovides only a value of 20%. This makes the former more attractive for risk-taker decision makers. 7. Financial Risk As it has been reviously mentioned, although stochastic models otimize the total exected erformance measure, they usually do not rovide any control of its variability over the different scenarios; i.e., they assume that the decision maker is risk neutral. However, different attitudes toward risk may be encountered. In this section, financial risk is reviewed and a recent aroach to manage it is described and alied to our roblem.

11 Ind. Eng. Chem. Res., Vol. 44, No. 3, Figure 6. Gantt chart of the stochastic solution. Figure 7. Deterministic vs stochastic solution. The financial risk associated with a lan under uncertainty is defined as the robability of not meeting a certain target rofit (maximization) or cost (minimization) level referred to as Ω. 17 For the two-stage stochastic roblem, the financial risk associated with a design x and target rofit Ω is therefore exressed by the following robability: Risk(x,Ω) ) P(Profit(x) < Ω) (64) where Profit(x) is the Profit after the uncertainty has been unveiled and a scenario is realized. The definition of Risk(x,Ω) can be rewritten with the hel of binary variables as follows: Risk(x,Ω) ) rob s z s (x,ω) (65) s where z s is a new binary variable which equals 1 in case Profit s < Ω and 0 otherwise: z s ) { 1ifProfit s < Ω 0 otherwise In the case of a discrete scenario, financial risk is given by the cumulative frequency obtained from the Profit histogram. Barbaro et al. 17 roved that minimization of risk at some rofit levels renders a tradeoff with exected rofit. A risk-averse decision maker will feel comfortable with low risk at low values of Ω, while a risk taker will refer to lower the risk at high values of Ω. The tradeoff lies in the fact that minimizing risk at low values of Ω (e.g., a loss) is in conflict with the minimization of risk at high values of Ω (e.g., large rofits) and vice versa.

12 568 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 From a mathematical rogramming oint of view, minimizing Risk(x,Ω) for a continuous range of rofit targets Ω results in an infinite multiobjective otimization roblem. Even though this model would be able to reflect the decision maker s intention, having an infinite otimization roblem is comutationally rohibitive. The authors, however, suggested aroximating the ideal infinite otimization aroach by a finite multiobjective roblem that only minimizes risk at some finite number of rofit targets and maximizes the exected rofit. This aroach gives rise to the following finite multiobjective formulation: subject to maximisee[b T ] ) rob s q T s y s - ct x s MinRisk(x,Ω 1 ) ) rob s z s1 s MinRisk(x,Ω i ) ) rob s z si s q T s y s - ct x g Ω i - U s i, s (66) q T s y s - ct x e Ω i + U s (1 - z si ) i, s (67) z si {0, 1} i, s (68) where y s reresents the otimal second-stage solution associated with the design x that corresonds to scenario s. In the above formulation, constraints 66 and 67 force the new integer variable z si to take a value of zero if the rofit for scenario s is greater than or equal to the target level (Ω i ) and a value of one otherwise. To do this, an uer bound of the rofit of each scenario (U s )is used. The value of the binary variables is then used to comute and enalize financial risk in the objective function. Such a rocedure generates a set of Pareto otimal solutions behaving in dissimilar ways under the uncertain environment from which the decision maker should choose the best one according to his/her references. Nevertheless, the inclusion of new integer variables reresents a major comutational limitation of the resulting formulation. In this work, a novel aroach is introduced aiming at the reduction of the comutational exenses associated with the revious rocedure. Within this framework, the multiobjective otimization aroach is relaced by the samle average aroximation method (SAA) as a way of generating the set of candidate solutions that exhibit different risk erformances. Furthermore, the concet of dominance in terms of exected rofit and risk associated to a discrete set of targets, in the way introduced by Barbaro et al., 17 and some standard risk erformance indexes are also alied for evaluating the solutions obtained and roviding decision suort in finding the one that reresents the right comromise between exected rofit and risk. The roosed framework to manage risk has similar features to the one suggested by Aseeri et al. 23 The SAA method is an aroach for solving stochastic otimization roblems by using Monte Carlo simulation. 24 In the SAA technique, the exected second-stage rofit (recourse function) in the objective function is aroximated by an average estimate of NS indeendent random samles of the uncertain arameters, and the resulting roblem is called aroximation roblem. Here, each samle corresonds to a ossible scenario, and so NS is the total number of scenarios considered. Then, the resulting aroximation roblem is solved reeatedly for M different indeendent samles (each of size NS) as a deterministic otimization roblem. In this way, the average of the objective function of the aroximation roblems rovides an estimate of the stochastic roblem objective. Notice that this rocedure may generate u to M different candidate solutions. To determine which of these M (or ossibly less) candidates is otimal in the original roblem, the values of the firststage variables corresonding to each candidate solution are fixed and the roblem is solved again using a larger number of scenarios NS >>NS in order to distinguish the candidates better. After solving these new roblems, the otimal solution of the original roblem (xˆ*) is determined. Therefore, xˆ* is given by the solution of the aroximate roblems that yields the highest objective value for the aroximation roblem with NS samles. For our secific roblem this algorithm would be as follows: Select NS, NS, M For m ) 1toM For s ) 1toNS Use Monte Carlo samling to generate an indeendent observation of the uncertain arameters. Define rob s ) 1/NS. Next s Solve roblem STOCINT with NS scenarios. Let the xˆm be the otimal first-stage solution. Next m For m ) 1toM For s ) 1toNS Use Monte Carlo samling to generate an indeendent observation of the uncertain arameters. Define rob s ) 1/NS. Next s Solve roblem STOCINT with NS scenarios, fixing xˆm as the otimal first-stage solution. Next m Use xˆ* ) argmax{obj(xˆm) m ) 1, 2,..., M} as the estimate of the otimal solution to the original roblem where Obj(xˆm) is the estimate of the otimal objective value. End. In our aroach it is roosed that the roblem for each scenario be solved and these deterministic results be used to obtain an uer bound of the roblem, in other words, NS ) 1 and M ) NS. Then, according to the SAA, the roblem is solved fixing the first-stage variables (schedules and rices) obtained in the revious runs, and the second-stage variables (sales and inventory rofiles) are comuted using the stochastic model for the NS scenarios. After that, the solutions are screened as follows. First, all the dominated solutions, namely those which are comletely dominated by at least another one, are disregarded. A solution x 1, which comrises a set of scheduling and ricing decision variables, associated to the objective function vector u ) [u 1, u 2,..., u n ], dominates other solution x 2, with its corresonding objective function vector v ) [v 1, v 2,..., v n ], if and only if i {1, 2,.., n},v i e u i i {1, 2,.., n} v i < u i

13 Ind. Eng. Chem. Res., Vol. 44, No. 3, Figure 8. Best risk curves. In other words, x 1 yields better or equal objective function values than those reorted by x 2 for all the objectives, and it erforms strictly better than the latter in at least one of them. In our case, we consider as objectives the exected rofit and the risk associated to a set of n - 1 targets; therefore u ) [E[B T ], -Risk(x,Ω 1 ),... -Risk(x,Ω i ),... -Risk(x,Ω n - 1 )] This imlies that if solution x 1 dominates x 2, its risk curve lies entirely to the right side of the risk curve of x 2. This roerty can be therefore used for discarding the dominated solutions from the original set of solutions comuted by the SAA. A risk curve belonging to the set of nondiscarded solutions (nondominated solutions) must satisfy the condition of intersecting at least at one oint all the nondominated curves. 17 As a result of this calculations, a set of nondominated solutions are obtained. Second, various measures used and/or introduced by Aseeri et al. 23 are automatically calculated, and finally, the solutions which erform better in terms of these measures as well as the one which exhibits maximum exected rofit are identified. The risk measures considered are as follows: (a) The Value at Risk or VaR, defined as the difference between the exected value and the rofit for a certain confidence interval usually set at 5%. 25,26 Solutions with low VaR erform better for low targets and therefore are likely to be chosen by risk-averse decision makers. (b) The Uside Potential (UP) or Oortunity Value (OV) roosed by Aseeri et al., 23 defined in a similar way to VaR but at the other end of the risk curve with a quantile of (1-) as the difference between the benefit corresonding to a risk of (1-) and the exected value of the rofit. Solutions with high OV may be chosen by risk-taker decision makers, since they exhibit better erformance for high targets. (c) The Risk Area Ratio (RAR) roosed also by Aseeri et al., 23 calculated as the ratio of the OortunityArea (O_Area), enclosed by the two curves (the curve under analysis and the curve corresonding to maximum exected rofit) above their intersection, to the RiskArea (R_Area), enclosed by the two curves below their intersection as stated by the following equation: RAR ) O Area R Area (69) The areas can be calculated by integrating the difference of risk between the two curves over B Ts. The closer this ratio is to one, the better the alternative solution is. This is indeed an indication of how significant the reduction in oortunity is comared to the small reduction in risk. The roosed aroach is therefore alied to the case study resented before. Of the 250 runs, only 34 render nondominated solutions, one of them being the deterministic solution comuted for the mean scenario to these nondominated set of solutions. The solution with highest exected rofit obtained by means of the decomosition aroach is equal to that comuted by the stochastic model for the given 250 scenarios. Moreover, the solution obtained by means of the iterative rocedure is also evaluated against the set of 250 scenarios and turns out to be dominated by the ones generated by the SAA. The risk curves of the best solutions in terms of the aforementioned risk measures are lotted together with the maximum exected rofit one and the uer bound curve as deicted by Figure 8. The associated Gantt charts are given in Figure 9, while Tables

14 570 Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 Figure 9. Gantt charts of the risk curves. 10 and 11 show the value of the risk measures and the decision variables for each curve. Best solutions in terms of the redefined risk erformance criteria: As it can be observed in Table 10, the solutions behave in very different ways under the uncertain environment. Solutions with low VaR exhibit high OV and vice-versa, thus showing the tradeoff between olicies of risk-taker and risk-averse decision makers. Besides, the solutions lead to dissimilar values of RAR which can be used so as to roerly evaluate the different alternatives. For instance, the RAR index suggests not to select the maximum OV solution since it exhibits a very high reduction in oortunity comared to the small increase in robabilities of high rofits. This conclusion can be also derived from the low exected rofit of the solution. Figure 8 shows also how solutions with low VaR lead to low robabilities of small benefits, while those with