A new robust optimization approach for scheduling under uncertainty II. Uncertainty with known probability distribution

Transcription

1 Coputers and Cheical Engineering 31 (2007) A new robust optiization approach for scheduling under uncertainty II. Uncertainty with nown probability distribution Stacy L. Jana, Xiaoxia Lin, Christodoulos A. Floudas Departent of Cheical Engineering, Princeton University, Princeton, NJ , USA Received 1 April 2005; received in revised for 3 May 2006; accepted 22 May 2006 Available online 1 August 2006 Abstract In this wor, we consider the proble of scheduling under uncertainty where the uncertain proble paraeters can be described by a nown probability distribution function. A novel robust optiization ethodology, originally proposed by Lin, Jana, and Floudas [Lin, X., Jana, S. L., & Floudas, C. A. (2004). A new robust optiization approach for scheduling under uncertainty: I. Bounded uncertainty. Coputers and Cheical Engineering, 28, ], is extended in order to consider uncertainty described by a nown probability distribution. This robust optiization forulation is based on a in ax fraewor and when applied to ixed-integer linear prograing (MILP) probles, produces robust solutions that are iune against data uncertainty. Uncertainty is considered in the coefficients of the objective function, as well as the coefficients and right-hand-side paraeters of the inequality constraints in MILP probles. Robust optiization techniques are developed for uncertain data described by several nown distributions including a unifor distribution, a noral distribution, the difference of two noral distributions, a general discrete distribution, a binoial distribution, and a poisson distribution. The robust optiization forulation introduces a sall nuber of auxiliary variables and additional constraints into the original MILP proble, generating a deterinistic robust counterpart proble which provides the optial/feasible solution given the (relative) agnitude of the uncertain data, a feasibility tolerance, and a reliability level. The robust optiization approach is then applied to the proble of short-ter scheduling under uncertainty. Using the continuous-tie odel for short-ter scheduling developed by Floudas and co-worers [Ierapetritou, M. G. & Floudas, C. A. (1998a). Effective continuous-tie forulation for short-ter scheduling: 1. Multipurpose batch processes. Ind. Eng. Che. Res., 37, ; Lin, X. & Floudas, C. A. (2001). Design, synthesis and scheduling of ultipurpose batch plants via an effective continuous-tie forulation. Cop. Che. Engng., 25, ], three of the ost coon sources of uncertainty in scheduling probles are explored including processing ties of tass, aret deands for products, and prices of products and raw aterials. Coputational results on several exaples and an industrial case study are presented to deonstrate the effectiveness of the proposed approach Published by Elsevier Ltd. Keywords: Process scheduling; Uncertainty; Robust optiization; MILP; Probability distribution 1. Introduction A significant body of wor has appeared in the literature over the past decade in the research area of production scheduling. Most of the existing wor assues that all data are of nown, constant values. However, in reality, uncertainty is coon in any scheduling probles due to the lac of accurate process odels and variability of process and environental data. Thus, an eerging area of Corresponding author. Tel.: ; fax: E-ail address: floudas@titan.princeton.edu (C.A. Floudas) /$ see front atter 2006 Published by Elsevier Ltd. doi: /j.copcheeng

2 172 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) research ais at developing ethods to address the proble of scheduling under uncertainty, in order to create reliable schedules which reain feasible in the presence of paraeter uncertainty (see the recent reviews by Floudas (2005) and Floudas and Lin (2004, 2005)). Different ethodologies can be used for the proble of scheduling under uncertainty including stochastic, probabilistic, and fuzzy prograing ethods (Sahinidis, 2004). Stochastic forulations incorporate uncertainty by taing advantage of the fact that probability distributions governing the data are nown or can be estiated. The goal then is to find an optial solution that is feasible for all, or alost all, of the instances of the uncertain paraeters while axiizing the expectation of soe function of the proble variables and the rando variables. The ost coonly studied stochastic prograing odels for scheduling probles are two-stage progras. For twostage progras, in the first stage, all data are assued to be nown. Then, a rando event occurs that affects the outcoe of the first stage and a recourse decision is ade in the second stage to copensate for any negative effects that ight have been experienced as a result of the first stage decision. The result of two-stage prograing odels is a single first stage policy followed by a collection of recourse decisions that indicate which second stage policy should be ipleented in response to each rando outcoe. Thus, as the nuber of uncertain paraeters increases, ore scenarios ust be considered for the recourse decisions, resulting in a uch larger proble. In fact, the nuber of scenarios increases exponentially with the nuber of uncertain paraeters. This ain drawbac liits the application of these approaches to solve practical probles with a large nuber of uncertain paraeters. Bassett, Peny, and Relaitis (1997) eployed stochastic ethods to consider uncertainty in processing ties, equipent reliability/availability, process yields, deands, and anpower changes of scheduling probles. They used Monte Carlo sapling to generate rando instances of the uncertainty and deterined a schedule for each instance. Then, distributions of aggregated properties are generated using the instances to infer operating policies that are robust. However, the forulation does not produce a robust schedule. Vin and Ierapetritou (2001) considered deand uncertainty in the short-ter scheduling of ultiproduct and ultipurpose batch plants. They introduced several etrics to evaluate the robustness of a schedule and proposed a ultiperiod prograing odel using extree points of the deand range as scenarios to generate a single sequence of tass with the inial average aespan over all scenarios. Balasubraanian and Grossann (2002) proposed a ultiperiod ixed-integer linear prograing (MILP) odel for scheduling ultistage flowshop plants with uncertain processing ties described by discrete or continuous probability distributions. The objective is the iniization of the expected aespan and a special branch and bound algorith is eployed based on a lower bound generated by an aggregated probability odel. Balasubraanian and Grossann (2004) also considered the proble of scheduling under deand uncertainty. They utilized a ultistage stochastic MILP odel where soe decisions are ade without respect to the uncertainty and others are ade upon realization of the uncertainty. They proposed an approxiation strategy which solves a series of two-stage odels within a shrining-horizon approach. In addition, Jia and Ierapetritou (2004) used the idea of inference-based sensitivity analysis for MILP probles to deterine the iportance of different paraeters and constraints in their scheduling odel. The forulation provides a set of alternative schedules for the range of uncertain paraeters under consideration. Bonfill, Bagajewicz, Espuña, and Puigjaner (2004) presented an approach to anage ris for scheduling with uncertain deands. They eployed a two-stage stochastic optiization odel which axiizes the expected profit and anages ris explicitly by considering a new objective as a control easure, leading to a ultiobjective optiization forulation. Subsequently, Bonfill, Espuña, and Puigjaner (2005) extended this forulation to consider uncertain processing ties. A two-stage stochastic approach is again utilized where a weighted su of the expected aespan and the expected wait ties is iniized and ris is easured using different robustness criteria. Ostrovsy, Datsov, Achenie, and Volin (2004) discussed extensions of the two-stage optiization odel to tae into account the possibility of accurately estiating soe of the uncertain paraeters. A split and bound approach is utilized to solve the proble and is based on a partitioning of the uncertain region and estiation of bounds on the objective function. There have also been attepts to transfor a stochastic odel to a direct deterinistic equivalent representation. Bertsias and Si (2004) defined a robust optiization forulation that allows for a degree of conservatis in every uncertain constraint. Their forulation is developed for uncertainty that is bounded and syetric and a large nuber of paraeters ust be uncertain in each constraint for the resulting forulation to be desirably tight, or achieve solutions that are not too conservative. In addition, Bertsias and Si (2003) also used this robust optiization forulation to incorporate data uncertainty for discrete optiization and networ flow probles where cost coefficients and the data in the constraints can be uncertain. The forulation allows for control over the degree of conservatis of the solution in ters of the probabilistic bounds on constraint violation. Probabilistic or chance constraints have also been used to incorporate uncertainty into scheduling probles. Chance constraints do not require all decisions to be feasible for every outcoe of the rando variables, but require feasibility with at least soe specified probability distribution. Orçun, Altinel, and Hortaçsu (1996) considered uncertain processing ties in batch processes and eployed chance constraints to account for the ris of violation of tiing constraints under certain conditions such as unifor distribution functions. In addition, Petov and Maranas (1997) proposed a stochastic odel for scheduling with uncertain product deands which involves the axiization of the expected profit where single or ultiple product deands are satisfied with a prespecified probability level using chance constraints. Wang (2004) presented a robust optiization ethodology based on fuzzy set theory for uncertain product developent projects where the uncertain paraeters in the odel are represented by fuzzy sets. A genetic algorith approach is used to solve the

3 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) proble where the easure of schedule robustness is based on qualitative possibility theory. Also, Balasubraanian and Grossann (2003) used the concept of fuzzy set theory to describe the uncertainty in scheduling probles. They proposed an MILP odel for uncertainty in processing ties for flowshop scheduling probles and new product developent process scheduling probles. An alternative approach for scheduling under uncertainty is reactive scheduling in which an existing schedule is adjusted upon realization of the uncertain paraeters or the occurrence of unexpected events. Due to the on-line nature of reactive scheduling, it is necessary to generate updated schedules in a tiely anner and often, heuristic approaches are utilized (e.g., Cott & Macchietto, 1989; Honop, Mocus, & Relaitis, 1999; Kanaaedala, Relaitis, & Venatasubraanian, 1994; Sanarti, Huercio, Espuña, & Puigjaner, 1996; Rodrigues, Gieno, Passos, & Capos, 1996; Vin & Ierapetritou, 2000). A recent review on scheduling approaches that includes reactive scheduling can be found in Floudas and Lin (2004). In this wor, we extend the robust optiization approach presented in Lin, Jana, and Floudas (2004) for scheduling under bounded uncertainty and bounded and syetric uncertainty to consider uncertainty described by a nown probability distribution function. The underlying atheatical fraewor is based on a robust optiization ethodology first introduced for linear prograing (LP) probles by and extended in Lin et al. (2004) and this wor for ixed-integer linear prograing probles. The approach produces robust solutions for uncertainties in both the coefficients and right-hand-side paraeters of the linear inequality constraints and can be applied to address the proble of production scheduling with uncertain paraeters. The rest of this paper is organized as follows. We will first review the proble of short-ter scheduling under paraeter uncertainty. Then the robust optiization approach is extended for the case of general MILP probles with uncertain paraeters in the inequality constraints described by nown probability distributions. Next, this approach is applied to the proble of short-ter scheduling with uncertainty in processing ties of tass, product deands, and aret prices. Finally, coputational results are presented, followed by concluding rears. 2. Robust optiization for MILP probles Consider the following generic ixed-integer linear prograing proble Min/Max c T x + d T y x,y s.t. Ex+ Fy = e Ax + By p x x x y = 0, 1. Assue that the uncertainty arises fro both the coefficients and the right-hand-side paraeters of the inequality constraints, naely, a l, b l and p l where l is the index of the uncertain inequality, is the index of the continuous ters, and is the index of the binary ters. Thus, we are concerned about the feasibility of the following inequality ã l x + b l y p l (2) where a l, b l, and p l are the noinal values of the uncertain paraeters and ã l, b l, and p l are the true values of the uncertain paraeters. Assue that for inequality constraint l, the true values of the uncertain paraeters are obtained fro their noinal values by rando perturbations (1) ã l = (1 + ɛξ l )a l b l = (1 + ɛξ l )b l (3) p l = (1 + ɛξ l )p l where ξ l, ξ l and ξ l are independent rando variables and ɛ>0 is a given (relative) uncertainty level. In this situation, we call a solution (x, y) robust if it satisfies the following (i) (x, y) is feasible for the noinal proble, and (ii) for every inequality l, the probability of violation of the uncertain inequality in Eq. (2) (i.e., the left-hand-side exceeds the right-hand-side) is at ost κ, { Pr ã l x + } b l y > p l + δ ax[1, p l ] κ (4)

4 174 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) where δ>0 is a given feasibility tolerance and is introduced to allow a sall aount of infeasibility in the uncertain inequality, and κ>0 is a given reliability level. Thus, κ represents the probability of violation of constraint l where κ = 0% indicates that there is no chance of constraint violation, yielding the ost conservative solution. As shown in the first part of this wor (Lin et al., 2004), the optial solution of an MILP progra ay becoe severely infeasible, that is, one or ore constraints are violated substantially, if the noinal data are slightly perturbed. This aes the noinal optial solution questionable. Thus, in this wor, our objective is to extend this robust optiization fraewor to be able to generate reliable solutions to the MILP progra in Eq. (1), which are iune against uncertainty that can be described by a nown probability distribution. This robust optiization ethodology was first introduced for linear prograing probles with uncertain linear coefficients by Ben-Tal and Neirovsi (2000) and was extended in Lin et al. (2004) to consider uncertainty in MILP probles. It is iportant to ephasize that the robust optiization approach is different than the chance-constrained prograing approach. Chance-constrained prograing probles use constraints of the following type Pr{g i (x, ξ) 0, i} p or Pr{g i (x, ξ) 0} p i, i (5) where x is a decision variable, ξ is a rando variable, i indicates a finite set of inequalities, Pr is a probability easure, and p is the given probability level such that 0 p 1. The first for describes the case where ultiple constraints hold with a single probability while the second for, tered individual chance constraints, describes the case where ultiple probabilistic constraints hold separately. Chance-constrained prograing was first defined by Charnes and Cooper (1959, 1962, 1963) where the authors consider individual chance constraints and rando variations are confined to the right-hand-side vector, or constant ter. In contrast, robust optiization techniques, which were originally proposed by Ben-Tal, Neirovsi, and co-worers (Ben-Tal & Neirovsi, 1998, 1999, 2000; Ben-Tal, Goryasho, Guslitzer, & Neirovsi, 2004) and independently by El Ghaoui and co-worers (El Ghaoui, 1997; El Ghaoui, Oustry, & Lebret, 1998) and later extended by Bertsias and Si (2003, 2004, in press) and Floudas and co-worers (Lin et al., 2004), see to deterine a robust feasible/optial solution to an uncertain proble. This eans that the optial solution should provide the best possible value of the original objection function and also be guaranteed to reain feasible in the range of the uncertainty set considered. Using this notion, the concept of robust optiization, as defined in these wors, involves the following uncertain linear prograing proble Min x s.t. c T x Ax b ξ [A, b, c] Z (6) where the data set ξ [A, b, c] varies in a given uncertainty set Z. Then the optial solution which gives the best possible value of the original objection is defined as sup c T x ξ Z (7) where sup is the supreu of a function. Note that if the axiu of a function exists, then it ust be finite and it occurs at the supreu of the function. Thus, the robust feasible/optial solution can be found by solving the following proble Min x Max ξ Z [ct x] s.t. A T x b 0 ξ [A, b, c] Z (8) called the robust counterpart proble. Note that this is a in ax proble. The uncertainty set Z is selected so as to balance the robustness and optiality of the solution. Thus, although the proposed robust optiization forulation uses individual probabilistic constraints, as seen in constraint (4), these constraints are eployed to guarantee the reliability of the solution (or robustness of the solution) and not as initial proble constraints. As can be seen in the subsequent section, the probabilistic constraints are used in the proofs of the individual theores deriving the deterinistic forulations for each probability distribution and not in the forulations theselves as in individual chance-constrained prograing Uncertainty with nown probability distribution If the probability distributions of the rando variables ξ l, ξ l and ξ l in the uncertain paraeters are nown, it is possible to obtain a ore accurate estiation of the probability easures involved. The MILP fro (1) can be rewritten as an uncertain MILP

5 as follows Min x, y s.t. Max ξ Z [ct x + d T y] Ex+ Fy = e Ãx + By p δ ax[1, p ] x x x y = 0, 1 ξ [A, B, p] Z S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) where the data set ξ [A, B, p] varies in a given uncertainty set Z, Ã, B, and p represent the true values of the uncertain coefficients, and δ 0 is an infeasibility tolerance introduced to allow a certain aount of infeasibility into the inequality constraint. The inequality can be written in expanded for as ã l x + b l y p l δ ax[1, p l ] (10) for every constraint l where ã l, b l, and p l are again the true values of the uncertain coefficients. Substituting the expressions for the true values of the uncertain coefficients given in constraint (3) (i.e., ã l, b l, and p l ), the uncertain inequality in (10) can be rewritten as follows (1 + ɛξ l )a l x + (1 + ɛξ l )b l y (1 + ɛξ l )p l δ ax[1, p l ]. (11) Rearranging ters, we get a l x + b l y p l + ɛ ξ l a l x + ξ l b l y ξ l p l δ ax[1, p l ] (12) Ml K l where M l and K l define the sets of uncertain paraeters a l and b l, respectively, for constraint l. Then, a solution (x, y) tothe original uncertain MILP given in Eq. (9) which satisfies this constraint is called reliable because it taes into account the axiu aount of uncertainty ξ Z and allows an aount of infeasibility δ. Now, to transfor the constraint into a deterinistic for, we instead consider the following forulation Pr a l x + b l y p l + ɛ ξ l a l x + ξ l b l y ξ l p l >δ ax[1, p l ] κ. (13) Ml K l This constraint enforces that the probability of violation of the uncertain inequality is at ost κ, where δ 0 is a given feasibility tolerance (i.e., aount of error allowed in the feasibility of constraint l) and κ 0 is a given reliability level (i.e., the probability of violation of constraint l where κ = 0 indicates that there is no chance of constraint violation). Thus, if we now a probability distribution function for the su of the rando variables, ξ = ξ l a l x + ξ l b l y ξ l p l (14) M l K l (9) we can use this inforation in the probabilistic constraint (13) to write a deterinistic for for the uncertain constraint which is alost reliable, depending on the value of κ. This is done using the definition of a probability distribution function and the following relationship F ξ (λ) = Pr{ξ λ} =1 Pr{ξ >λ}=1 κ (15) to replace the stochastic eleents in constraint (12), generating a deterinistic constraint that is alost reliable for the given uncertainty level, ɛ, infeasibility tolerance, δ, and reliability level, κ. The final for of the deterinistic constraint (or robust counterpart proble) is siply deterined using the inverse distribution function (quantile) of the rando variable ξ Fξ 1 (1 κ) = f (λ, a l x, b l y, p l ). (16) Thus, the additional constraints in the robust counterpart (RC) proble can be written as a l x + b l y + ɛf (λ, a l x, b l y, p l ) p l + δ ax[1, p l ], l (17) where ɛ is a given uncertainty level, δ is a given infeasibility tolerance, and λ is deterined fro κ using constraint (15) and the probability distribution function for ξ.

6 176 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) In the following sections, robust optiization ethods are developed for several of the ost coon probability distributions including the unifor distribution, noral distribution, difference of noral distributions, general discrete distribution, binoial distribution, and poisson distribution Uncertainty with unifor probability distribution Assue that there is only one uncertain paraeter in each constraint, which can be one of the following ã l = (1 + ɛξ l )a l or b l = (1 + ɛξ l )b l (18) or p l = (1 + ɛξ l )p l where ξ l is a rando variable with unifor distribution in the interval [ 1, 1]. Theore 1. Given an uncertainty level (ɛ), an infeasibility tolerance (δ), and a reliability level (κ), to generate robust solutions, the following (ɛ, δ, κ)-robust counterpart (RC[ɛ, δ, κ]) of the original uncertain MILP proble can be derived. Min/Max x,y,u s.t. c T x + d T y Ex+ Fy = e Ax + By p a lx + b ly + ɛ(1 2κ) a l u p l + δ ax[1, p l ] l L c u x u l L c, M l a lx + b ly + ɛ(1 2κ) b l y p l + δ ax[1, p l ] l L b a lx + b ly p l ɛ(1 2κ) p l +δax[1, p l ] l L r x x x y = 0, 1 (19) where L c,l b,l r are the set of inequalities with uncertainty in the coefficients of continuous variables, the coefficients of binary variables, and the right-hand-side paraeters, respectively. Proof. Let (x, y) satisfy the following: If a l is the uncertain paraeter, then a l x + b l y + ɛ(1 2κ) a l u p l + δ ax[1, p l ], (20) u x u, (21) or if b l is the uncertain paraeter, then a l x + b l y + ɛ(1 2κ) b l y p l + δ ax[1, p l ], (22) or if p l is the uncertain paraeter, then a l x + b l y p l ɛ(1 2κ) p l +δax[1, p l ]. (23) It follows that if a l is the uncertain paraeter, Pr a l x + ã l x + b l y >p l + δ ax[1, p l ] { = Pr a l x + ɛξ l a l x + } b l y >p l + δ ax[1, p l ] Pr {ξ l x > (1 2κ)u } Pr {ξ l x > (1 2κ) x } (note: we only need to consider κ 0.5) = 1 Pr {ξ l x (1 2κ) x } = 1 (1 κ) = κ,

7 or if b l is the uncertain paraeter, Pr a l x + b l y + b l y >p l + δ ax[1, p l ] S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) { = Pr a l x + } b l y + ɛξ l b l y >p l + δ ax[1, p l ] Pr {ξ l > (1 2κ)} = 1 Pr {ξ l (1 2κ)} = κ, or if p l is the uncertain paraeter, { Pr a l x + } b l y > p l + δ ax[1, p l ] { = Pr a l x + } b l y >p l + ɛξ l p l +δax[1, p l ] Pr { ξ l > (1 2κ)} = κ. Note that this forulation for uncertainty described by a unifor probability distribution results in an MILP proble and includes a set of auxiliary variables, u Uncertainty with noral probability distribution Suppose that the distributions of the rando variables ξ l, ξ l and ξ l in (3) are all standardized noral distributions with zero as the ean and one as the standard deviation. Then, the distribution of ξ defined in (14) is also a noral distribution, with zero as the ean and M a2 l l x2 + K b2 l l y + p 2 l as the standard deviation. Theore 2. Given an uncertainty level (ɛ), an infeasibility tolerance (δ), and a reliability level (κ), to generate robust solutions, the following (ɛ, δ, κ)-robust counterpart (RC[ɛ, δ, κ]) of the original uncertain MILP proble can be derived. Min/Max x,y s.t. c T x + d T y Ex+ Fy = e Ax + By p a lx + b ly + ɛλ M a2 l l x2 + K b2 l l y + p 2 l p l + δ ax[1, p l ] l x x x y = 0, 1 where λ = Fn 1 (1 κ) and F 1 n is the inverse distribution function of a rando variable with standardized noral distribution. Thus, λ and κ are related as follows κ = 1 F n (λ) κ = 1 Pr{ξ λ} λ ( 1 x 2 ) κ = 1 exp dx 2π 2 where ξ is a rando variable with standardized noral distribution. Proof. Let (x, y) satisfy the following a l x + b l y + ɛλ al 2 x2 + bl 2 y + p 2 l p l + δ ax[1, p l ] (25) M l K l where λ = Fn 1 (1 κ) and F 1 n is the inverse distribution function of a rando variable with standardized noral distribution. Then { Pr ã l x + } b l y > p l + δ ax[1, p l ] = Pr a l x + ɛ ξ l a l x + b l y + ɛ ξ l b l y >p l + ɛξ l p l +δax[1, p l ] M l K l (24)

8 178 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) Pr ξ l a l x + ξ l b l y ξ l p l / al 2 x2 + bl 2 y + p 2 l >λ Ml K l M l K l = 1 Pr ξ l a l x + ξ l b l y ξ l p l / al 2 x2 + bl 2 y + p 2 l λ Ml K l M l K l = 1 F n (λ) = 1 (1 κ) = κ. Note that M l ξ l a l x + K l ξ l b l y ξ l p l Ml a2 l x2 + Kl b2 l y + p 2 l is also a rando variable with standardized noral distribution. This forulation results in a convex MINLP proble, but can still be solved efficiently using a ixed-integer nonlinear solver (e.g., DICOPT (Viswanathan & Grossann, 1990), MINOPT (Schweiger & Floudas, 1998)) Uncertainty with difference of noral probability distributions Suppose that the distributions of rando variables ξ l, ξ l and ξ l in (3) are all represented as the difference of two noral rando variables, each with a given ean, μ, and a given standard deviation, σ. Then, the rando variables can be represented by ξ l = A l ξ lx B l ξ ly, ξ l = A l ξ lx B l ξ ly, ξ l = A l ξ lx B l ξ ly (26) where each has a ean μ and a standard deviation σ. Furtherore, the overall rando variable ξ defined in (14) is also given by a noral distribution and has a ean and standard deviation of μ = M l a l x (A l μ lx B l μ ly ) + K l b l y (A l μ lx B l μ ly ) p l (A l μ lx B l μ ly ) (27) σ = al 2 x2 (A2 l σ2 lx + B2 l σ2 ly ) + bl 2 y2 (A2 l σ2 lx + B2 l σ2 ly ) + p2 l (A2 l σ2 lx + B2 l σ2 ly ). (28) M l K l We can then standardize the rando variable ξ so that it has zero as the ean and one as the standard deviation as follows ξ std = (ξ μ) σ. (29) Theore 3. Given an uncertainty level (ɛ), an infeasibility tolerance (δ), and a reliability level (κ), to generate robust solutions, the following (ɛ, δ, κ)-robust counterpart (RC[ɛ, δ, κ]) of the original uncertain MILP proble can be derived. Min/Max x,y s.t. c T x + d T y Ex+ Fy = e Ax + By p a lx + b ly + ɛ(λ σ + μ) p l + δ ax[1, p l ] x x x y = 0, 1. l (30) where λ = Fn 1 (1 κ) and F 1 n is the inverse distribution function of a rando variable with standardized noral distribution and μ and σ are described as given in Eqs. (27) and (28). Proof. Let (x, y) satisfy the following a l x + b l y + ɛ(λ σ + μ) p l + δ ax[1, p l ] (31)

9 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) where λ = Fn 1 (1 κ) and F 1 n is the inverse distribution function of a rando variable with a standardized noral distribution. Then { Pr ã l x + } b l y > p l + δ ax[1, p l ] = Pr a l x + ɛ ξ l a l x + b l y + ɛ ξ l b l y >p l + ɛξ l p l +δax[1, p l ] M l K l Pr ξ l a l x + ξ l b l y ξ l p l μ / σ>λ Ml K l = 1 Pr ξ l a l x + ξ l b l y ξ l p l μ / σ λ = 1 F n(λ) = 1 (1 κ) = κ. Ml K l This forulation results in a convex MINLP proble, but can still be solved efficiently using a ixed-integer nonlinear solver (e.g., DICOPT (Viswanathan & Grossann, 1990), MINOPT (Schweiger & Floudas, 1998)) Uncertainty with general discrete probability distribution Suppose that the distributions of the rando variables ξ l, ξ l and ξ l in (3) are all given by general discrete distributions. Then the distribution of ξ defined in (14) is also a discrete distribution. Theore 4. Given an uncertainty level (ɛ), an infeasibility tolerance (δ), and a reliability level (κ), to generate robust solutions, the following (ɛ, δ, κ)-robust counterpart (RC[ɛ, δ, κ]) of the original uncertain MILP proble can be derived. Min/Max x,y s.t. c T x + d T y Ex+ Fy = e Ax + By p a lx + ( b ly + ɛ λ a Ml l u + ) λ b Kl l y λ p l p l + δ ax[1, p l ] u x u x x x y = 0, 1 l (32) where κ = 1 F(λ) and where F is the distribution function of a general discrete rando variable. Proof. Let (x, y) satisfy the following a l x + ɛ λ a l u + M l u x u b l y + ɛ K l λ b l y p l + ɛλ p l +δ ax[1, p l ], (33) (34) where M l and K l are the set of indices of the x and y variables, respectively, with uncertain coefficients in the lth inequality constraint and F(λ) = 1 κ and κ>0is a given reliability level. Then { Pr ã l x + } b l y > p l + δ ax[1, p l ] = Pr a l x + ɛ ξ l a l x + b l y + ɛ ξ l b l y >p l + ɛξ l p l +δax[1, p l ] M l K l Pr ξ l a l x + ξ l b l y ξ l p l > λ a l u + λ b l y λ p l M l K l M l K l

10 180 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) = 1 Pr ξ l a l x + ξ l b l y ξ l p l λ a l u + λ b l y λ p l M l K l M l K l = 1 F(λ) = 1 (1 κ) = κ. Note that this forulation for uncertainty described by a general discrete distribution results in an MILP proble and includes a set of auxiliary variables, u Uncertainty with binoial probability distribution Assue that in each inequality constraint l, there is at ost one uncertain paraeter, which can be the coefficient of a continuous variable, or the coefficient of a binary variable, or the right-hand-side paraeter. The true value of the uncertain paraeter is obtained fro its noinal value by rando perturbation ã l = (1 + ɛξ l )a l or b l = (1 + ɛξ l )b l (35) or p l = (1 + ɛξ l )p l where ξ l is a discrete rando variable with a binoial distribution. Theore 5. Given an uncertainty level (ɛ), an infeasibility tolerance (δ), and a reliability level (κ), to generate robust solutions, the following (ɛ, δ, κ)-robust counterpart (RC[ɛ, δ, κ]) of the original uncertain MILP proble can be derived. Min/Max c T x + d T y x,y,u s.t. Ex+ Fy = e Ax + By p a lx + b ly + ɛλ a l u p l + δ ax[1, p l ] l L c u x u l L c, M (36) l a lx + b ly + ɛλ b l y p l + δ ax[1, p l ] l L b a lx + b ly p l ɛλ p l +δax[1, p l ] l L r x x x y = 0, 1 where L c,l b,l r are the set of inequalities with uncertainty in the coefficients of continuous variables, the coefficients of binary variables, and the right-hand-side paraeters, respectively. Proof. Let (x, y) satisfy the following: If a l is the uncertain paraeter, then a l x + b l y + ɛλ a l u p l + δ ax[1, p l ], (37) u x u, (38) or if b l is the uncertain paraeter, then a l x + b l y + ɛλ b l y p l + δ ax[1, p l ], (39) or if p l is the uncertain paraeter, then a l x + b l y p l ɛλ p l +δax[1, p l ]. (40) where λ = F 1 (1 κ) and F 1 is the inverse distribution function of a discrete rando variable with a binoial distribution. It follows that if a l is the uncertain paraeter, Pr a l x + ã l x + b l y >p l + δ ax[1, p l ] { = Pr a l x + ɛξ l a l x + } b l y >p l + δ ax[1, p l ] Pr {ξ l x >λu } = 1 Pr {ξ l x λu } = 1 (1 κ) = κ,

11 or if b l is the uncertain paraeter, Pr a l x + b l y + b l y >p l + δ ax[1, p l ] S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) { = Pr a l x + } b l y + ɛξ l b l y >p l + δ ax[1, p l ] Pr {ξ l >λ} = 1 Pr {ξ l λ} = κ, or if p l is the uncertain paraeter, { Pr a l x + } b l y > p l + δ ax[1, p l ] { = Pr a l x + } b l y >p l + ɛξ l p l +δax[1, p l ] Pr { ξ l >λ} = κ. Note that this forulation for uncertainty described by a binoial distribution results in an MILP proble and includes a set of auxiliary variables, u Uncertainty with poisson probability distribution The derivation for uncertainty with a poisson distribution is analogous to that of uncertainty with a binoial distribution, as given in Eq. (36) with the only difference being that F 1 is the inverse distribution function of a discrete rando variable with a poisson distribution. In the discussion above, for siplicity, we have assued that there is a single coon uncertainty level (ɛ), infeasibility tolerance (δ), and reliability level (κ) in each MILP or convex MINLP proble with uncertain paraeters. However, the proposed robust optiization techniques can easily be extended to account for the ore general case in which the uncertainty level varies fro one paraeter to another and the infeasibility tolerance and reliability level are dependent on the constraint of interest. Furtherore, note that for each type of uncertainty addressed above, one additional constraint is introduced for each inequality constraint with uncertain paraeter(s) and auxiliary variables are added if needed. Because the transforation is carried out at the level of constraints, in principle, the various robust optiization techniques presented can be applied to a single MILP or convex MINLP proble involving different types of uncertainties. More specifically, for each inequality constraint, as long as all of its uncertain paraeters are of the sae type, an additional constraint that corresponds to the uncertainty type can be introduced to obtain the deterinistic robust counterpart proble. It should be pointed out that the aforeentioned robust optiization ethodology circuvents any need for explicit or iplicit discretization or sapling of the uncertain data, avoiding an undesirable increase in the size of the proble. Thus, the proposed ethodology is potentially capable of handling probles with a large nuber of uncertain paraeters. In addtion, the resulting conservatis of a robust solution is deterined by the values of the uncertainty level (ɛ), infeasibility tolerance (δ), and reliability level (κ) used in the uncertain inequalities. As the uncertainty level increases, the uncertain inequalities becoe ore difficult to satisfy resulting in a ore conservative robust solution with a worse objective function value. In constrast, as the infeasibility tolerance increases, the uncertain inequalities tolerate a higher level of infeasibility, generating less conservative solutions with better objective function values. Siilarly, as the reliability level increases (or the probability of violation increases), the aount of violation allowed in the uncertain inequalities increases, generating less conservative solutions with better objective function values. In this way, paraetric studies of the effects of each of these paraeters can be carried out, deterining the tradeoffs between the level uncertainty and reliability with the quality and conservatis of the robust solution. 3. Robust optiization for scheduling under uncertainty The robust optiization ethodology proposed in the previous section can now be used to study the effects of uncertainty in short-ter scheduling probles. In order to odel this scheduling proble, we use the continuous-tie forulation developed by Floudas and co-worers (Floudas & Lin, 2004, 2005; Ierapetritou & Floudas, 1998a, 1998b; Ierapetritou, Hené, & Floudas, 1999; Jana, Lin, & Floudas, 2004; Lin & Floudas, 2001; Lin, Floudas, Modi, & Juhasz, 2002; Lin, Chajais, & Floudas, 2003), which results in MILP odels (see Appendix A for the coplete scheduling forulation). This continuous-tie forulation has also been extended to deterine the ediu-ter production scheduling of a large-scale, industrial batch plant utilizing actual plant operating data (Jana, Floudas, Kallrath, & Vorbroc, 2006a, 2006b). In addition, a coparative study of several continuous-tie forulations for short-ter scheduling can be found in Shai, Jana, and Floudas (in press). Let us consider uncertainty in three classes of paraeters that participate in short-ter scheduling probles, naely: (i) uncertainty in processing

12 182 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) ties/rates of tass, (ii) uncertainty in aret deands for products, and (iii) uncertainty in aret prices of products and/or raw aterials Proble stateent for short-ter scheduling The proble of short-ter scheduling for cheical processes under uncertainty is defined as follows. Given (i) the production recipes (i.e., the processing ties for each tas in the suitable units and the aount of aterial required for the production of each product), (ii) the available equipent and the ranges of their capacities, (iii) the aterial storage policy, (iv) the production requireents, and (v) the tie horizon under consideration, we want to deterine (i) the optial sequence of tass taing place in each unit, (ii) the aount of aterial being processed at each tie in each unit, and (iii) the processing tie of each tas in each unit, so as to optiize a perforance criterion, for exaple, to iniize the aespan or to axiize the overall profit while taing into account uncertainty inherent in soe of the process paraeters. The ost coon sources of uncertainty in scheduling probles are (i) the processing ties of tass, (ii) the aret deands for products, and (iii) the prices of products and/or raw aterials. An uncertain paraeter can be described using a discrete or continuous distribution and in soe cases, only liited nowledge about the distribution is available. In the best situation, the distribution function for the uncertain paraeter is given, for instance, as a noral distribution with nown ean and standard deviation or as a unifor distribution in a given range. In this wor, we will focus on uncertainty characterized by a nown probability distribution function including cobinations of probability distributions within a single proble Uncertainty in processing ties The paraeters of processing ties/rates of tass appear in the duration constraint and appear as linear coefficients of the binary variable (i.e., α ij ) and the continuous variable (i.e., β ij ) as follows T f (i, j, n) T s (i, j, n) = α ij wv(i, n) + β ij B(i, j, n) (41) where wv(i, n) is a binary variable indicating whether or not tas (i) starts at event point (n), B(i, j, n) is a continuous variable deterining the batch-size of the tas, and T s (i, j, n) and T f (i, j, n) are continuous variables representing the starting and finishing tie of the tas, respectively. Note that this is an equality constraint. Thus, in order to apply the robust optiization techniques proposed in Section 2 for inequality constraints with uncertain paraeters, two separate approaches are developed. Approach 1 In the first approach, the duration constraint is relaxed to an inequality constraint T f (i, j, n) T s (i, j, n) α ij wv(i, n) + β ij B(i, j, n). (42) Consequently, the variable T f (i, j, n) represents the lower bound on the finishing tie of the tas, instead of the exact finishing tie as deterined by the original duration constraint. Using this odified duration constraint, the various robust optiization techniques can be readily applied to consider uncertainty in the paraeters α ij and β ij. For exaple, consider a batch tas with fixed processing tie represented by paraeter α ij. Then the true value of the processing tie can be represented in ters of the noinal processing tie as follows α ij = (1 + ɛξ αij )α ij (43) where ξ αij is a rando variable with nown distribution. For the case where the uncertainty is unifor in the interval [ 1, 1], then according to Theore 1, to obtain the deterinistic robust counterpart proble, the following constraint is added to the original scheduling odel T s (i, j, n) T f (i, j, n) + [1 + ɛ(1 2κ)]α ij wv(i, n) δ. (44) As another exaple, consider the case where the uncertainty is characterized by a standardized noral distribution. Then, according to Theore 2, to obtain the deterinistic robust counterpart proble, the following constraint is added to the original scheduling odel T s (i, j, n) T f (i, j, n) + [1 + ɛλ]α ij wv(i, n) δ (45) where λ = F 1 (1 κ) and Fn 1 is the inverse distribution function of a rando variable with a standardized noral distribution. In addition, for the case where the uncertainty is characterized by a discrete binoial distribution, then according to Theore 5, to obtain the deterinistic robust counterpart proble, the following constraint is added to the original scheduling odel T s (i, j, n) T f (i, j, n) + [1 + ɛλ]α ij wv(i, n) δ (46) where λ = F 1 (1 κ) and Fn 1 is the inverse distribution function of a rando variable with a binoial distribution.

13 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) Approach 2 In the second approach, the original duration constraint (41) is eliinated fro the scheduling odel and the variable T f (i, j, n) is substituted into the other tiing constraints using the following expression T f (i, j, n) = T s (i, j, n) + α ij wv(i, n) + β ij B(i, j, n). (47) The tiing constraints which include the variable T f (i, j, n) are the sequencing constraints and the tie horizon constraints T s (i, j, n + 1) T f (i, j, n) i I, j J i,n N, n N, (48) T s (i, j, n + 1) T f (i,j,n) H[1 wv(i,n)] j J, i, i I j,i i,n N, n N, (49) T s (i, j, n + 1) T f (i,j,n) H[1 wv(i,n)] j, j J, i I j,i I j,i i,n N, n N, (50) T f (i, j, n) H, i I, j J i,n N. (51) After the substitution, they becoe T s (i, j, n + 1) T s (i, j, n) + α ij wv(i, n) + β ij B(i, j, n) i I, j J i,n N, n N, (52) T s (i, j, n + 1) T s (i,j,n) + α i j wv(i,n) + β i j B(i,j,n) H[1 wv(i,n)] j J, i, i I j, i i,n N, n N, (53) T s (i, j, n + 1) T s (i,j,n) + α i j wv(i,n) + β i j B(i,j,n) H[1 wv(i,n)] j, j J, i I j,i I j,i i, n N, n N, (54) T s (i, j, n) + α ij wv(i, n) + β ij B(i, j, n) H, i I, j J i,n N. (55) Now the uncertain paraeters α ij and β ij participate only in these inequality constraints and the robust optiization techniques can be readily applied as in approach Uncertainty in product deands The product deands (i.e., de s ) appear as the right-hand-side paraeters in the deand constraints STF(s) de s, s S p (56) where STF(s) is a continuous variable representing the aount of state (s) accuulated at the end of the tie horizon and S p is the set of final products. If we consider an uncertain product deand represented by paraeter de s, then the true value of the product deand can be represented in ters of the noinal product deand as follows de s = (1 + ɛξ s )de s (57) where ξ s is a rando variable with nown distribution. For the case of unifor uncertainty in the product deands, then according to Theore 1, the constraint to be added to the original scheduling odel to derive the deterinistic robust counterpart proble is STF(s) de s (1 + ɛ(1 2κ) δ). (58) In addition, in the case of noral uncertainty, according to Theore 2, the constraint to be added to the original scheduling odel to derive the deterinistic robust counterpart proble is STF(s) de s (1 + ɛλ δ) (59) where λ = Fn 1 (1 κ) and F 1 n is the inverse distribution function of a rando variable with standardized noral distribution. Moreover, in the case of discrete uncertainty, according to Theore 4, the constraint to be added to the original scheduling odel to derive the deterinistic robust counterpart proble is STF(s) de s (1 + ɛλ δ) (60) where λ = F 1 (1 κ) and F 1 n is the inverse distribution function of a rando variable with a general discrete distribution.

14 184 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) Uncertainty in aret prices The aret prices (i.e., p s ) participate in the objective function for the calculation of the overall profit Maxiize Profit = p s STF(s) p s STI(s) (61) s S p s S r where S p and S r are the sets of final products and raw aterials, respectively, and STI(s) and STF(s) are continuous variables representing the initial aount of state (s) at the beginning and the final aount of state (s) at the end, respectively. The objective function can be expressed in an equivalent way as follows Maxiize Profit s.t. Profit s S p p s STF(s) (62) s S r p s STI(s). Now the uncertain paraeters p s appear as linear coefficients ultiplying the continuous variables STF(s) and STI(s) in an inequality constraint and the robust optiization techniques can be readily applied. For exaple, if the uncertainty is norally distributed p s = (1 + ɛξ s )p s (63) where ξ s is a standardized noral rando variable, then, according to Theore 2, the deterinistic robust counterpart proble can be obtained by introducing the following constraint to the original scheduling odel Profit p s STF(s) p s STI(s) ɛλ p 2 s STF(s)2 + p 2 s STI(s)2 + δ (64) s S p s S r s S p s S r where λ = F 1 (1 κ) and F 1 n 4. Coputational studies is the inverse distribution function of a rando variable with standardized noral distribution. In this section, the robust optiization forulation is applied to four exaple probles. All the exaples are ipleented with GAMS 2.50 (Brooe, Kendric, Meeraus, & Raan, 2003) on a 3.20 GHz Linux worstation. The MILP probles are solved using CPLEX 8.1 while the MINLP probles are solved using DICOPT (Viswanathan & Grossann, 1990) Exaple 1: Uncertainty with a poisson distribution in the processing ties Consider the following exaple process that was first presented by Kondili, Pantelides, and Sargent (1993) and was also used as the otivating exaple in the Part 1 paper (Lin et al., 2004) on bounded uncertainty. Two products are produced fro three feeds according to the State-Tas Networ shown in Fig. 1. The STN utilizes three different types of tass which can be perfored in four different units. The corresponding data for the exaple including suitabilities, capacities, processing ties, and storage liitations are given in Table 1. The objective is to axiize the profit fro sales of products anufactured in a tie horizon of 12 h. Assue that the uncertainty in the processing ties has a poisson distribution with a paraeter value of 5 and let us consider an uncertainty level (ɛ) of 5%, an infeasibility tolerance level (δ) of 20%, and a reliability level (κ) of 24% (corresponding to a λ value of 6). By solving the RC[ɛ, δ, κ] proble, a robust schedule is obtained, as shown in Fig. 3, which taes into account uncertainty in the processing ties. The noinal schedule can be seen in Fig. 2. Copared to the noinal solution which is obtained at the noinal values of the processing ties, the robust solution exhibits very different scheduling strategies. For exaple, even the sequences of tass in the two reactors in Fig. 3 deviates significantly fro Fig. 1. State-tas networ for exaple 1.

15 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) Table 1 Data for exaple 1 Units Capacity Suitability Processing tie Heater 100 Heating 1.0 Reactor 1 50 Reaction1, 2, 3 2.0, 2.0, 1.0 Reactor 2 80 Reaction1, 2, 3 2.0, 2.0, 1.0 Separator 200 Separation 2.0 States Storage capacity Initial aount Price Feed A Unliited Unliited 0.0 Feed B Unliited Unliited 0.0 Feed C Unliited Unliited 0.0 Hot A IntAB IntBC IpureE Product 1 Unliited Product 2 Unliited Fig. 2. Optial solution with noinal processing ties (profit = ). Fig. 3. Robust solution with uncertain processing ties (profit = ). those in the noinal solution in Fig. 2. The robust solution ensures that the robust schedule obtained is feasible with the specified uncertainty level, infeasibility tolerance, and reliability level. However, the resulting profit is reduced, fro to , which reflects the effect of uncertainty on overall production. A coparison of the odel and solution statistics for the noinal and robust solutions can be found in Table 2. Fig. 4 suarizes the results of the RC proble with several different cobinations of levels of uncertainty and infeasibility at increasing values of the reliability level. It is shown that at a given reliability level, the axial profit that can be achieved decreases as the uncertainty level increases, which indicates ore conservative scheduling decisions because of the existence of uncertainty. Also, at a given reliability level, the axial profit increases as the infeasibility tolerance level increases, which eans ore aggressive scheduling arrangeents can be incorporated if violations of related tiing constraints can be tolerated to a larger extent. In addition, at a given uncertainty level and infeasibility tolerance, the profit increases as the reliability level increases, eaning that the probability of violation of the uncertain constraint allows for ore aggressive scheduling. These results are consistent with intuition and other approaches; however, with the robust optiization approach, the effects of uncertainty and the trade-offs between conflicting objectives are quantified rigorously and efficiently. Table 2 Model and solution statistics for exaple 1 Noinal solution Robust solution Profit CPU tie (s) Binary variables Continuous variables Constraints

16 186 S.L. Jana et al. / Coputers and Cheical Engineering 31 (2007) Fig. 4. Profit vs. reliability level at different uncertainty and infeasibility levels in exaple 1. Fig. 5. Optial solution with noinal product deands (aespan = 8.007). Fig. 6. Robust solution with uncertain product deands (aespan = 8.174) Exaple 2: Uncertainty with a unifor distribution in the product deands In this exaple, we consider uncertainty with a unifor distribution in the product deands for the sae process given in exaple 1. However, the objective function is the iniization of the aespan for a given deand of 70 for Product 1 and 80 for Product 2. The uncertainty level (ɛ) is 10%, the infeasibility tolerance (δ) is 5%, and the reliability level (κ) is 0%. The noinal schedule is shown in Fig. 5 with a aespan of The robust schedule is obtained by solving the robust counterpart proble, as shown in Fig. 6, and the corresponding aespan is By executing this schedule, the aespan is guaranteed to be at ost with a probability of 100% in the presence of the 10% uncertainty in the deands of the products. A coparison of the odel and solution statistics for the noinal and robust solutions can be found in Table 3. Fig. 7 suarizes the results of the RC proble with several different cobinations of levels of uncertainty and infeasibility at increasing values of the reliability level. It is shown that at a given reliability level, the iniu aespan increases as the uncertainty level increases, which indicates ore conservative scheduling decisions that tae ore tie because of the existence of uncertainty in the deands. Also, at a fixed reliability level, the iniu aespan decreases as the infeasibility tolerance level increases, which eans ore aggressive scheduling arrangeents can be incorporated if violations of related deand constraints Table 3 Model and solution statistics for exaple 2 Noinal solution Robust solution Maespan CPU tie (s) Binary variables Continuous variables Constraints