Operational Planning of Large-Scale Industrial Batch Plants under

260

Ind. Eng. Chem. Res. 2010, 49, 260–275

Operational Planning of Large-Scale Industrial Batch Plants under Demand Due Date and Amount Uncertainty: II. Conditional Value-at-Risk Framework Peter M. Verderame and Christodoulos A. Floudas* Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544

A novel framework based on conditional value-at-risk theory has been applied to the problem of operational planning for large-scale industrial batch plants under demand due date and amount uncertainty. The nominal planning with production disaggregation model has been extended by means of conditional value-at-risk theory to address the objectives of providing a daily production profile that not only is a tight upper bound on the production capacity of the plant but also is immune to the various forms of demand uncertainty. An industrial case study was conducted to demonstrate the viability of the proposed approach, which involves the novel application of conditional value-at-risk theory to the problem of operational planning under demand due date and amount uncertainty. A comparative study that juxtaposes the proposed operational planning model and the robust operational planning with production disaggregation model presented in part I of this series of articles (Verderame and Floudas Ind. Eng. Chem. Res. 2009, 48, 7214) is introduced as well. 1. Introduction The operational planning of large-scale industrial batch plants entails determining the production profile for the chemical plant over a time horizon of several months. The uncertain nature of customer orders needs to be explicitly taken into account in order to ensure the efficient allocation of plant resources. In part I of this series of articles,1 a robust optimization framework, originally presented by Lin et al.2 and Janak et al.,3 was extended and applied to the problem of operational planning of largescale industrial batch plants under demand due date and amount uncertainty. It was demonstrated that the proposed robust operational planning with production disaggregation model (PPDM), based on the nominal PPDM developed by Verderame and Floudas4 and later extended to multisite production and distribution networks by Verderame and Floudas,5 can accomplish the two primary objectives of an operational planning model by providing a production profile that represents a tight upper bound on the production capacity of the plant in question and also addressing the aforementioned forms of demand uncertainty. The robust optimization framework has the ability to capture the uncertain nature of the demand parameters through the formulation of probabilistic demand constraints that are transformed into their deterministic equivalents used to augment the nominal operational planning model. Although the proposed robust operational PPDM1 represents a novel contribution to the field of operational planning, there exist other alternative approaches to dealing with demand uncertainty at the operational planning level that are able to capture the stochastic nature of the various demand parameters. Petkov and Maranas6 applied chance constraint programming within a multiperiod planning and scheduling framework for multiproduct batch plants with uncertain demand profiles due at the end of each period. Ahmed and Sahinidis7 developed a two-stage stochastic planning under uncertainty model that minimizes the actual cost of the first-stage model and the expected cost of the second-stage model while also minimizing the variance of the expected cost for the stochastic inner problem. Gupta et al.8,9 formulated a two-stage stochastic supply chain planning model with an outer deterministic production * To whom correspondence should be addressed. Tel.: 1-609-2584595. E-mail: [email protected].

site component and an inner recourse supply chain management component that is dependent on the realization of customer demand. Wu and Ierapetritou10 presented a hierarchical approach to planning and scheduling under uncertainty. This approach models uncertainty by means of multistage programming where the overall horizon is discretized into time periods based on the given system parameters’ degrees of certainty which vary inversely with the time horizon. Colvin and Maravelias11 developed a multistage stochastic programming model for clinical trial planning in new drug development. You and Grossmann12 utilized chance constraint programming for the design of responsive supply chains under demand uncertainty. You et al.13 developed a two-stage stochastic linear programming approach for the risk management of a global supply chain at the planning level, and their studies indicated that the explicit consideration of system uncertainty can lead to cost reduction. Despite the aforementioned contributions to the field, planning under uncertainty remains a challenging problem according to the reviews by Sahinidis14 and Li and Ierapetritou.15 Based on the work of Artzner et al.,16 Ogryczak,17 and Rockafellar and Uryasev,18,19 conditional value-at-risk (CVAR) theory poses a potentially attractive alternative to the robust optimization framework and other uncertainty approaches, and it has been used extensively in the financial sector to mitigate risk, as noted by Krokhmal et al.20 Both Rockafellar and Uryasev18 and Mansini et al.21 have applied CVAR to the portfolio management problem originally presented by Markowitz.22 Conditional value-at-risk theory represents an advance over established value-at-risk (VAR) theory, which has been explicitly described in the works of Staumbaugh,23 Pritzker,24 Jorion,25 and Duffiean and Pan,26 and it has been documented that CVAR has several important advantages over VAR. Both value-at-risk theory and conditional value-at-risk theory seek to guard against unfavorable realization of uncertain parameters by going beyond expectation evaluation when expressing the uncertainty of system parameters. For example, in the portfolio management problem, the investor attempts to select those commodities that will maximize the portfolio’s profit; however, the values of these commodities are stochastic. To guard against potential losses, the investor defines a loss function, f(x,u), where x represents a decision variable (i.e., the

10.1021/ie900925k CCC: $40.75  2010 American Chemical Society Published on Web 10/19/2009

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010

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The aforementioned literature examples demonstrate the potential of the CVAR approach.

Figure 1. Conditional value-at-risk versus value-at-risk.

vector of obtained commodities) and u is a stochastic vector (i.e., the vector of random commodity values) representing uncertainty and having a probability distribution of p(u). Both value-at-risk theory and conditional value-at-risk theory can be used to constrain the level of the loss function depending on the investor’s degree of risk aversion. Value-at-risk can be defined as the maximum loss that is expected to be exceeded with a probability of (1 - ω), whereas conditional value-atrisk is the expected loss given that the loss is greater than the value-at-risk for a given confidence level ω. As shown in Figure 1, conditional value-at-risk is the average of the probabilistic loss function’s tail defined by the lower and upper bounds of the value-at-risk and the maximum loss. Conditional value-atrisk represents a more pessimistic loss level when compared to value-at-risk, so it is often the preferable representation of loss. Conditional value-at-risk features the important property of maintaining convexity regardless of the type of probability distribution used, whereas value-at-risk exhibits inherent theoretical and algorithmic difficulties within an optimization model when the probability distribution is not normal or log-normal, because the resulting problem becomes nonconvex.20 Given its attractive convexity property and more pessimistic level of risk evaluation, CVAR theory will be adapted and applied to the demand uncertainty problem at the operational planning level. Because of the aforementioned characteristics, conditional value-at-risk theory has the potential to be applied to several different types of optimization problems sharing a common objective of guarding against the unfavorable realization of uncertain system elements. For instance, conditional value-atrisk was used by Jabr27 to formulate a robust self-scheduling model for a power producing company attempting to provide power bids ahead of market trends. Cabero et al.28 developed a medium-term risk management model for a hydrothermal generation company considering uncertainty in fuel prices, power demands, water inflows, and electricity prices while measuring risk by means of conditional value-at-risk theory. Gotoh and Takano29 applied conditional value-at-risk to the classic newsvendor problem dealing with maintaining inventory for one time period while facing uncertain demand. Tsang et al.30 utilized conditional value-at-risk theory when considering the capacity investment problem in the vaccine industry. Yamout et al.31 addressed the shortcomings of using sensitivity and scenario analysis to deal with the allocation and management of uncertain water resources by applying CVAR to the given problem. Barbaro and Bagajewicz32,33 demonstrated that the use of financial risk management can enhance the effectiveness of a proposed planning formulation by explicitly taking into account system uncertainty beyond an expectation evaluation.

The remainder of this article takes the following form. A description of the problem dealing with demand due date and amount uncertainty is presented, followed by an explanation of how conditional value-at-risk theory can be utilized to explicitly model the aforementioned forms of demand uncertainty within the nominal planning with production disaggregation model. An industrial case study is conducted to demonstrate the viability of the proposed approach, and the given results are compared to those generated by the robust operational PPDM.1 Finally, discussion along with some concluding remarks, which highlight the key contributions of this article (part II of the series), is presented. 2. Problem Description The large-scale multipurpose and multiproduct batch plant under investigation was studied previously by Janak et al.34,35 and Verderame and Floudas.4 The plant has the capability of producing hundreds of different products that belong to either of two product classes. For the made-to-order (bulk) product class, the customer supplies the plant with intermediate demand due dates that specify the amount of a product that needs to be sent to market on a given day. The supplied intermediate demand due dates have uncertainty associated with both the requested amount of product and the day of demand realization. It should be noted that the possibility exists for new bulk product demand due dates that are not related to an existing order to arise within the operational planning time horizon; however, this form of demand uncertainty is not taken into account at the planning level. Instead, it is more appropriate to handle the arrival of new customer orders at the scheduling level by means of online optimization techniques. As shown by Janak et al.,35 reactive scheduling can be performed so as to take into account new customer orders as they manifest themselves and adjust the existing plant schedule and production profile accordingly. The objective of the proposed operational planning model is to generate a plant production profile that explicitly accounts for the uncertainty associated with existing product orders, and the given production profile can be adjusted later at the scheduling level to take into account the arrival of new product orders by means of an online optimization technique. For an overview of online optimization theory and algorithms, see Grotschel et al.36 For the made-to-stock (packed) product class, the customer supplies the plant with weekly demand totals that should be satisfied by the end of the given week. For the packed products, there is uncertainty associated with the amount of demand requested at the end of each week. The state-task network (STN) for the plant can be seen in Figure 2 where the rectangles denote the various unit operations occurring within the plant and the circles denote the different states, with F, I, and P standing for feed, intermediate, and product, respectively. A thorough description of the various production recipes for the given classes of final states can be found in the work of Janak et al.34,35 The objective of this article is to extend the nominal PPDM by means of conditional value-at-risk theory to take into account the aforementioned forms of demand uncertainty at the operational planning level and determine the plant’s daily production profile, which should be a tight upper bound on the production capacity of the plant.

262


Figure 2. Plant state-task network (STN).

Property 1 indicates that the minimization of Fω(x,ξ) in terms

3. Demand Uncertainty via Conditional Value-at-Risk Theory The objective of this article is to replace the packed and bulk demand constraints present in the nominal operational PPDM4 with conditional value-at-risk (CVAR) constraints that will, in turn, aim at addressing demand uncertainty regardless of the type of amount and date distributions related to the uncertain demand due date parameters. Prior to addressing the specific application of conditional value-at-risk theory to demand due date and amount uncertainty, the general approach of conditional value-at-risk theory is presented with a focus on its theoretical underpinnings. 3.1. Theoretical Development. When applying conditional value-at-risk theory, as well as value-at-risk theory, one must first define a loss function, f(x,u), where x represents a decision vector and u is a stochastic vector representing uncertainty and having a probability distribution of p(u). Having generically defined the loss function, the value-at-risk problem, which determines the minimum value-at-risk, ξω, such that the probability of the loss function exceeding ξω is less than or equal to (1 - ω), can be formulated as ξω ) minξ ξ s.t.

∫

f(x,u)eξ

p(u) du g ω

(1)

where ω is a user-specified confidence level with a value between 0 and 1. Having defined the ω-value-at-risk, ξω, by means of eq 1, the conditional expectation of the loss function being greater than ξω is expressed as Φω(x) )

1 (1 - ω)

∫

f(x,u)>ξω

f(x, u) p(u) du

(2)

Following the work of Rockafellar and Uryasev,18,19 a special function, Fω(x,ξ), where ξ represents a given loss level, can be defined (eq 3). Fω(x,ξ) has the properties given by eqs 4a-4d, and because of these properties, it suffices to use Fω(x,ξ) instead of Φω(x) when applying conditional valueat-risk theory. Fω(x, ξ) ) ξ +

1 (1 - ω)

∫

u∈Rn

max[0, f(x, u) - ξ] p(u) du (3)

Property 1: Φω(x) ) min Fω(x, ξ)

(4a)

Property 2: ξω ) left end point of arg min Fω(x, ξ)

(4b)

Property 3: Φω(x) ) Fω[x, ξω(x)]

(4c)

Property 4: min Φω(x) ) min Fω(x, ξ)

(4d)

ξ

ξ

x

x,ξ

of ξ transforms the special function, Fω(x,ξ), into the conditional expectation of the loss function, Φω(x). Property 2 states that the ω-value-at risk can be ascertained by means of Fω(x,ξ). Property 3 conveys the fact that Fω(x,ξ) is equal to Φω(x) when the generic loss variable, ξ, is replaced by the ω-value-at-risk [ξω(x)], which can be expressed as a function of x. Property 4 demonstrates that the minimization of Φω(x) in terms of x is equivalent to the minimization of Fω(x,ξ) in terms of x and ξ. For a more detailed analysis of the aforementioned four properties, consult the work of Rockafellar and Uryasev.18,19 Fω(x,ξ) can be approximated by replacing the integral by a summation over a number of scenarios, where sc stands for a given scenario and πsc is the probability that a given scenario will occur F˜ω(x, ξ) ) ξ +

1 (1 - ω)

∑π

sc

max[0, f(x, sc) - ξ]

sc

(5) The loss function scenarios as denoted by f(x,sc) can be determined in part by sampling the underlying probability distribution of the stochastic vector u. Assuming that the scenarios have equal probabilities of occurring, eq 5 can be rewritten in the form ∼

Fω(x, ξ) ) ξ +

1 (1 - ω)|Sc|

∑ max[0, f(x, sc) - ξ] sc

(6) where |Sc| is the number of scenarios. To include eq 6, representing the scenario-based conditional value-at-risk, in a generic mixed-integer linear programming problem, the inner maximization term must be reformulated as


z(sc) g f(x, sc) - ξ ∀sc

(7)

where z(sc) is a positive variable representing the adjusted loss function for a given scenario (sc). With the aforementioned reformulation (eq 7), the scenariobased conditional value-at-risk [i.e., F∼ω(x,ξ)] can be restricted by means of eq 8 to be below some threshold defined by the parameter risk_aversion. Overall, eqs 7 and 8 are used to represent and bound the scenario-based conditional value-atrisk within an optimization model. The inclusion of eqs 7 and 8 in an optimization model helps to restrict the evaluation of the system’s variables in accordance with the level of risk aversion expressed by the parameter risk_aversion, which is a user-specified entity representing the user’s tolerance of risk. If the user has a low risk tolerance, then the value of risk_aversion should be similarly low when compared to that of a user with a high risk tolerance. ξ+

1 (1 - ω)|Sc|

∑ z(sc) e risk_aversion

(8)

sc

3.2. Nomenclature. The indices, sets, parameters, and variables listed in this section are required when modeling demand uncertainty in the operational planning with production disaggregation model, which was presented in part I1 and the work of Verderame and Floudas4 and is summarized in the Appendix. Indices d ) days m ) months s ) states sc ) scenarios w ) weeks Sets D ) days in the overall planning horizon M ) months in the overall planning horizon Sc ) demand due date parameters scenarios Sp ) states which are final products Spb ) states that are bulk final products Spp ) states that are packed final products W ) weeks in the overall planning horizon Parameters initial_mm ) first day of month m initial_ww ) first day of week w prices ) price for state s rs,d ) demand for state s on day d (intermediate demand due date parameter) r_scs,d,sc ) demand for state s on day for scenario sc (intermediate demand due date scenario parameter) term_mm ) last day of month m term_ww ) last day of week w δ ) user-specified risk aversion parameter ω ) user-specified probability level Continuous Variables sl1a(s,d) ) daily underproduction demand slack variable for bulk products sl1b(s,d) ) daily overproduction demand slack variable for bulk products sl2a(s,w) ) weekly underproduction demand slack variable for packed products tot(s,d) ) amount of state s produced on day d zbulk_over(m,sc) ) bulk overproduction loss function variable for month m and scenario sc zbulk_under(w,sc) ) bulk underproduction loss function variable for week w and scenario sc

263

zpack_over(m,sc) ) packed overproduction loss function variable for month m and scenario sc zpack_under(m,sc) ) packed underproduction loss function variable for month m and scenario sc ξbulk_over(m) ) acceptable bulk overproduction loss level for month m ξbulk_under(w) ) acceptable bulk underproduction loss level for week w ξpack_over(m) ) acceptable packed overproduction loss level for month m ξpack_under(m) ) acceptable packed underproduction loss level for month m 3.3. Bulk Product Demand Uncertainty. Bulk products have two forms of demand uncertainty that need to be taken into account. Both the realized date and the required amount associated with each bulk product demand due date parameter, rs,d, defined in part I1 (see also the Appendix) are uncertain. The realized due date and amount related to each demand due date parameter follow user-specified probability distributions. For a given scenario (sc), a demand due date parameter scenario, r_scs,d,sc, is generated by simultaneously sampling the date distribution and amount distribution associated with each demand due parameter (rs,d). Once all of the requisite scenarios have been generated for each demand due date parameter, the following bulk product daily underproduction and overproduction CVAR demand constraints can be generated. To maintain computational tractability, both the underproduction and overproduction loss functions represent an aggregation of the nominal daily underproduction and overproduction demand constraints present in the nominal PPDM, as shown in eqs 9 and 10, respectively. Instead of writing underproduction and overproduction loss functions for each bulk product on a daily basis, the underproduction and overproduction loss functions shown in eqs 11 and 12, respectively, are written on weekly and monthly bases, respectively, for the entire bulk product class. The bulk product underproduction loss function is written on a weekly basis so as to stress the importance of satisfying customer orders over violating inventory restrictions. The statespecific terms found in the bulk product underproduction loss function (eq 11) are weighted by the respective price of the bulk product in order to emphasize the importance of meeting customer demand for profitable products. The state-specific terms found in the bulk product overproduction loss function (eq 12) are not weighted by their respective prices because the inventory cost for each bulk product is the same regardless of how expensive the product is on the open market. The underproduction loss function for a given week (w) and scenario (sc) takes on a positive value when the total weighted production of bulk product is less than the total weighted demand for the scenario, and the overproduction loss function for a given month (m) and scenario (sc) takes on a positive value when the total production is greater than the total demand for the scenario. It should be noted that the proposed aggregation scheme has the potential of compromising solution quality when compared to a disaggregated model; however, as demonstrated in the computational study (section 5), the proposed framework generates a favorable production profile when compared to the production profile generated by the robust operational PPDM, which is a disaggregated demand constraint model. d')d

d')d

d')1

d')1

∑ tot(s, d') + sl1a(s, d)g ∑ r

s,d'

∀s|s ∈ Sp ∩ Sbp, ∀d (9)

264


d')d

∑

d')d

tot(s, d') - sl1b(s, d)e

d')1

∑r

s,d'

∀s|s ∈ Sp ∩ Sbp, ∀d

d')1

(10) where the intermediate demand due date parameter rs,d represents the amount of state s required on day d and the variable tot(s,d) signifies the amount of state s produced on day d.

∑

fbulk_under(w, sc) )

∑

prices[r_scs,d,sc -

s∈Sp∩Spb initial_wwedeterm_ww

tot(s, d)] ∀w, ∀sc

fbulk_over(m, sc) )

∑

∑

(11)

[tot(s, d) -

s∈Sp∩Spb initial_mmedeterm_mm

r_scs,d,sc] ∀m, ∀sc

(12)

where fbulk_under(w,sc) and fbulk_over(m,sc) are the bulk product underproduction and overproduction loss functions, respectively, for a given scenario (sc). The bulk underproduction and overproduction loss functions are used in conjunction with the bulk underproduction and overproduction acceptable loss variables [ξbulk_under(w) and ξbulk_over(m), respectively] to define the loss function underproduction and overproduction variables [zbulk_under(w,sc) and zbulk_over(m,sc), respectively] as shown in eqs 13 and 14, respectively. If zbulk_under(w,sc) takes on a positive value, then the plant underproduced beyond the acceptable threshold for a given week (w) and scenario (sc), and similarly, if zbulk_over(m,sc) is positive for a given month (m) and scenario (sc), then the plant overproduced beyond the acceptable threshold for the given time period and scenario. zbulk_under(w, sc) g

∑

∑

prices[r_scs,d,sc -

a probabilistic confidence level with bounds of 0 and 1. Equations 13-16 replace the nominal daily underproduction and overproduction demand constraints (eqs 9 and 10) in the operational planning with production disaggregation model so as to take into account bulk product demand uncertainty by means of conditional value-at-risk theory. 3.4. Packed Product Demand Uncertainty. Having presented the bulk product underproduction and overproduction CVAR demand constraints, we now reformulate the packed product demand constraints (eqs 17 and 18) into their respective overproduction and underproduction CVAR demand constraints. For the packed product class, the date of demand realization is considered to be deterministic in nature. For a given scenario (sc), the demand due date parameter scenario, r_scs,d,sc, is generated by sampling the underlying probability distribution that the given demand due date parameter’s (rs,d) amount obeys. Having indicated how the requisite scenarios for each demand due date parameter are to be generated, the packed product monthly underproduction and overproduction loss functions can be defined as in eqs 19 and 20, respectively

∑

tot(s, d) + sl2a(s, w) g

determ_ww

∑

zbulk_over(m, sc) g

∑

∑

5 6 initial_m

∑

bulk_under(w)

+

w

1 (1 - ω)|Sc|

[tot(s, d) -

fpack_under(m, sc) )

bulk_over(m)

m

+

1 (1 - ω)|Sc|

d

bulk_over(m, sc)

m

sc

δ

e

∑ ∑r

s,d

s∈Sp∩Spb

∑

prices[r_scs,d,sc -

∑

(19)

[tot(s, d) r_scs,d,sc] ∀m, ∀sc

(14)

bulk_under

∑ ∑z

∑

s∈Sp∩Spp initial_mmedeterm_mm

s s,d

∑ξ

∑

tot(s, d)] ∀m, ∀sc

fpack_over(m, sc) )

sc

s∈Sp∩Spb

(18)

(13)

(w, sc) e ∑ ∑z (15) δ ∑ ∑ price r w

∑

initial_mmedeterm_mm


Once the requisite loss function variables [zbulk_under(w,sc) and zbulk_over(m,sc)] have been defined, the encapsulated CVAR constraints for the underproduction and overproduction demand constraints can be written in the respective forms

∑ξ

rs,d ∀s|s ∈ Sp ∩ Spp, ∀m

tot(s, d)e

medeterm_mm


r_scs,d,sc] - ξbulk_over(m) ∀m, ∀sc

(17)

determ_ww


tot(s, d)] - ξbulk_under(w) ∀w, ∀sc

rs,d ∀s|s ∈ Sp ∩ Spp, ∀w

(16)

d

The left-hand side of each encapsulated CVAR constraint represents the scenario-based approximation of the conditional value-at-risk for the particular loss function, and the right-hand side is the acceptable loss threshold as specified by the user, which is partially dependent on the historical demand distribution; the price of each of the products under investigation; and the user-specified parameter δ, which can take on any value between 0 and 1 and is used to express the user’s level of risk aversion. As defined in section 3.1, the parameter ω represents

(20)

where fpack_under(m,sc) and fpack_over(m,sc) are the packed product underproduction and overproduction loss functions, respectively, for a given scenario (sc). The packed product loss functions represent an aggregation of the original packed product demand constraints on both the time and state levels in order to maintain computational tractability, and the terms comprising the two loss functions are weighted by their respective packed product price in accordance with the same standards used to generate the bulk product loss functions. Once again, the proposed aggregation scheme has the potential of compromising solution quality; however, the computational study (section 5) validates the demand aggregation approach. The packed product underproduction and overproduction loss functions (fpack_under(m,sc) and fpack_over(m,sc), respectively], along with with packed product underproduction and overproduction acceptable loss variables [ξpack_under(m) and ξpack_over(m), respectively], are used to generate a lower bound on the packed product underproduction and overproduction loss function variables [zpack_under(m,sc) and zpack_over(m,sc), respectively] as shown in eqs 21 and 22. zpack_under(m, sc) g

∑

∑

prices[r_scs,d,sc -


tot(s, d)] - ξpack_under(m) ∀m, ∀sc

(21)


zpack_over(m, sc) g

∑

∑

265

[tot(s, d) -


r_scs,d,sc] - ξpack_over(m) ∀m, ∀sc

(22)

The underproduction and overproduction loss function variables are further constrained by the encapsulated underproduction and overproduction CVAR demand constraints shown in eqs 23 and 24, respectively.

∑ξ

pack_under(m)

+

m

1 (1 - ω)|Sc|

(m, sc) e ∑ ∑z (23) δ ∑ ∑ price r pack_under

m

sc

s s,d

s∈Sp∩Spp

∑ξ

pack_over(m)

m

+

1 (1 - ω)|Sc|

d

∑ ∑z

pack_over(m, sc)

m

sc

δ

e

∑ ∑r

s,d

s∈Sp∩Spp

(24)

d

The scenario-based approximation of the conditional valueat-risk represented by the left-hand side of the underproduction and overproduction CVAR demand constraints (eqs 23 and 24, respectively) should be less than or equal to the maximum allowable risk to which the user is willing to be exposed. This is expressed as a fraction of the potential packed profit for the underproduction case and a fraction of the potential packed production for the overproduction case. Equations 21-24 replace eqs 17 and 18 so that the operational PPDM explicitly takes into account packed product demand uncertainty by means of conditional value-at-risk theory. Overall, eqs 13-16 and eqs 21-24 have replaced the nominal demand constraints represented by eqs 9, 10, 17, and 18 in the formulation of the operational CVAR-PPDM. Also, the objective function for the operational CVAR-PPDM now contains only the reactor underutilization penalization and gross profit maximization terms as a result of the elimination of all of the demand slack variables present in the nominal operational PPDM. 3.5. Additional Production Constraints. Because of computational tractability issues, the CVAR demand constraints have been written in an aggregated time basis and in terms of bulk and packed product classes as opposed to individual states. Additional production constraints need to be added to the operational CVAR-PPDM in order to ensure an even production of all products having customer orders within the time horizon. Equations 25 and 26 bound the production of each final product based on the minimum and maximum demand due date parameter scenarios for the state s on day d. Equation 27 mandates that at least 80% of the total demand for a given final product state s be satisfied within the time horizon.

∑ tot(s, d) e ∑ max r_sc

s,d,sc

∀s|s ∈ Sp

(25)

∑ tot(s, d) g ∑ min r_sc

s,d,sc

∀s|s ∈ Sp

(26)

d

d

d

d

sc

sc

∑ tot(s, d) g 54 ∑ r

s,d

d

∀s|s ∈ Sp

(27)

d

The complete CVAR-PPDM model is summarized in the Appendix. 4. Sample Average Approximation The proposed approach to modeling demand uncertainty explicitly at the operational planning level utilizes scenario-

Figure 3. Sample average approximation algorithm. Table 1. Progress of the Sample Average Approximation Algorithm pack_under

pack_over

bulk_under

bulk_over

Iteration 1 average std dev z

93711.8 0.0 ∞

979.8 121.9 -0.035

39648.9 9249.2 -0.278

595.8 6.0 12.028

7115.9 7095.5 4.223

587.7 5.9 13.583

17723.9 7894.1 2.424

592.9 4.6 16.370

22316.1 11130.0 1.326

591.2 5.8 13.205

32260.7 8988.0 0.536

596.2 4.8 14.982

6854.6 7213.9 4.190

589.6 5.4 14.507

Iteration 2 average std dev z

87464.3 0.0 ∞

917.7 130.1 0.444 Iteration 3

average std dev z

87464.6 54.6 114.422

847.7 124.0 1.030 Iteration 4

average std dev z

87464.3 0.0 ∞

787.8 128.7 1.458 Iteration 5

average std dev z

87484.1 489.9 12.711

712.2 116.1 2.269

average std dev z

87473.0 334.2 18.670

713.4 127.0 2.063

Iteration 6

based conditional value-at-risk theory. The issue of adequately sampling the given probability space and ensuring that the obtained solution is feasible when considering the complete uncertainty space for the bulk and packed customer orders arises as a result. To maintain computational tractability, only a finite number of scenarios can be considered within the operational CVAR-PPDM, and therefore, an iterative approach needs to be adopted to guarantee that the final production profile satisfies the conditional value-at-risk demand constraints when considering the entire uncertainty space. The sample average approximation (SAA) developed by Wang and Ahmed37 provides such an iterative framework for determining a valid, feasible upper bound on the original minimimization problem when considering the entire uncertainty space. Figure 3 presents an overview of the given SAA algorithm. The first step in the algorithm is to generate a finite set of scenarios for both the packed and bulk demand due date

266


parameters and apply the operational CVAR-PPDM model to obtain a candidate production profile and acceptable loss function variable values. A larger, independent set of demand due date parameter scenarios is then considered, and the averages and standard deviations of the CVAR demand constraints’ left-hand side are calculated using the larger set of demand due date parameter scenario values along with the obtained production profile and loss function variable values. A z transformation is performed for each of the CVAR demand constraints by subtracting the average left-hand-side value of the CVAR demand constraint from the original righthand side representing the user’s degree of risk aversion, and the given quantity is divided by the standard deviation of the left-hand side for the CVAR demand constraint. If the z-transformed quantity is large, then it is unlikely in a probabilistic sense that the given CVAR demand constraint would be violated by the production profile and acceptable loss function variables values currently under evaluation. As a termination criterion, the z-transformed quantity for each of the CVAR demand constraints needs to be greater than or equal to 2, indicating that the right-hand side is two standard deviation greater than the average of the left-hand side. If the given termination criterion is not satisfied, then the operational CVAR-PPDM is solved again with another finite set of demand due date parameter scenarios; however, the right-hand-side values of those CVAR demand constraints corresponding to the z-transformed quantities that were not greater than or equal to two are reduced in order to encourage the generation of a production profile and acceptable loss variable values which will satisfy the termination criterion when considering the original right-hand side-side values and a larger, independent set of demand due date parameter scenarios. The averages, standard deviations, and z-transformed quantities for the CVAR demand constraints can be calculated by means of eqs 28-44. Bulk Underproduction LHS_bulk_undersc )

bulk underproduction case, and z_trans_bulk_under is the z-transformed quantity for the bulk underproduction case. Bulk Overproduction LHS_bulk_oversc ) 1 (1 - w)

+

∑ max(0, { ∑

∑

×


m

[tot(s, d) - r_scs,d,sc] - ξbulk_over(m)})

(32)

∑ LHS_bulk_over

sc

LHS_bulk_over_avg )

sc

(33)

|Sc|

LHS_bulk_over_stdev )

∑ (LHS_bulk_over_avg - LHS_bulk_over

2 sc)

sc

(34)

(|Sc|-1)

z_trans_bulk_over ) Orig_RHS_bulk_over - LHS_bulk_over_avg LHS_bulk_over_stdev

(35)

where LHS_bulk_oversc is the left-hand-side value of the bulk overproduction CVAR demand constraint for scenario sc, LHS_bulk_over_avg is the average left-hand-side value for the bulk overproduction CVAR demand constraint, LHS_bulk_over_stdev is the standard deviation of the left-hand side for the bulk overproduction CVAR demand constraint, Orig_RHS_bulk_over is the original risk tolerance level for the bulk overproduction case, and z_trans_bulk_over is the z-transformed quantity for the bulk overproduction case. Packed Underproduction LHS_pack_undersc )

∑ξ

pack_under(m)

+

m

∑ξ

bulk_under(w)

1 (1 - w)

+

∑ max(0, { ∑

∑

∑ max(0, { ∑

(28)

sc

LHS_pack_under_avg ) (29)

|Sc|

LHS_bulk_under_stdev ) 2

sc)

sc

(|Sc|-1)

(30)

∑ LHS_pack_under sc

(37)

|Sc|

LHS_pack_under_stdev )

∑ (LHS_pack_under_avg - LHS_pack_under

2

sc)

sc

(|Sc|-1)

(38)

z_trans_pack_under )

z_trans_bulk_under ) Orig_RHS_bulk_under - LHS_bulk_under_avg LHS_bulk_under_stdev

(36)

sc

∑ LHS_bulk_under

∑ (LHS_bulk_under_avg - LHS_bulk_under

×

prices[r_scs,d,sc - tot(s, d)] - ξpack_under(m)})

×

sc

LHS_bulk_under_avg )

∑


m


w

prices[r_scs,d,sc - tot(s, d)] - ξbulk_under(w)})

bulk_over(m)

m

w

1 (1 - w)

∑ξ

(31)

where LHS_bulk_undersc is the left-hand-side value of the bulk underproduction CVAR demand constraint for scenario sc, LHS_bulk_under_avg is the average left-hand-side value for the bulk underproduction CVAR demand constraint, LHS_bulk_under_stdev is the standard deviation of the left-hand side for the bulk underproduction CVAR demand constraint, Orig_RHS_bulk_under is the original risk tolerance level for the

Orig_RHS_pack_under - LHS_pack_under_avg LHS_pack_under_stdev

(39)

where LHS_pack_undersc is the left-hand-side value of the packed underproduction CVAR demand constraint for scenario sc, LHS_pack_under_avg is the average left-hand-side value for the packed underproduction CVAR demand constraint, LHS_pack_under_stdev is the standard deviation of the lefthand-side for the packed underproduction CVAR demand constraint, Orig_RHS_pack_under is the original risk tolerance level for the packed underproduction case, and z_trans_pack-


_under is the z-transformed quantity for the packed underproduction case. Packed Overproduction LHS_pack_oversc )

∑ξ

pack_over(m)+

m

1 (1 - ω)

∑ max(0, { ∑ m

∑

[tot(s, d) -


r_scs,d,sc] - ξpack_over(m)})

(40)

∑ LHS_pack_over

sc

LHS_pack_over_avg )

sc

(41)

|Sc|

LHS_pack_over_stdev )

∑ (LHS_pack_over_avg - LHS_pack_over

sc)

2

sc

(|Sc|-1)

(42)

z_trans_pack_over ) Orig_RHS_pack_over - LHS_pack_over_avg LHS_pack_over_stdev

(43)

where LHS_pack_oversc is the left-hand-side value of the packed overproduction CVAR demand constraint for scenario sc, LHS_pack_over_avg is the average left-hand-side value for the packed overproduction CVAR demand constraint, LHS_pack_over_stdev is the standard deviation of the left-hand side for the packed overproduction CVAR demand constraint, Orig_RHS_pack_over is the original risk tolerance level for the packed overproduction case, and z_trans_pack_over is the z-transformed quantity for the packed overproduction case. 5. Computational Study 5.1. CVAR Optimization Framework. To demonstrate the viability of the proposed approach, the operational CVARPPDM has been applied to the plant in question for a 3-month time horizon. For the detailed plant characteristics, the reader is directed to the work of Janak et al.34,35 The proposed operational CVAR-PPDM can be applied regardless of the due

Figure 4. CVAR aggregate bulk product planning period production totals.

267

date distribution and/or the amount distribution related to the intermediate demand due date parameters (rs,d) because the distributions come into play only when the demand due date parameter scenarios (r_scs,d,sc) are generated and have no effect on the CVAR-PPDM constraints. For the given computational study, however, the amounts related to the demand due parameters for both the bulk and packed product classes were assumed to follow a normal distribution with a mean of the nominal demand due date value and a standard deviation that is horizon-dependent. The time horizon was divided into seven planning periods, each having a time duration of approximately 2 weeks, in order to generate a tiering scheme for the standard deviation, σd, as shown in eq 44. For the bulk products, the due date realization was assumed to follow a uniform discrete distribution with bounds of (1 day from the nominal day, whereas the due date was considered to be deterministic for the packed products. The normal distribution case was chosen because it is perhaps the most likely distribution for the demand due date parameters’ amount to follow, given that demand is typically affected by several stochastic elements and, according to the central limit theorem,6 the sum of these stochastic elements should, in the limit follow, a normal distribution. σd ) [0.05 + 0.025(p - 1)]rs,d ∀d, ∀p|initial_pp e d e term_pp (44) where p stands for planning period and initial_pp and term_pp are the first and last days in the given planning period, respectively. All of the 531 supplied demand due parameters (rs,d) were considered to be uncertain, with the average supplied customer orders for bulk and packed products being 19.208 and 21.667 mu, respectively. One thousand scenarios were used in the application of the operational CVAR-PPDM, and 60 000 scenarios were utilized in the implemented SAA algorithm to ensure that the final planning level production profile represents a solution that is a valid, feasible upper bound. The reactor underutilization penalty coefficient γ, which is also found in the objective function of the operational PPDM presented in part I,1 was given a value of 10 to reflect the relative importance of the objective function term’s and to remain consistent with part I.1 The CVAR parameters ω and δ were assigned values of 0.01 and 0.15, respectively, to ensure the generation of a

268


Figure 5. CVAR aggregate bulk product planning period profit totals.

Figure 6. CVAR aggregate packed product planning period production totals.

Figure 7. CVAR aggregate packed product planning period profit totals.


269

Figure 8. CVAR Gantt chart for 13 bottleneck reactors for 3-month time horizon.

relatively conservative production profile. It should be noted that the planning model was implemented using GAMS 22.538 and solved with CPLEX39 on a 3.2 GHz Linux workstation. A relative optimality tolerance equal to 10% was used as the termination criterion, along with a 5-h time limit and an integer solution limit of 200 for the given planning model. The aforementioned solution tolerances were selected based on the objective of finding a high-quality solution within a finite amount of time so as to demonstrate the proposed framework’s ability to address industrially relevant operational planning problems. The executed model contained 113 199 equations, 111 858 continuous variables, and 41 490 binary variables. The operational CVAR-PPDM was applied six times within the

presented SAA algorithm before a valid, feasible upper bound was determined, and Table 1 shows the algorithm’s progression. Each time a given CVAR demand constraint did not meet the aforementioned termination criterion, the right-hand side of the given constraint was reduced by 1% from its previous value before the operational CVAR-PPDM was executed again. The conservative level of 1% was chosen to maintain model feasibility for every iteration of the SAA algorithm. The final production profile generated by the execution of the operational CVAR-PPDM within the SAA algorithm was supplied to the medium-term scheduling framework developed by Janak et al.34 in order to demonstrate that the supplied production profile provides a tight upper bound on the produc-

270


Figure 9. Robust aggregate bulk product planning period production totals.

Figure 10. Robust aggregate bulk product planning period profit totals.

tion capacity of the plant, because the given medium-term scheduling framework rigorously models the production capacity of the plant by means of utilizing the continuous-time unitspecific event-based short-term scheduling model developed by Floudas and co-workers.40-50 The application of the scheduling model to each scheduling subhorizon entailed the use of an optimality tolerance of 0.001% as the termination criterion, along with a 3-h time limit and an integer solution limit of 40, and the given medium-term scheduling model was implemented using GAMS 22.538 and solved with CPLEX39 on a 3.2 GHz Linux workstation. For a given scheduling subhorizon of 3 days, the medium-term scheduling model contained 6198 binary variables, 35 769 continuous variables, and 344 320 constraints. 5.1.1. Planning and Scheduling Optimization Results. Figures 4 and 5 show the planning period aggregate production and profit totals, respectively, for the bulk products at both the

planning and scheduling levels. Figures 6 and 7 provide analogous analysis for the packed products. Figures 4-7 demonstrate that the operational CVAR-PPDM provides a tight upper bound on the production capacity of the plant because of the relatively small and primarily positive difference between the planning level’s aggregate production/profit totals and the scheduling level’s production/profit totals for both bulk and packed product classes. The Gantt chart (Figure 8) generated for the 3-month time horizon also indicates that the 13 bottleneck reactors in the plant were highly utilized. 5.2. Robust Optimization Framework. To provide a basis of comparison for the operational CVAR-PPDM, the robust operational PPDM computational study presented in part I1 of this series of articles was used. For a detailed explanation of the robust operational PPDM computational study, please see part I.1


271

Figure 11. Robust aggregate packed product planning period production totals.

Figure 12. Robust aggregate packed product planning period profit totals.

5.2.1. Planning and Scheduling Optimization Results. For ease of comparison, the planning and scheduling optimization results for the robust operational PPDM computational study outlined in part I1 are presented again here. Figures 9 and 10 show the bulk class production and profit planning period totals, and Figures 11 and 12 provide analogous treatment for the packed product class. Together with the Gantt chart in Figure 13, Figures 9-12 demonstrate that the robust operational PPDM also provides a relatively tight upper bound on the production capacity of the plant in question. Table 2 presents a comparison of the differences between the planning and scheduling planning period production and profit totals for the operational CVARPPDM and the robust operational PPDM applications. This table demonstrates that, even though both planning approaches provide acceptable upper bounds on plant production capacity, the robust operational PPDM, on average, provides a tighter upper bound on packed product production whereas the

operational CVAR-PPDM, on average, provides a tighter upper bound on bulk production. 6. Robust Optimization Technique versus Conditional Value-at-Risk Theory The robust optimization technique and the scenario-based conditional value-at-risk approach presented in parts I1 and II, respectively, of this series of articles represent two different ways of explicitly addressing demand due date and amount uncertainty at the operational planning level. The robust optimization technique has the ability to maintain the same functional form for the underproduction and overproduction demand constraints without any aggregation. On the other hand, the scenario-based conditional value-at-risk approach requires aggregation of the demand constraints on both the time and state level; however, as evidenced by the viable production profile

272


Figure 13. Robust Gantt chart for 13 bottleneck reactors for 3-month time horizon.

Table 2. CVAR and Robust Production/Profit Difference Comparison robust bulk

CVAR

packed

bulk

packed

Production average std dev

85.887 56.252

195.104 91.626

73.565 75.536

212.247 146.362

Profit average std dev

4334.94 3156.45

19433.67 8593.24

4802.42 4886.68

25667.90 19341.54

generated by the CVAR-PPDM, the chosen levels of demand aggregation helped to maintain computational tractability while still allowing for a high level of solution quality. The presented scenario-based conditional value-at-risk approach has the ability to adapt to any probability distribution under consideration without having to change model constraints, whereas the application of the robust optimization technique is distributiondependent, requiring model reformulation. A potentially detrimental characteristic of the operational CVAR-PPDM is the increased computational time caused by the application of the presented sample average approximation algorithm, which is


used to ensure that the generated production profile represents a solution that is a valid, feasible upper bound. On the other hand, the robust operational planning model needs to be applied only once to generate the desired production profile, which can then be supplied to the scheduling level. The planning and scheduling optimization results indicate that both proposed planning models are viable alternative approaches. Ultimately, both operational planning approaches will need to be integrated within a planning and scheduling framework, and it will be the objective of subsequent research to determine which of the presented operational planning models effectively and efficiently interfaces with the medium-term scheduling model developed by Janak et al.34,35 7. Conclusions In this article (part II of a series), a framework based on the novel application of conditional value-at-risk theory to the problem of demand due date and amount uncertainty at the operational planning level has been presented. The operational CVAR-PPDM in conjunction with a SAA algorithm has been demonstrated to provide a production profile that is a tight upper bound on the production capacity of the plant in question. The operational CVAR-PPDM has also been compared to the robust operational PPDM presented in part I of the series,1 and it has been demonstrated that both approaches represent viable alternatives and are novel contributions to the field of operational planning under demand uncertainty. It will be the goal of subsequent research to apply both planning models in a integrated planning and scheduling framework in order to decern which planning approach is preferable. Acknowledgment The authors gratefully acknowledge financial support from the National Science Foundation (CMMI-0856021).

Cleanup ) batch changeover downtime Fixedtimeus ) batch processing time for state s in reactor u H ) lower bound on aggregate reactor processing time Hstd ) storage duration for day d initial_mm ) first day of month m prices ) price for state s rs,d ) demand for state s on day d (intermediate demand due date parameter) term_mm ) last day of month m term_ww ) last day of week w Tud ) total available processing time in reactor u on day d Continuous Variables Area(s,d) ) inventory area for state s on day d sl1a(s,d) ) daily underproduction demand slack variable for bulk products sl1b(s,d) ) daily overproduction demand slack variable for bulk products sl2a(s,w) ) weekly underproduction demand slack variable for packed products slt(u) ) processing time underutilization slack variable for reactor u tot1(u,s,d) ) amount of state s produced on day d in reactor u tot(s,d) ) amount of state s produced on day d Tu(u,d) ) total time reactor u is utilized on day d Binary Variable y(u,s,d) ) assigns the production of state s to reactor u on day d A.1.2. PPDM Model.

{ ∑∑ [∑∑

min R

s∈Sp∩Spb

sl1a(s, d)prices + β

d

Area(s, d) +

s∈Sp∩Spb

Indices d ) days m ) months s ) states u ) reactors w ) weeks Sets D ) days in the overall planning horizon M ) months in the overall planning horizon Sp ) states that are final products Spb ) states that are bulk final products Spp ) states that are packed final products Su ) states that can be produced in a given reactor Us ) reactors that can produce a given state s W ) weeks in the overall planning horizon Parameters Capmax us ) maximum batch size for state s in reactor u Capmin us ) minimum batch size for state s in reactor u

∑ ∑ sl2a(s, w)price

s∈Sp∩Spp

d

γ

Appendix A A.1. Operational PPDM. The operational PPDM requires the indices, sets, parameters, and variables listed in section A.1.1. Using the above notation, the operational PPDM is composed of the objective function and constraints presented in section A.1.2. For a detailed exposition of the nominal operational PPDM, see part I1 A.1.1. Nomenclature.

273

∑ slt(u) - ∑ tot(s, d)price u

s∈Sp

] }

s

d

+

s

(45)

subject to

∑

[Fixedtimeusy(u, s, d)] + Cleanup

s∈Sp∩Su

∑

y(u, s, d) )

s∈Sp∩Su

Tu(u, d) ∀u, ∀d Tu(u, d) e Tud ∀u, ∀d|d ) 1

(46) (47)

d)d'-1

∑

Tu(u, d') e Tud' +

[Tud - Tu(u, d)] ∀u, ∀d'|d' > 1

d)1

(48)

∑ T (u, d) + slt(u) g H u

∀u

(49)

d

Capminus y(u, s, d) e tot1(u, s, d) e Capmaxus y(u, s, d) ∀u, ∀s|s ∈ Sp, ∀d

∑ tot1(u, s, d) ) tot(s, d)

∀s|s ∈ Sp, ∀d

(50) (51)

u∈Us d')d

d')d

d')1

d')1

∑ tot(s, d') + sl1a(s, d) g ∑ r

s,d'

∀s|s ∈ Sp ∩ Sbp, ∀d (52)

274


d')d

∑

d')d

tot(s, d') - sl1b(s, d) e

d')1

∑r

∀s|s ∈ Sp ∩ Sbp, ∀d

s,d'

∑

zpack_under(m, sc) g

d')1

zpack_over(m, sc) g

∑

tot(s, d) + sl2a(s, w) g

∑

rs,d ∀s|s ∈ Sp ∩

∀w

∑ξ (55)

∑

+

m

1 (1 - ω)|Sc|

determ_ww

5 6 initial_m

[tot(s, d) -

r_scs,d,sc] - ξpack_over(m) ∀m, ∀sc

pack_under(m)

Spp,

∑

∑

pack_under

m

sc

s s,d

tot(s, d) e

∑

∑ξ

pack_over(m)

rs,d ∀s|s ∈ Sp ∩

Spp,

∀m

(56)

+

m

1 (1 - ω)|Sc|

pack_over(m, sc)

m

sc

δ

∑ slt(u) - ∑ tot(s, d)price ]

(57)

s

s∈Sp

∑

y(u, s, d) )

Tu(u, d) e Tud ∀u, ∀d|d ) 1

(58) (59)

d)d'-1

∑

Tu(u, d') e Tud' +

[Tud - Tu(u, d)] ∀u, ∀d'|d' > 1

d)1

(60)

∑ T (u, d) + slt(u) g H

∀u

u

(61)

d

Capminus y(u, s, d) e tot1(u, s, d) e Capmaxus y(u, s, d) ∀u, ∀s|s ∈ Sp, ∀d

∑ tot1(u, s, d) ) tot(s, d)

∀s|s ∈ Sp, ∀d

(62) (63)

u∈Us

zbulk_under(w, sc) g

∑

∑

prices[r_scs,d,sc -


tot(s, d)] - ξbulk_under(w) ∀w, ∀sc

zbulk_over(m, sc) g

∑

∑

(64)

[tot(s, d) -


r_scs,d,sc] - ξbulk_over(m) ∀m, ∀sc

∑ξ

bulk_under(w)

+

w

1 (1 - ω)|Sc|

(65)

(w, sc) e ∑ ∑z (66) δ ∑ ∑ price r bulk_under

w

sc

s s,d

s∈Sp∩Spb

∑ξ

bulk_over(m)

m

(72)

∑ tot(s, d) g ∑ min r_sc

s,d,sc

∀s|s ∈ Sp

(73)

d

d

sc

sc

+

1 (1 - ω)|Sc|

d

∑ ∑z

bulk_over(m, sc)

m

sc

δ

e

∑ ∑r

s,d

s∈Sp∩Spb

d

∀s|s ∈ Sp

(74)

d

Literature Cited

s∈Sp∩Su

Tu(u, d) ∀u, ∀d

(71)

∀s|s ∈ Sp

d

s,d

[Fixedtimeusy(u, s, d)] + Cleanup

s,d

d

s,d,sc

d

d

s∈Sp∩Su

e

∑ tot(s, d) e ∑ max r_sc

∑ tot(s, d) g 54 ∑ r

subject to

∑

∑ ∑r

s∈Sp∩Spp

A.2. Operational CVAR-PPDM. The operational CVARPPDM consists of the following objective function and constraints:

u

d

∑ ∑z

initial_mmedeterm_mm

min[γ

(69)

(m, sc) e ∑ ∑z (70) δ ∑ ∑ price r s∈Sp∩Spp

medeterm_mm

(68)


(54)

determ_ww

prices[r_scs,d,sc -

tot(s, d)] - ξpack_under(m) ∀m, ∀sc

(53) Area(s, d) ) sl1b(s, d)Hstd ∀s|s ∈ Sp ∩ Sbp, ∀d

∑


(67)

(1) Verderame, P. M.; Floudas, C. A. Operational Planning of LargeScale Industrial Batch Plants under Demand Due Date and Amount Uncertainty: I. Robust Optimization Framework. Ind. Eng. Chem. Res. 2009, 48, 7214. (2) Lin, X.; Janak, S. L.; Floudas, C. A. A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty. Comput. Chem. Eng. 2004, 28, 1069. (3) Janak, S. L.; Lin, X.; Floudas, C. A. A new robust optimization approach for scheduling under uncertainty: II. Uncertainty with known probability distribution. Comput. Chem. Eng. 2007, 31, 171. (4) Verderame, P. M.; Floudas, C. A. Integrated Operational Planning and Medium-Term Scheduling for Large-Scale Industrial Batch Plants. Ind. Eng. Chem. Res. 2008, 47, 4845. (5) Verderame, P. M.; Floudas, C. A. Operational planning framework for multisite production and distribution networks. Comput. Chem. Eng. 2009, 33, 1036. (6) Petkov, S. B.; Maranas, C. D. Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty. Ind. Eng. Chem. Res. 1997, 36, 4864. (7) Ahmed, S.; Sahinidis, N. Robust Process Planning under Uncertainty. Ind. Eng. Chem. Res. 1998, 37, 1883. (8) Gupta, A.; Maranas, C. D.; McDonald, C. Mid-term supply chain planning under demand uncertainty: Customer demand satisfaction and inventory management. Comput. Chem. Eng. 2000, 24, 2613. (9) Gupta, A.; Maranas, C. D. A Two-Stage Modeling and Solution Framework for Multisite Midterm Planning under Demand Uncertainty. Ind. Eng. Chem. Res. 2000, 39, 3799. (10) Wu, D.; Ierapetritou, M. Heirarchical approach for production planning and scheduling under uncertainty. Chemical Engineering and Processing 2007, 46, 1129. (11) Colvin, M.; Maravelias, C. T. A stochastic programming approach for clinical trial planning in new drug development. Comput. Chem. Eng. 2008, 32, 2626. (12) You, F.; Grossmann, I. E. Design of responsive supply chains under demand uncertainty. Comput. Chem. Eng. 2008, 32, 3090. (13) You, F.; Wassick, J.; Grossmann, I. Risk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms. AIChE J. 2009, 55, 931. (14) Sahinidis, N. V. Optimization under uncertainty: State-of-the-art and opportunities. Comput. Chem. Eng. 2004, 28, 971. (15) Li, Z.; Ierapetritou, M. Process scheduling under uncertainty: Review and challenges. Comput. Chem. Eng. 2008, 32, 715. (16) Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D. Coherent Measure of Risk. Math. Finance 1999, 9, 203.

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010 (17) Ogryczak, W. Stochastic Dominance Relation and Linear Risk Measures. In Financial ModellingsProceedings of the 23rd Meeting of the EURO Working Group on Financial Modelling; Skulimowski, A. M. J., Ed.; Progress & Business Publishers: Cracow, Poland, 1999; pp 191-212. (18) Rockafellar, R. T.; Uryasev, S. Optimization of Conditional Valueat-Risk. J. Risk 2000, 2, 21. (19) Rockafellar, R. T.; Uryasev, S. Conditional value-at-risk for general loss distributions. J. Banking Finance 2002, 26, 1443. (20) Krokhmal, P.; Uryasev, S.; Zrazhevsky, G. Risk Management for Hedge Fund Portfolios: A Comparative Analysis of Linear Rebalancing Strategies. J. AlternatiVe InVestments 2002, 5, 10. (21) Mansini, R.; Ogryczak, W.; Speranza, M. G. Conditional value at risk and related linear programming models for portfolio optimization. Ann. Oper. Res. 2007, 152, 227. (22) Markowitz, H. M. Portfolio Selection. J. Finance 1952, 37, 77. (23) Staumbaugh, F. Risk and Value-at-risk. Eur. Manage. J. 1996, 14, 612. (24) Pritzker, M. Evaluating Value-at-Risk Methodologies. J. Financial SerVices Res. 1997, 12, 201. (25) Jorion, P. Value-at-Risk: The New Benchmark for Controlling Market Risk; McGraw-Hill: New York, 1997. (26) Duffie, D.; Pan, J. An Overview of Value-at-Risk. J. DeriVatiVes 1997, 4, 7. (27) Jabr, R. A. Robust Self-Scheduling under Price Uncertainty Using Conditional Value-at-Risk. IEEE Trans. Power Syst. 2005, 20, 1852. (28) Cabero, J.; Baillo, A.; Cerisola, S.; Ventosa, M.; Alcalde, A. G.; Peran, F.; Relano, G. author. IEEE Trans. Power Syst. 2005, 20, 1379. (29) Gotoh, J.; Takano, Y. Newsvendor solutions via conditional valueat-risk minimization. Eur. J. Oper. Res. 2007, 179, 80. (30) Tsang, K. H.; Samsati, N. J.; Shah, N. Capacity Investment Planning for Multiple Vaccines under Uncertainty 2: Financial Risk Analysis. Food Bioprod. Process. 2007, 85, 129. (31) Yamout, G. M.; Hatfield, K.; Romeijn, H. E. Comparison of new conditional value-at-risk-based manangement models for optimal allocation of uncertain water supplies. Water Resour. Res. 2007, 43, W07430. (32) Barbaro, A.; Bagajewicz, M. J. Managing Financial Risk in Planning under Uncertainty. AIChE J. 2004, 50, 963. (33) Barbaro, A.; Bagajewicz, M. J. Use of Inventory and Option Contracts to Hedge Financial Risk in Planning under Uncertainty. AIChE J. 2004, 50, 990. (34) Janak, S. L.; Floudas, C. A.; Kallrath, J.; Vormbrock, N. Production Scheduling of a Large-Scale Industrial Batch Plant. I. Short-Term and Medium-Term Scheduling. Ind. Eng. Chem. Res. 2006, 45, 8234. (35) Janak, S. L.; Floudas, C. A.; Kallrath, J.; Vormbrock, N. Production Scheduling of a Large-Scale Industrial Batch Plant. II. Reactive Scheduling. Ind. Eng. Chem. Res. 2006, 45, 8253.

275

(36) Gro¨tschel, M., Krumke, S. O., Rambau J., Eds.; Online Optimization of Large Scale Systems; Springer: Berlin, 2001. (37) Wang, W.; Ahmed, S. Sample Average Approximation of Expected Value Constrained Stochastic Programs. Oper. Res. Lett. 2008, 36, 515. (38) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R. GAMS: A User’s Guide; GAMS: San Francisco, CA, 2003. (39) CPLEX, ILOG CPLEX 9.0 User’s Mannual; ILOG, Inc.: Mountain View, CA, 2005. (40) Ierapetritou, M. G.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes. Ind. Eng. Chem. Res. 1998, 37, 4341. (41) Ierapetritou, M. G.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 2. Continuous and Semicontinuous Processes. Ind. Eng. Chem. Res. 1998, 37, 4360. (42) Ierapetritou, M. G.; Hene’, T. S.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 3. Multiple Intermediate Due Dates. Ind. Eng. Chem. Res. 1999, 38, 3446. (43) Lin, X.; Floudas, C. Design, Synthesis and Scheduling of Multipurpose Batch Plants via an Effective Continuous-Time Formulation. Comput. Chem. Eng. 2001, 25, 665. (44) Lin, X.; Floudas, C. A.; Modi, S.; Juhasz, N. M. Continuous-Time Optimization Approach for Medium-Range Production Scheduling of a Multiproduct Batch Plant. Ind. Eng. Chem. Res. 2002, 41, 3884. (45) Janak, S. L.; Lin, X.; Floudas, C. A. Enhanced Continuous-Time Unit-Specific Event-Based Formulation for Short-Term Scheduling of Multipurpose Batch Processes: Resource Constraints and Mixed Storage Policies. Ind. Eng. Chem. Res. 2004, 43, 2516. (46) Shaik, M. A.; L, J. S.; Floudas, C. A. Continuous-Time Models for Short-Term Scheduling of Multipurpose Batch Plants: A Comparative Study. Ind. Eng. Chem. Res. 2006, 45, 6190. (47) Shaik, M. A.; Floudas, C. A. Unit-specific event-based continuoustime approach for short-term scheduling of batch plants using RTN framework. Comput. Chem. Eng. 2008, 32, 260. (48) Janak, S. L.; Floudas, C. A. Improving unit-specific event based continuous-time approaches for batch processes: Integrality gap and task splitting. Comput. Chem. Eng. 2008, 32, 913. (49) Shaik, M. A.; Floudas, C. A.; Kallrath, J.; Pitz, H. Production scheduling of a large-scale industrial continuous plant: Short-term and medium-term scheduling. Comput. Chem. Eng. 2009, 33, 670. (50) Shaik, M. A.; Floudas, C. A. Novel Unified Modeling Approach for Short-Term Scheduling. Ind. Eng. Chem. Res. 2009, 48, 2947.

ReceiVed for reView June 5, 2009 ReVised manuscript receiVed August 28, 2009 Accepted September 4, 2009 IE900925K

Operational Planning of Large-Scale Industrial Batch Plants under

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