Robust Scenario Formulations for Strategic Supply Chain Optimization

Article pubs.acs.org/IECR

Robust Scenario Formulations for Strategic Supply Chain Optimization under Uncertainty Kyle McLean and Xiang Li* Department of Chemical Engineering, Queen’s University, 19 Division Street Kingston, Ontario, K7L 3N6 Canada S Supporting Information *

ABSTRACT: Strategic supply chain optimization (SCO) problems are often modeled as two-stage optimization problems, in which the first-stage variables represent decisions on the development of the supply chain and the second-stage variables represent decisions on the operations of the supply chain. When uncertainty is explicitly considered, the problem becomes an intractable infinite-dimensional optimization problem, which is usually solved approximately using a scenario or a robust approach. This article proposes a novel synergy of the scenario and robust approaches for strategic SCO under uncertainty. Two ̈ robust scenario formulation and affinely adjustable robust scenario formulation. It is formulations are developed, namely, naive shown that both formulations can be reformulated into tractable deterministic optimization problems if the uncertainty is bounded by the infinity norm and the uncertain equality constraints can be reformulated into deterministic constraints without any assumption about the uncertainty region. Case studies of a classical farm planning problem and an energy and bioproduct SCO problem demonstrate the advantages of the proposed formulations over the classical scenario formulation. The proposed formulations not only can generate solutions with guaranteed feasibility or indicate infeasibility of a problem, but also can achieve optimal expected economic performance with smaller numbers of scenarios.

1. INTRODUCTION Supply chain optimization (SCO) is a set of approaches utilized to efficiently integrate suppliers, manufacturers, warehouses, and stores, so that merchandise is produced and distributed in the right quantities, to the right locations, and at the right time, to minimize system-wide costs while satisfying service level requirements.1 SCO has emerged as a major research direction in the process systems engineering (PSE) community in the past decade; in the context of PSE, it is sometimes called enterprise-wide optimization if the emphasis is placed on the manufacturing stage.2 SCO problems can be categorized into three levels of problems, namely, strategic, tactical, and operational problems, that are associated with the design, long-term/midterm planning, and short-term operation, respectively, of supply chains. This article focuses on strategic SCO problems, specifically, the optimal design of supply chain networks. As in other PSE areas, mathematical programming has been the major tool for SCO research; for strategic SCO problems, mixed-integer linear programming (MILP) has been widely employed for systematic solution. Integer variables are often used to represent design decisions, such as the number and locations of manufacturing sites or distribution centers and network connections. Continuous variables can represent either design decisions, such as the capacities of manufacturing equipment and warehouses, or long-term operational decisions that are needed for the estimation of supply chain performance, such as material allocation and production planning. Linear models are used for two reasons. The first reason stems from the belief that linear models are often acceptable for problems at the strategic level, that is, nonlinear models with higher fidelity might not be significantly better than linear models for decision making at the strategic level. The second reason is that © 2013 American Chemical Society

MILP problems are generally computationally inexpensive compared to mixed-integer nonlinear programming (MINLP) problems. Recently, MILP models have been applied for the optimization of biofuel supply chains,3−6 pharmaceutical supply chains,7,8 and oil or gas supply chains,9,10 among others. Readers can refer to several recent survey articles for more relevant work in the PSE literature.11−13 At any decision-making stage in SCO, there are always factors that are not known exactly but might significantly impact the supply chain performance, such as customer demands, material and product prices, and plant yields. Thus, uncertainty is always a concern in SCO, especially at the strategic level, in which the decisions involve large investment costs and influence the whole life cycle of the supply chain. Recently, there has been growing interest in systematically incorporating environmental sustainability considerations into SCO decision making,12−14 which introduces more sources of uncertainty in the supply chain model, such as emission and waste generation, collection and recovery of returned products, and so on. If decisions are made without addressing different sources of uncertainty, such decisions can lead to poor economic or sustainability performance and, in the worst case, can fail to meet stringent performance criteria (e.g., customer service level). When model uncertainty is addressed explicitly, the SCO problem can be formulated into a stochastic programming problem.15 A two-stage stochastic program with recourse is a typical stochastic programming model. In this model, the firstReceived: Revised: Accepted: Published: 5721

November 12, 2012 March 12, 2013 March 19, 2013 March 19, 2013 dx.doi.org/10.1021/ie303114r | Ind. Eng. Chem. Res. 2013, 52, 5721−5734

Industrial & Engineering Chemistry Research

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formulations that guarantee the feasibility of the solution and also provide near-optimal solutions. In section 2, a general twostage formulation for strategic SCO under uncertainty is presented, and three approximating formulations in the literature are introduced. Then, the two robust scenario ̈ robust scenario formulation and formulations, namely, naive affinely adjustable robust scenario formulation, are proposed in sections 3 and 4, respectively. In addition, it will be demonstrated that these formulations can be equivalently transformed into tractable optimization problems if the uncertainty is bounded by the infinity norm. The advantages of the proposed formulations over expected value formulation and scenario formulation are demonstrated through case studies in section 5. The article ends with concluding remarks in section 6.

stage variables represent decisions that have to be made before the realization of uncertainty, the second-stage variables present decisions that can be made after the realization of uncertainty, and the objective is to optimize the expected value of a performance function. This model applies to strategic SCO problems, where the uncertainty realization is known only after the development of the supply chain network. Naturally, then, the design decisions are the first-stage variables, and the operational decisions are the second-stage decisions. When there are infinite uncertainty realizations (which is often the case), the two-stage stochastic program with recourse is an intractable infinite-dimensional optimization problem, so it is usually approximated with a scenario formulation for a practical solution. The scenario formulation involves only a finite (yet large) number of uncertainty realizations, so it is a regular mathematical programming problem over a finite-dimensional space. The scenario formulation retains the flexibility of choosing different second-stage decisions according to different realizations of uncertainty, and it often achieves a good estimation of the expected performance and returns reasonable solutions. The scenario formulation is widely employed in SCO research, and recent work includes applications in oil supply chains,9 chemical supply chains,16 biorefinery supply chains,17 and carbon capture and storage networks.18 Again, readers can refer to several recent survey articles for more relevant work in the PSE literature.11−13 However, the scenario formulation cannot guarantee the feasibility of the solution (for the problem before the scenario approximation) in general, because not all of the uncertainty realizations are examined. Including more realizations for each uncertain factor in the scenario formulation can reduce the chance of generating infeasible solutions, but it can also increase the size of the formulation dramatically and make the problem computationally intractable. The difficulty of ensuring feasibility can be overcome in a robust optimization19−23 framework, which was originally developed to achieve guaranteed feasibility against a given set of uncertainty realizations. The basic idea of robust optimization is to address the “worst-case uncertainty realization” (for which the constraints of the problem are most likely to be violated) instead of all or a finite number of predefined uncertainty realizations. With appropriate assumptions, the worst-case uncertainty realization can be taken into account within a tractable convex optimization framework.19,23 The disadvantage of robust optimization is that it usually cannot optimize the expected performance, because only the worst-case uncertainty realization (sometimes also the nominal uncertainty realization) is considered in the problem. As a result, robust optimization has been widely used for problems in which feasibility is more important than optimality for a solution, such as robust control problems.24 Although the application of robust optimization for solving two-stage or multistage stochastic SCO problems has been reported in the mathematical programming and operations research literature,22,25 it has been paid little attention in SCO (especially strategic SCO) research in the PSE community.14 According to the preceding discussion, scenario formulation and robust formulation have complementary features, that is, the advantage of scenario formulation is the disadvantage of robust formulation and vice versa. Thus, this article proposes a novel synergy of scenario formulation and robust formulation for solving two-stage stochastic programs with recourse that come from strategic SCO. This leads to two robust scenario

2. TWO-STAGE FORMULATIONS FOR STRATEGIC SUPPLY CHAIN OPTIMIZATION A general form of the two-stage formulation with recourse is given by Problem P min c Tx + Eξ ∈Ξ[Q (x , ξ)] x∈X

(1a)

subject to Ax ≤ b

(1b)

In problem P, x ∈ X ⊂  denotes first-stage (continuous or integer) decision variables. In the context of strategic supply chain optimization under uncertainty, these variables are design decisions for the supply chain, such as the decision on whether to develop a unit in the supply chain network, the capacity of a manufacturing plant or warehouse, or the means of transportation. c ∈ nx denotes the costs associated with the first-stage decisions, such as the investment cost of a manufacturing plant. A ∈ m1× nx , b ∈ m1, and constraint 1b includes limitations on the first-stage decisions, such as topology relationships among the units in the supply chain network and capacity limits of plants and warehouses. For simplicity, we do not explicitly write out the equality constraints here, as they can be expressed by paired inequality constraints. The cost for second-stage decisions is Q(x,ξ) = min{q(ξ)T y:T(ξ)x + Wy ≤ h(ξ)}, where ξ ∈ Ξ denotes the parameters that are not known exactly when making the firststage decisions (but will be realized when making second-stage decisions), such as material and product prices, customer demands, and plant yields. y ∈ ny denotes continuous secondstage decisions; in the context of strategic supply chain optimization under uncertainty, these variables are long-term operational decisions for the supply chain, such as material allocation for different manufacturing plants, plant production rates, and material and product transportation plans. q(ξ) ∈ ny , T(ξ) ∈ m2 × nx , W ∈ m2 × ny , and h(ξ) ∈ m2 are parameters for the second stage, where ξ in parentheses after the parameters indicates that the parameters are dependent on the realization of uncertainty. Note that W is assumed to be independent of the uncertain parameters here; that is, problem P has fixed recourse.15 Eξ∈Ξ[Q(x,ξ)] denotes the expected second-stage cost over different realizations of ξ. As problem P is, in general, intractable, it is usually transformed (approximately) into another tractable optimization problem for practical solution. One approach to nx

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nominal value (usually set to be the expected value) of that parameter or variable. Problem R is generally a challenging problem because of its bilevel optimization structure, but it can be equivalently reformulated into a tractable single-level convex optimization problem with mild assumptions on the uncertainty set Ξ.20 However, problem R obtains feasible results and efficient computation for the solution at the cost of losing optimality. On one hand, the second-stage decisions y are assumed to be constant over the different realizations of uncertainty, which introduces significant conservativeness for optimization (as, in reality, different second-stage decisions can be made for different uncertainty realizations). On the other hand, a nominal (or sometimes the worst-case) second-stage cost is involved in the objective function, which is usually different from the expected second-stage cost. A variant of problem R for two-stage stochastic programming might be a chance-constrained optimization formulation, where the feasibility of the constraints is required for most of the scenarios (with a predefined confidence level) instead of all scenarios, such as that proposed by Bosch et al.28 To reduce the conservativeness in robust optimization, an affinely adjustable robust formulation22,25 has been proposed in the literature, where the second-stage decisions are allowed to be adjusted for different realizations of uncertainty according to an affine function of the uncertain parameters. This idea leads to the following approximation of problem P Problem AAR

approximate problem P is to reduce the number of uncertainty realizations in the formulation, that is, to address only a finite number of selected realizations of uncertainty in the optimization. The selected uncertainty realizations are usually called scenarios, and the resulting formulation is called scenario formulation. A typical scenario formulation is given by Problem S s

min c Tx +

xy ∈,..., Xy 1 s

∑ pω qω Tyω

(2a)

ω=1

subject to Ax ≤ b

Tωx + wyω ≤ hω ,

ω = 1, ..., s

where the uncertainty-dependent parameters are characterized by s scenarios (indexed by subscript ω) and the relevant probabilities pω. Accordingly, s groups of second-stage variables are explicitly optimized for the uncertainty realizations simultaneously in the scenario formulation. This formulation has a computational advantage, as it is a deterministic MILP that can be solved by state-of-the-art commercial solvers. In addition, the formulation is featured with a decomposable structure, which can be exploited by Benders decomposition/Lshaped method26,27 for efficient solution. When the number of scenarios is 1 and the uncertain parameters realize their expected values in this scenario, problem S becomes so-called expected value formulation. The disadvantage of scenario formulation lies in the fact that the uncertainty might not be fully characterized by the scenarios, so the decisions made through the formulation might be infeasible or suboptimal. Increasing the number of scenarios can improve the quality of the solution, but it also introduces a much heavier computational burden. In addition, it is usually hard to identify how many scenarios are adequate for a reliable solution. Another approach to approximate problem P is motivated by applications in which feasibility is much more important than optimality for a solution. In this approach, a constraint of the optimization addresses the worst-case scenario (i.e., the scenario in which the constraint is most likely to be violated) instead of a finite number of predetermined scenarios, and the objective of the optimization involves only the worst-case or nominal scenario. The resulting problem is usually called a robust optimization problem.23 A robust optimization formulation to approximate problem P can be written in the form Problem R min c Tx + q ̅ Ty

min c Tx + q ̅ T(Uξ ̅ + v)

x∈X ,U ,v

subject to Ax ≤ b ξ∈Ξ

i = 1, ..., m2

max{ti T(ξ)x + wi Ty − hi(ξ)} ≤ 0, ξ∈Ξ

(3b)

i = 1, ..., m2

(4c)

where the affine relationship y = Uξ + v is assumed and the second-stage decisions are optimized through the optimization of matrix U ∈ ny × nξ and vector v ∈ ny . Although such an affine relationship cannot fully reflect the flexibility in determining the second-stage decisions, it is less conservative than fixing the second-stage decisions for all uncertainty realizations. Problem AAR involves more decision variables than does problem R, but it has more flexibility in choosing the second-stage decisions, which leads to improved optimality of the solution. This problem can also be reformulated into a tractable, single-level, convex optimization problem with mild assumptions on the uncertainty set Ξ.20 However, problem AAR might still lead to poor solutions, because the assumption of affine dependence of second-stage decisions on uncertainty realization is restrictive and the objective function still involves the nominal instead of expected second-stage cost. In summary, two-stage problem P can be approximated by a scenario formulation S, a robust formulation R, or an affinely adjustable robust formulation RS for practical solution. The scenario formulation cannot guarantee the feasibility and optimality of its solution for problem P, whereas the two robust formulations yield solutions that are feasible for problem P but might be very conservative. To achieve better solutions, two hybrid formulations are developed in the next two sections,

subject to Ax ≤ b

(4b)

max{ti T(ξ)x + wi T(Uξ + v) − hi(ξ)} ≤ 0,

(3a)

x∈X ,y

(4a)

(3c)

For convenience of explanation, the group of second-stage constraints is expressed as m2 individual constraints in inequality 3c, where vectors ti , wi ∈ ny (i = 1, ..., m2) are obtained from disassembling matrices T and W; that is, [t1 ··· tm2]T = T, [w1 ··· wm2]T = W, hi ∈  (i = 1, ..., m2), [h1 ··· hm2]T = h. A bar over a parameter or variable indicates the 5723



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δi,ω > 0. Note that the uncertain parameters in uncertainty subregion Ξω are now described in different subregions Ξ1,ω, T T Ξ2,ω, ..., Ξm2,ω. [td,i,ω ξi,ω ] = [tTi,ω hi,ω], where td, i , ω ∈ nx + 1 − lξ ,i contains deterministic elements in [ti,ωT hi,ω] and ξi,ω contains the uncertain elements. [xd,iT xu,iT] = [xT −1], where xd, i ∈ nx + 1 − lξ ,i contains the elements in [xT −1] that are

by combining scenario formulation with robust formulation and scenario formulation with affinely adjustable robust formulation, respectively.

̈ ROBUST SCENARIO FORMULATION 3. NAIVE ̈ Naive robust scenario formulation results from the synergy of scenario formulation S and robust formulation R. In this formulation, the infinite number of uncertainty realizations are grouped into a finite number of scenarios, and each scenario contains a set of uncertainty realizations instead of one uncertainty realization. Thus, all uncertainty realizations are included in only a finite number of scenarios. This formulation is given by Problem RS

associated with td,i,ω and x u, i ∈ lξ ,i contains the elements in [xT −1] that are associated with ξi,ω. If we separate the “deterministic part” of the inequality from the maximization operation, then inequality 6 becomes td, i , ω Txd, i + wi Tyω + max {ξi , ω Tx u, i} ≤ 0, ξi , ω ∈Ξi , ω

i = 1, ..., m2 ; ω = 1, ..., s

s T

min c x +

x∈X , y1,..., ys

∑ pω qω̅

T

yω

ω=1

Proposition 1: Given the optimization problem

(5a)

max(ϕ − ϕ ̅ )T x

subject to Ax ≤ b

subject to || M(ϕ − ϕ ̅ )||∞ ≤ δ

ξω ∈Ξω

(5d)

td, i , ω Txd, i + wi Tyω + δi , ω ||(Mi , ω−1)T x u, i || ≤ 0,

Here, the uncertainty region Ξ is covered by s uncertainty subregions Ξs for the s scenarios, that is, ∪sω=1Ξ ⊃ Ξ. As a result, a scenario is associated with a set of uncertainty realizations instead of a single uncertainty realization. Therefore, deterministic inequalities 2c of the scenario formulation become the “robust inequalities” 5c of problem RS, and the second-stage cost coefficient qω in scenario formulation S becomes the nominal value of the coefficient qω̅ in problem RS. Note that all equality constraints are explicitly expressed in problem RS as eq 5d, because equality constraints have to be satisfied for all realizations of uncertainty instead of a single worst-case realization. Obviously, the solution of problem RS is feasible for problem P, no matter how many scenarios are involved in the problem. In addition, problem RS is less conservative than problem R because the second-stage decisions are allowed to be different for different scenarios (although they have to be constant for the different uncertainty realizations in one scenario); in fact, problem R can be viewed as a special case of problem RS, in which only one scenario is considered. Next, we demonstrate that problem RS can be equivalently transformed into a tractable problem if the uncertainty is bounded by the infinity norm. First, consider inequality constraints 5c. For convenience of discussion, we rewrite constraints 5c to distinguish the uncertain and deterministic parameters in an inequality as

1

i = 1, ..., m2 ; ω = 1, ..., s

(9)

Inequality 9 can be equivalently reformulated into no more than 2lξ,i linear inequalities. The reformulation is explained in Appendix B. Note that, usually, lξ,i ≪ nξ (which means that only a small portion of uncertain parameters is present in an inequality), and in this case, the reformulation does not lead to a much larger problem. Second, consider equality constraints in 5d. Again, to distinguish uncertain and deterministic parameters, we rewrite the constraints in the form T (eq) (eq) T (eq) (eq) T (td,(eq) j , ω) xd, j + (ξj , ω ) x u, j + (wj ) yω = 0, (eq) (eq) ξj(eq) , ω ∈ Ξ j , ω ; j = 1, ..., m 2 ; ω = 1, ..., s

(10)

Here, Ξ(eq) j,ω denotes the uncertainty subregion for the uncertain parameters involved in the equation indexed by j and ω, and we do not need any assumption on this subregion for the T (eq) T (eq) T (eq) subsequent reformulation. [(t(eq) d,j,ω) (ξj,ω ) ] = [(tj,ω ) hj,ω ], (eq) T (eq) where td,j,ω contains the deterministic elements in [(t(eq) j,ω ) hj,ω ] (eq) (eq) T (eq) T and ξj,ω contains the uncertain elements. [(xd,j ) (xu,j ) ] = T [xT 1], where x(eq) d,j contains the elements in [x −1] that are (eq) associated with t(eq) and x contains the elements in [xT −1] d,j,ω u,j (eq) that are associated with ξj,ω . It can be found that

max {td, i , ω Txd, i + ξi , ω Tx u, i + wi Tyω } ≤ 0,

ξi , ω ∈Ξi , ω

i = 1, ..., m2 ; ω = 1, ..., s

(8b)

where M is invertible and δ ≥ 0, the optimal objective value is δ∥(M−1)Tx∥1. Proof: See Appendix A. Proposition 1 implies that inequality 7 can be written as

(5c)

T (eq) T (eq) [t (eq) j , ω (ξω)] x + (wj , ω ) yω − hj , ω (ξω) = 0,

ξω ∈ Ξω ; j = 1, ..., m2(eq); ω = 1, ..., s

(8a)

ϕ

(5b)

max {ti , ω T(ξω)x + wi Tyω − hi , ω(ξω)} ≤ 0, i = 1, ..., m2 ; ω = 1, ..., s

(7)

x u,(eq) j = 0, (6)

j = 1, ..., m2(eq)

(11)

must hold for eq 10, because otherwise, the “uncertain part” of T (eq) eq 10, (ξ(eq) j,ω ) xu,j , has different values for different realizations (eq) of ξj,ω . Accordingly, the deterministic part of eq 10 has to be 0, that is

Here, the uncertainty subregion Ξi,ω = {ξi,ω:||Mi,ω(ξi,ω − ξ̅i,ω)||∞ ≤ δi,ω} ⊂ lξ ,i , lξ,i ≤ nξ (as a constraint might not involve all uncertain parameters), Mi,ω is an invertible weighting matrix, 5724



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T (eq) (eq) T (td,(eq) j , ω) xd, j + (wj ) yω = 0,

max {ti , ω T(ξω)x + wi T(Uωξω + vω) − hi , ω(ξω)} ≤ 0,

ξω ∈Ξω

j = 1, ..., m2(eq); ω = 1, ..., s

(12)

i = 1, ..., m2 ; ω = 1, ..., s

Therefore, eq 10 can be equivalently transformed into eqs 11 and 12, without any assumption about the uncertainty region. To the best of our knowledge, this transformation has not been discussed in the literature before. According to the preceding discussion, problem RS can be equivalently transformed into the following problem RS_IN, if the uncertainty is bounded by the infinity norm

T (eq) T (eq) [t (eq) j , ω (ξω)] x + (wj , ω ) (Uωξω + vω) − hj , ω (ξω) = 0,

ξω ∈ Ξω ; j = 1, ..., m2(eq); ω = 1, ..., s

s x∈X , y1,..., ys

∑ pω qω̅ Tyω

(13a)

ω=1

subject to Ax ≤ b

(13b)

td, i , ω Txd, i + wi Tyω + δi , ω ||(Mi , ω−1)T x u, i || ≤ 0, 1

i = 1, ..., m2 ; ω = 1, ..., s

x u,(eq) j = 0,

(13c)

j = 1, ..., m2(eq)

(13d)

T (eq) (eq) T (td,(eq) j , ω) xd, j + (wj ) yω = 0,

j = 1, ..., m2(eq); ω = 1, ..., s

E ξω[qω̅ T(Uωξω + vω)] = qω̅ T[UωE ξω(ξω) + vω]

(13e)

= qω̅ T(Uωξω̅ + vω)

For convenience, we do not expand inequalities 13c into linear inequalities in the preceding formulation. In practice, these inequalities need to be expressed as linear inequalities for solution (as explained above), which makes the size of the problem larger. However, if each inequality involves only a small number of uncertain parameters (which is often true), problem RS_IN will not be much more difficult to solve than the scenario formulation S.

max {td, i , ω Txd, i + ξi , ω Tx u, i + wi T(Uωξω + vω)} ≤ 0,

ξω ∈Ξω

i = 1, ..., m2 ; ω = 1, ..., s

x ∈ X , U1, v1, ···, Us , vs

∑ pω Eξ [qω̅ T(Uωξω + vω)] ω

ω=1

td, i , ω Txd, i + wi Tvω + max {ξω T(Pi , ω Tx u, i + Uω Twi)} ≤ 0, ξω ∈Ξω

i = 1, ..., m2 ; ω = 1, ..., s

(17)

Also, according to proposition 1, inequality 17 can be written in the form td, i , ω Txd, i + wi Tvω + δω ||(Mω−1)T (Pi , ω Tx u, i + Uω Twi)||

1

≤ 0,

i = 1, ..., m2 ; ω = 1, ..., s

(18)

Inequality 18 can be expressed by no more than 2nξ groups of linear inequalities as explained in Appendix B. Similarly, eq 14d can be rewritten as T (eq) (eq) T (eq) (eq) T (td,(eq) j , ω) xd, j + (ξj , ω ) x u, j + (wj ) (Uωξω + vω) = 0,

(14a)

(eq) (eq) ξj(eq) , ω ∈ Ξ j , ω ; j = 1, ..., m 2 ; ω = 1, ..., s

subject to Ax ≤ b

(16)

where we assume the uncertainty is bounded using the infinity norm, that is, Ξω = {ξ:∥Mω(ξω − ξ̅ω)∥ω ≤ δω}, Mω is invertible, δω > 0. For convenience of discussion, let ξi,ω = Pi,ωξω. Then, inequality 16 becomes

s

c Tx +

(15)

where ξ̅ω is the expected value of ξω for scenario ω. Second, consider constraints 14c and 14d. As in section 3, inequality 14c can be written as

4. AFFINELY ADJUSTABLE ROBUST SCENARIO FORMULATION Problem RS can be overly conservative, because it assumes the same second-stage decisions for different uncertainty realizations in one scenario (whereas, in reality, different second-stage decisions might be made for different uncertainty realizations in one scenario). To reduce the conservativeness, an affinely adjustable robust scenario formulation is proposed here, in which the second-stage decisions are allowed to be adjusted affinely with respect to the realizations of uncertainty in a scenario. Although the affine relationships cannot fully reflect the flexibility in determining the second-stage decisions, they are less restrictive than using the same second-stage decisions for different uncertainty realizations in one scenario. According to the discussion in section 2, the affine adjustment can be expressed by yω = (Uωξω + vω) for scenario ω, where Uω, vω are constant coefficients for the affine relationship for scenario ω. Therefore, the affinely adjustable robust scenario formulation can be written in the form min

(14d)

In this formulation, the second-stage decisions are optimized through the optimization of Uω and vω. Note that the maximization operation in inequality 14c is over all the uncertain parameters, as each constraint therein involves all uncertain parameters (because of the dependence of secondstage decisions on all uncertain parameters). By using different affine functions for different uncertainty subregions, problem AARS essentially uses a piecewise-affine function over the entire uncertainty region for the adjustment of second-stage decisions. As a result, it is less conservative than problem AAR, which uses a single affine function over the entire uncertainty region. If only one scenario is considered, problem AARS degrades into problem AAR. Next, we demonstrate that problem AARS can also be equivalently transformed into a tractable problem if the uncertainty is bounded by the infinity norm. First, the expected second-stage cost in the objective function

problem RS_IN min c Tx +

(14c)

(19)

Let ξj,ω = P(eq) j,ω ξω. Then, eq 19 becomes

(14b) 5725



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scenario formulations can be transformed into computationally tractable problems RS_IN and AARS_IN as described in sections 3 and 4, respectively. All problems were modeled using GAMS 22.9.3,29 and they were solved on a machine with a 3.40 GHz CPU and Linux operating system using CPLEX 12.4.30 A relative termination criterion of 10−4 was used for all problems. 5.1. Problem 1: Farm Planning. 5.1.1. Problem Statement: Nominal Case. Problem 1 is a classical farm planning problem from the stochastic programming literature.15 In this problem, a farmer needs to plan the allocation of his land area for raising three crops: wheat, corn, and sugar beets. The goal of the planning is to achieve the best overall profit while reserving a certain amount of wheat and corn for feeding cattle. If the harvested wheat or corn is not sufficient for feeding cattle, it can be purchased on the market at a relatively high price. Sugar beets are the most profitable crop among the three, but the selling price drops after a certain amount (i.e., the quota on sugar beet production) of sugar beets has been sold. Although it is not a typical SCO problem, the farm planning problem has features similar to thise of the strategic supply chain planning problem of interest. In addition, this problem is fairly simple, so it is easy to go into more detail when analyzing the solutions. Without considering the uncertainty, the problem can be formulated as the optimization problem

T (eq) (eq) T (eq) (eq) T (eq) T (td,(eq) j , ω) xd, j + {(x u, j ) P j , ω + (wj ) Uω}ξω + (wj ) vω

= 0,

(eq) (eq) ξj(eq) , ω ∈ Ξ j , ω ; j = 1, ..., m 2 ; ω = 1, ..., s

(19b)

which is equivalent to T (eq) (eq) T (x u,(eq) j ) P j , ω + (wj ) Uω = 0,

j = 1, ..., m2(eq); ω = 1, ..., s

(20)

T (eq) (eq) T (td,(eq) j , ω) xd, j + (wj ) vω = 0,

j = 1, ..., m2(eq); ω = 1, ..., s

(21)

According to the preceding discussion, if the uncertainty is bounded by the infinity norm, then problem AARS can be reformulated into the form Problem AARS_IN s

min

x ∈ X , U1, v1, ···, Us , vs

c Tx +

∑ pω qω̅ T(Uωξω̅ + vω)

(22a)

ω=1

subject to Ax ≤ b

(22b)

td, i , ω Txd, i + wi Tvω + δω ||(Mω−1)T (Pi , ω Tx u, i + Uω Twi)||

1

≤ 0,

i = 1, ..., m2 ; ω = 1, ..., s

min −

T (eq) (eq) T (x u,(eq) j ) P j , ω + (wj ) Uω = 0,

j = 1, ..., T (eq) (td,(eq) j , ω) xd, j

m2(eq); +

ω = 1, ..., s

T (w(eq) j ) vω

c p, mxm +

∑

∑

(cb, mym − cs, mwm)

m ∈Ω1

(cs,hmwmh + cs,l mwml ) (23a)

m ∈Ω 2

(22d)

subject to

∑

= 0,

j = 1, ..., m2(eq); ω = 1, ..., s

∑ m ∈Ω

(22c)

xm ≤ L

(23b)

m ∈Ω

(22e)

Ymxm + ym − wm ≥ Fm ,

Compared to the robust scenario formulation RS_IN, problem AARS_IN involves more decision variables and constraints, so it is more difficult to solve. However, this formulation will lead to much better solutions, as shown in the next section.

wmh + wml ≤ Ymxm , wmh ≤ Q m ,

5. CASE STUDIES The purpose of the case studies is to evaluate the solutions of ̈ robust scenario and affinely adjustable the proposed naive robust scenario formulations in comparison with the solutions of scenario formulation and expected value formulation (which is a special scenario formulation with one scenario that involves the expected values of the uncertain parameters). All uncertain parameters are assumed to be independently and uniformly distributed, so we give only the ranges of the uncertain parameters when explaining the subsequent case study problems. As this article is not focused on scenario generation, we use a simple approach to construct the scenarios for different formulations. The range of each uncertain parameter is divided into ns subintervals, and the uncertainty region is divided into nns ξ subregions (where nξ is the total number of uncertain parameters), which lead to nns ξ scenarios. ̈ robust scenario or affinely adjustable robust scenario For naive formulations, each scenario addresses the relevant uncertainty subregion; for scenario formulation, each scenario addresses the mean values of the uncertainty parameters over the relevant uncertainty subregion. These uncertainty subregions can be readily represented using the infinity norm, so the two robust

xm ≥ 0,

m ∈ Ω1

m ∈ Ω2

m ∈ Ω2

m∈Ω

ym , wm ≥ 0,

m ∈ Ω1

wmh , wml ≥ 0,

m ∈ Ω2

(23c) (23d) (23e) (23f) (23g) (23h)

where the index m indicates different crops and the index sets are Ω = {wheat, corn, sugar beets}, Ω1 = {wheat, corn}, and Ω2 = {sugar beets}. The first-stage decision variables (the planning/design variables), xm (acre), are the areas planned for the three crops. The second-stage decision variables (the operational variables), ym (t), wm (t), whm (t), and wlm (t), are the amounts of wheat and corn to be purchased on the market, the amounts of wheat and corn to be sold on the market, the amount of sugar beets above the quota to be sold on the market, and the amount of sugar beets below the quota to be sold on the market, respectively. Objective 23a is to minimize the overall cost associated with planting the crops and purchasing the wheat and corn. The parameters in the problem are detailed in Appendix C. 5.1.2. Results and Discussion. Uncertain Case A. In this case, the amounts of wheat and corn required for feeding cattle 5726



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are uncertain; specifically, Fwheat = 300 ± 300 t, Fcom = 340 ± 320 t. The scenarios are sampled as explained in the next paragraph. In addition, the selling price of sugar beets below the quota is changed to 27 $/t. Table 1 summarizes the results of expected value formulation ̈ robust scenario formulation EV, scenario formulation S, naive

gives a poor prediction because if its inherent conservativeness (as explained in section 3). Although formulation AARS_IN provides the best performance, it leads to the largest problem size and longest solution time when the same number of scenarios is considered for formulations S, RS_IN, and AARS_IN. (Note that the definitions of a scenario for formulation S and for formulations RS_IN/AARS_IN are different.) Therefore, one might want to determine whether formulation S or RS_IN might achieve equivalently good performance with increased numbers of scenarios. Therefore, we show the expected profits predicted and achieved with formulations S, RS_IN, and AARS_IN using increased numbers of scenarios in Figures 1−3, respectively. In

Table 1. Solution Results of Problem 1 in Uncertain Case A formulationa number of scenarios number of variables number of constraints solution time (s) crop area allocation result (acre) wheat corn sugar beets predicted expected profitb ($) achieved expected profitc ($)

EV

S

1 12 6 0.02

9 60 38 0.02

RS_IN 9 60 38 0.02

AARS_IN 9 366 506 0.06

120 113 267 30600 20700

200 113 187 25933 24833

240 149 111 −9400 25733

240 149 111 25733 25733

a

EV, expected value formulation; S, scenario formulation; RS_IN, robust scenario formulation RS_IN; AARS_IN, affinely adjustable robust scenario formulation. bExpected profit predicted by the formulation at its solution. cExpected profit that can be achieved with the obtained area allocation, as estimated using 992 = 9801 sampled uncertainty realizations. Figure 1. Predicted and achieved expected profits with formulation S: Uncertain case A of problem 1.

RS_IN (with uncertainty bounded by the infinity norm), and affinely adjustable robust scenario formulation AARS_IN (with uncertainty bounded by the infinity norm). The results include formulation sizes, solution times, optimal decisions obtained (i.e., crop area allocation results), predicted expected profit, and achieved expected profit. Here, the predicted expected profit is the expected profit predicted by the formulation if the firststage decisions obtained by the formulation are implemented, and the achieved expected profit is the expected profit that can actually be achieved if the first-stage decisions are implemented. To estimate the achieved expected profit, the expected secondstage cost is approximated by the average second-stage cost over a large number of sampled uncertainty realizations. In this article, 99 realizations of each uncertain parameter were sampled for the estimation of the achieved expected profit for all case studies. Therefore, 992 = 9801 sampled uncertainty realizations were used for this case study. In Table 1, it can be seen that formulation EV obtains the lowest achieved expected profit, even though it predicts a much higher value. This is because this formulation allocates most of the available area for sugar beets (which are most profitable), leaving wheat and corn areas that are just enough for the expected cattle feeding needs. Therefore, in the uncertainty realizations with higher cattle feeding needs, the farmer might have to purchase wheat and corn from the market at high prices, which can significantly reduce the overall expected profit. Formulation S considering 9 scenarios achieves a better expected profit, because it allocates more wheat area to hedge against higher cattle feeding needs. However, it still overestimates the expected profit it can achieve (because of its inability to consider all uncertainty realizations). The two robust scenario formulations RS_IN and AARS_IN achieve the highest expected profit by allocating sufficient wheat and corn areas. In addition, formulation AARS_IN gives a perfect prediction of the expected profit, whereas formulation RS_IN

Figure 1, it can be seen that, as the number of scenarios increases, the expected profit predicted by formulation S approaches the achieved expected profit. The predicted and achieved expected profits converge at $25,733 with 225 scenarios, and this profit is exactly the same as that predicted and achieved with formulation AARS_IN using 9 scenarios. Note that formulation S with 225 scenarios involves 1356 variables and 902 constraints, whereas formulation AARS_IN with 9 scenarios involves only 366 variables and 506 constraints. Thus, formulation S can achieve the same performance as formulation AARS_IN at the expense of solving a larger problem. Figure 2 shows that formulation RS_IN achieves consistently good expected performance with different numbers of scenarios, and its prediction of the

Figure 2. Predicted and achieved expected profits with formulation RS_IN: Uncertain case A of problem 1. 5727



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expected profit improves as the number of scenarios increases. Figure 3 shows that formulation AARS_IN provides consistently good predicted and achieved expected profits for different numbers of scenarios.

Table 2. Solution Results of Problem 1 in Uncertain Case B formulationa EV number of scenarios numbers of variablesc number of constraints solution time (s) crop area allocation result (acres) wheat corn sugar beets predicted expected profitd ($) achieved expected profite ($)

Figure 3. Predicted and achieved expected profits with formulation AARS_IN: Uncertain case A of problem 1. (Note that the predicted and achieved expected profits overlap.)

S

RS_IN

1

27

12167

3/10

3/114

6

b

AARS_IN

27

27

3/48674

3/168

3/816

110

48670

110

1946

0.03

0.03

109.86

0.05

0.06

120 115 265

140 135 225

150 145 205

150 145 205

150 145 205

78200

69700

65450

47010

65450

infeasible

infeasible

65450

65450

65450

a

EV, expected value formulation; S, scenario formulation; RS_IN, robust scenario formulation RS_IN; AARS_IN, affinely adjustable robust scenario formulation. bScenario formulation S keeps generating infeasible allocation results until the number of scenarios is increased to 233 = 12167. cNumber of integer variables/number of continuous variables. dExpected profit predicted by the formulation at its solution. e Expected profit that can be achieved with the obtained area allocation, as estimated using 992 = 9801 sampled uncertainty realizations.

Uncertain Case B. In this case, the yields of the crops are uncertain and range within ±20% of their nominal values; specifically, Ywheat = 2.5 ± 0.5 t/acre, Ycorn = 3.0 ± 0.6 t/acre, Ysugar beets = 20 ± 4 t/acre. Wheat and corn can no longer be purchased on the market, so there might be insufficient wheat or corn to feed cattle if the yield of wheat or corn is lower than anticipated. In addition, the allocated planting areas of crops need to be multiples of 5 acres, so an allocation decision is modeled in the optimization as 5 acres times an integer variable. Table 2 summarizes the results of different formulations. Formulation EV leads to an infeasible area allocation. Formulation S leads to infeasible result as well when 27 scenarios are considered, and it achieves a feasible result (which is also optimal) when the number of scenarios considered is increased to 12167. Formulations RS and RS_IN achieve the optimal result with 27 scenarios, whereas formulation RS gives a poor prediction. As in uncertain case A, formulation S needs to solve a larger problem to achieve the same performance as formulation AARS_IN, and in this case, the solution time for formulation S is several orders of magnitude longer than that for formulation AARS_IN. 5.2. Problem 2: Energy and Bioproduct Supply Chain. 5.2.1. Problem Statement: Nominal Case. This energy and bioproduct supply chain optimization problem is adapted from Č uček et al.31 The supply chain involves four layers. At layer 1, different biomass materials are harvested from 10 supply zones and then sent to up to 6 preprocessing centers. At layer 2, the materials go through different preprocessing procedures (e.g., drying, compaction, and collection) in the preprocessing centers and are then sent to up to 3 main plants. At layer 3, materials are converted into different final products at the main processing plants. A number of technologies are available for the main processing. At layer 4, the final products are shipped to 3 demand locations, including 2 local cities and 1 export location. The superstructure of the supply chain network is shown in Figure 4. The dashed line denotes a railway that joins the preprocessing centers with the processing plants. Export location j3 is located north of the region shown in the figure.

The goal of the strategic SCO is to determine the optimal configuration of the supply chain network and the technologies used in the processing plants, such that the total profit is maximized and the customer demands at the three demand locations are satisfied. The first-stage decisions are whether specific units or technologies are to be included in the supply chain and are represented by binary variables in the optimization. The second-stage decisions are material or product flows that determine the operation of the supply chain and are represented by continuous variables in the optimization. The variables and parameters for this problem are listed in the Nomenclature section, and the descriptions and values of the parameters are provided in the Supporting Information. We give the deterministic optimization model (that does not explicitly address uncertainty) next. The amount of each biomass material that can be harvested at each supply zone is subject to the capacity of that supply zone qiL1 ≤ HYpiAi ,pi , ,pi

∀ pi ∈ PI, ∀ i ∈ I

(24a)

and the mass balance for the materials harvested and sent for preprocessing is given by qiL1 = ,pi

∑

qiL1,L2 , , m ,pi

∀ pi ∈ PI, ∀ i ∈ I (24b)

m∈M

The material flows going through the preprocessing centers are subject to preprocessing capacities as ≤ qpiL1,L2,UPymL2 , ∑ qiL1,L2 , m ,pi i∈I

5728

∀ m ∈ M , ∀ pi ∈ PI (24c)



Article

Figure 4. Superstructure of the energy and bioproduct supply chain network.

∑∑

qiL1,L2 ≤ q L2,UPymL2 , , m ,pi

(24d)

i ∈ I pi ∈ PI

=

i∈I

, ∑ qmL2,L3 , n ,pi

∑

=

∑

qnL2,L3 ≤ qtL3,UPynL3, t , ,pi, t

+

c inv = (24g)

+

∑

∑ ∑ (ctfix,inv,L3ynL3,t n∈N t∈T

ctvar,inv,L3qnL2,L3 ) ,pi, t

(24m)

objective = revenue − total costs = (24h)

∑∑ ∑

qnL3,L4 c price + , j o ,pp pp

o

n ∈ N j ∈ J pp ∈ PP

(qnL3,L4 c price) − , j e ,pp pp

∑∑

∑∑ ∑

0.9

n ∈ N j e ∈ J pp ∈ PP

qiL1 c − c tr − c op − c inv ,pi pi

i ∈ I pi ∈ PI

(24n)

∀ n ∈ N , ∀ pp ∈ PP

The overall optimization model can be express as

j∈J

maximize objective

and they are also bounded by certain amounts of customer demands that must be satisfied (i.e., contracted demands) as well as the capacities of the markets at the demand locations ≤ Dem UP ∑ qnL3,L4 j ,pp, , j ,pp

c fix,inv,L2ymL2 +

Then, the total profit to be maximized is

(24i)

Dem LO j ,pp ≤

(24l)

pi ∈ PT

The final products sent to demand locations are limited by the products generated in the processing center

pi ∈ PIP t ∈ PIPT

∑ m∈M

∀ n ∈ N , ∀ pi ∈ PI, ∀ pp ∈ PP, ∀ t ∈ T , ∀

, ∑ qnL3,L4 , j ,pp

op,L3 L2,L3 c pi, t qn ,pi, t

and the total investment cost for the preprocessing centers and processing plants is

qnL2,L3 f conv,L3 = qnL2,L3 , ,pi, t pi,pp, t ,pi,pp, t

qnL2,L3 ≥ ,pi,pp, t

∑ ∑ ∑ n ∈ N pi ∈ PI t ∈ PT

∀ n ∈ N, ∀ t ∈ T

(pi, pp, t ) ∈ PIPT

c piop,L2qiL1,L2 , m ,pi

i ∈ I m ∈ M pi ∈ PI

The materials are converted into the products in the processing plants at specific conversion rates

∑ ∑

∑∑ ∑

c op =

(24f)

pi ∈ PT

(24k)

The total operating cost for preprocessing centers and processing plants is

∀ n ∈ N , ∀ pi ∈ PI

t ∈ PT

road,L3,L4 tr ,L3,L4 L3,L4 DnL3,L4 c pp qn , j ,pp ,j f n,j

n ∈ N j ∈ J pp ∈ PP

∀ m ∈ M , ∀ pi ∈ PI

and the processing is subject to the capacities of the technologies

∑

∑∑ ∑

+

The materials sent to the main processing plants are processed using different technologies

m∈M

road,L2,L3 tr ,L2,L3 L2,L3 DmL2,L3 c pi qm , n ,pi ,n f m,n

m ∈ M n ∈ N pi ∈ PI

(24e)

qnL2,L3 , ,pi, t

∑ ∑ ∑

+

n∈N

qmL2,L3 , n ,pi

road,L1,L2 tr ,L1,L2 L1,L2 DiL1,L2 c pi qi , m ,pi ,m f i,m

i ∈ I m ∈ M pi ∈ PI

and the inlet and outlet material flows of the preprocessing centers are subject to a mass balance that takes into account the loss of mass during preprocessing f conv,L2 ∑ qiL1,L2 , m ,pi pi

∑∑ ∑

c tr =

∀m∈M

subject to

constraints 24a−24n all continuous variables are nonnegative

(25)

Note that this problem involves a large number of equality constraints. When uncertainty needs to be addressed within an affinely adjustable robust formulation, these equality constraints cannot be transformed efficiently into deterministic constraints using the classical approach in the literature20 (which was primarily developed for inequality constraints). The approach

∀ j ∈ J, ∀ p ∈ P

n∈N

(24j)

The total transportation cost considering distance and road conditions is 5729



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supply chain network, the expected profits achieved by formulations S, RS_IN, and AARS_IN are more than 15% better than that achieved by formulation EV. In addition, formulation S overestimates the expected profit because of its incomplete consideration of uncertainty, whereas formulations RS_IN and AARS_IN underestimate the expected profit because of their inherent conservativeness. Uncertain Case B. In this case, both the lower and upper demand limits for electricity are assumed to be uncertain; LO specifically, Demj,electricity = 90000 ± 56000 MWh, and UP Demj,electricity = 500000 ± 10000 MWh. In addition, the capacity of incineration in the processing plants is qL3,UP incineration = 290000 t/year. Table 4 summarizes the results of each formulation. It can be seen that formulation EV and formulation S with 25 scenarios lead to infeasible supply chain configurations, whereas formulations RS_IN and AARS_IN lead to a feasible and optimal supply chain configuration. Although it is highly conservative, formulation RS_IN even gives a very good prediction of the expect profit in this case. The key difference between the two supply chain configurations is the number of plants that are equipped with incineration technology (for electric power generation). The infeasible configuration has incineration only in processing plant 2, and the optimal configuration has incineration in both processing plant 1 and 3. When the number of scenarios addressed in formulation S increased to 2809, this formulation leads to the optimal configuration; however, this result is obtained through the solution of a very large-scale MILP problem, which takes more than 46 h. Thus, formulation S is obviously outperformed by formulation AARS_IN, which requires less than 1 h to obtain the optimal configuration. Uncertain Case C. In this case, both the lower demand limit for electricity and the yield of corn stover are assumed to be uncertain; specifically, DemLO j,electricity = 80000 ± 18000 MWh, and HYcorn stover = 840 ± 300 t/(km2·year). In addition, the capacity of incineration in the processing plants is qL3,UP incineration = 500000 t/year. These parameters were selected such that no feasible configuration exists for the given superstructure of the

developed in section 4 can be used to transform the equality constraints in a systematic and efficient manner. 5.2.2. Results and Discussion. Uncertain Case A. In this case, the upper demand limits for electricity and the yield of corn stover are assumed to be uncertain; specifically, DemUP j,electricity = 200000 ± 150000 MWh, and HYcorn stover = 840 ± 300 t/(km2·year). In addition, the capacity of incineration in the processing plants is qL3,UP incineration = 390000 t/ year, and the availability of corn planting area is increased as indicated in Table 1 of the Supporting Information. Table 3 summarizes the results of each formulation. Whereas all of the formulations lead to feasible configurations of the Table 3. Solution Results of Problem 2 in Uncertain Case A formulationa number of scenarios numbers of variablesb number of constraints solution time (s) preprocessing centers to be developed processing technologies to be appliedc dry grind digestion incineration sawing predicted expected profitsd (million $) achieved expected profitse (million $)

EV

S

RS_IN

AARS_IN

1 18/670 376 0.16 1, 2, 4, 5, 6

25 18/16078 9088 4.02 1, 2, 4, 5, 6

25 18/16078 9088 4.85 1, 2, 3, 4, 5, 6

25 18/59428 84013 2420.25 1, 2, 4, 5, 6

1, 3 1 1 3 78.66

1, 3

1, 3

1, 3 3 76.23

1, 3 1 1, 3 3 65.29

65.77

76.00

75.98

76.00

1, 3 3 75.08

a

EV, expected value formulation; S, scenario formulation; RS_IN, robust scenario formulation RS_IN; AARS_IN, affinely adjustable robust scenario formulation. bNumber of binary variables/number of continuous variables. cProcessing plant(s) at which processing technology is to be applied. dExpected profit predicted by the formulation at its solution. eExpected profit that can be achieved with the obtained area allocation, as estimated using 992 = 9801 sampled uncertainty realizations.

Table 4. Solution Results of Problem 2 in Uncertain Case B formulationa EV number of scenarios numbers of variablesc number of constraints solution time (s) preprocessing centers to be developed processing technologies to be appliedd dry grind digestion incineration sawing predicted expected profitse (million $) achieved expected profitsf (million $)

RS_IN

AARS_IN

1 18/670 376 0.17 1, 2, 3, 4, 6

25 18/16078 9088 21.80 1, 2, 3, 4, 6

S 2809b 18/1803406 1019680 166890.05 1, 2, 3, 5, 6

25 18/16078 9088 13.75 1, 2, 3, 5, 6

25 18/59428 84013 2245.73 1, 2, 3, 5, 6

2 1 2 3 57.01 infeasible


2 1 1, 3 3 57.01 57.01

2 2 1, 3 3 56.80 57.01

2 1 1, 3 3 57.01 57.01

a EV, expected value formulation; S, scenario formulation; RS_IN, robust scenario formulation RS_IN; AARS_IN, affinely adjustable robust scenario formulation. bScenario formulation S keeps generating infeasible planning results until the number of scenarios is increased to 532 = 2809. cNumber of binary variables/number of continuous variables. dProcessing plant(s) at which processing technology is to be applied. eExpected profit predicted by the formulation at its solution. fExpected profit that can be achieved with the obtained area allocation, as estimated using 992 = 9801 sampled uncertainty realizations.

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Table 5. Solution Results of Problem 2 in Uncertain Case C formulationa EV number of scenarios number of variablesb number of constraints solution time (s) preprocessing centers to be developed processing technologies to be appliedc dry grind digestion incineration sawing predicted expected profitsd (million $) achieved expected profitse (million $)

S

RS_IN

AARS_IN

1 18/670 376 0.16 1, 2, 3, 4, 6

9 18/5806 3280 2.99 1, 2, 3, 4, 6

2500 18/1605028 907513 61044.77 1, 2, 3, 4, 6

9 18/5806 3280 0.13 −

9 18/21412 30301 113.01 −




−

−

infeasibility indicated −

infeasibility indicated −

a EV, expected value formulation; S, scenario formulation; RS_IN, robust scenario formulation RS_IN; AARS_IN, affinely adjustable robust scenario formulation. bNumber of binary variables/number of continuous variables. cProcessing plant(s) at which processing technology is to be applied. d Expected profit predicted by the formulation at its solution. eExpected profit that can be achieved with the obtained area allocation, as estimated using 992 = 9801 sampled uncertainty realizations.

structure similar to that of scenario formulation, which can be exploited by either classical Benders decomposition/L-shaped method26,27 or multicut Benders decomposition32,33 for efficient solution. There are three interesting directions for extending this work in the future. The first is to address other or more general uncertainty regions, such as ellipsoidal uncertainty regions20 or uncertainty regions bounded by general norms.21 A challenging issue is how to generate a reasonable uncertainty subregion for each scenario such that the overall uncertainty region is covered by the union of the uncertainty subregions. The second is to extend this framework to nonlinear models. Although the resulting MINLP problem is, in general, difficult to solve (especially to global optimality), its decomposable structure can be exploited for efficient global optimization using the recently developed nonconvex generalized Benders decomposition method.34 The third is to extend the work to multistage problems, for which reduction of the numbers of scenarios is even more important for practical applications.

supply chain network, as we wanted to investigate whether the optimization formulations can identify the infeasibility. Table 5 summarizes the results of each formulation. It can be seen that formulations RS_IN and AARS_IN with 9 scenarios indicate the infeasibility of the problem within fairly short times. Formulation EV and formulation S with 9 scenarios do not identify the infeasibility and report infeasible configurations. Formulation S does not report infeasibility even when it addresses 2500 uncertainty realizations (which takes almost 17 h to solve). This result demonstrates that, when a feasible solution does not exist, formulations RS_IN and AARS_IN can report infeasibility efficiently and effectively.

6. CONCLUSIONS In this article, a novel framework is proposed to solve two-stage stochastic programs with recourse that arise in strategic SCO under uncertainty. The framework integrates the classical scenario approach and a robust approach for addressing uncertainty, and it leads to two robust scenario formulations, namely, naiv̈ e robust scenario formulation and affinely adjustable robust scenario formulation. In these two formulations, a scenario represents a group of uncertainty realizations instead of a single uncertainty realization. It is shown that both formulations can be equivalently transformed into tractable deterministic optimization problems if the uncertainty is bounded by the infinity norm. A novel transformation of uncertain equality constraints is also presented that does not rely on any assumption about the uncertainty region. The case study results demonstrate that the proposed formulations effectively avoid infeasible solutions, or report the infeasibility of the problem, when small numbers of scenarios are considered. They also outperformed the classical scenario formulation by generating optimal solutions with smaller numbers of scenarios and shorter solution times in the case studies (except in uncertain case A of problem 2, for which the three formulations achieved similar expected profits). In addition, the affinely adjustable robust scenario formulation outperformed the naiv̈ e robust scenario formulation by consistently providing good predictions of the expected profit to be achieved, although this formulation generally led to larger optimization problems and required longer solution times. Note that the proposed formulations have a decomposable

■

APPENDIX A. PROOF OF PROPOSITION 1 Let M(ϕ − ϕ̅ ) = z = (z1, ..., zn), (M−1)T x = r = (r1, ..., rn), then problem 8a with constraint 8b becomes max z Tr

(A.1a)

z

subject to || z ||∞ ≤ δ

(A.1b)

A feasible solution of this optimization problem is ⎧ δ , if ri ≥ 0 zi = ⎨ ⎩−δ , if ri < 0 ⎪

⎪

(A.2)

With this solution n

z Tr =

∑ δ|ri| = δ || r ||1 i=1

(A.3)

so δ∥r∥1 is a lower bound on the optimal objective value. On the other hand, when ∥z∥∞ ≤ δ 5731


Industrial & Engineering Chemistry Research n

z Tr =

n

Article

n

∑ ziri ≤

∑ |zi||ri| ≤

∑ δ|ri| = δ || r ||1

i=1

i=1

i=1

Table C.1. Problem 1 Parameters and Their Values for the Deterministic Formulation

(A.4)

crop, m

so δ∥r∥1 is an upper bound on the optimal objective value. Therefore, δ∥r∥1 is the optimal objective value of problem A.1a with constraint A.1b, and thus, δ∥(M−1)Tx∥1is the optimal objective value of problem 8a with constraint 8b.

total land area, L (acre) planting cost, cp,m ($/acre) purchase price, cb,m ($/t) selling price, cs,m ($/t) selling price (under quota), chs,m ($/t) selling price (over quota), cls,m ($/t) yield, Ym (t/acre) reserved for feeding cattle, Fm (t) quota on production, Qm (t)

■

APPENDIX B. REFORMULATION OF A 1-NORM IN AN INEQUALITY CONSTRAINT For convenience, we discuss the reformulation of the inequality constraint || z ||1 ≤ 0 (B.1)

■

where z = (z1 , ..., zn) ∈ n. Note that n

|| z ||1 =

n

mod(⌊j /2i − 1⌋,2)

max {∑ ( −1) ∑ |zi| = j = 1,...,2 −1 n

i=1

zi }

i=1

i−1

max {∑ ( −1)mod(⌊j /2

j = 0,...,2n − 1

⌋,2)

zi } ≤ 0

i=1

i−1

∑ (−1)mod(⌊j /2

⌋,2)

zi ≤ 0,

■ ■ ■

ACKNOWLEDGMENTS This work was supported by Queen’s University Research Initial Grant.

−z1 + z 2 + z 3 ≤ 0,

DDGS = dried distillers grains with solubles MSW = municipal solid waste

z1 − z 2 + z 3 ≤ 0,

−z1 − z 2 + z 3 ≤ 0, z1 + z 2 − z 3 ≤ 0,

−z1 + z 2 − z 3 ≤ 0, z1 − z 2 − z 3 ≤ 0, −z1 − z 2 − z 3 ≤ 0,

Sets and Subsets

I = set of supply zones, I = {i1, ..., i10} M = set of preprocessing centers, M = {m1, ..., m6} N = set of process plants, N = {n1, n2, n3} J = set of demand locations, J = {j1, j2, j3} Je = subset of export location, Je = {j3} Jo = subset of local demand locations, Jo = {j1, j2} P = set of all materials, P = {corn, corn stover, wood chips, manure, MSW, timber, electricity, heat, bioethanol, boards, digestate, DDGS} T = set of technology options, T = {dry grind, digestion, sawing, incineration} PI = subset of raw materials, PI = {corn, corn-stover, wood chips, MSW, manure, timber} PP = subset of produced products, PP = {electricity, heat, bioethanol, digestate, DDGS, boards} PT = pairs of intermediated products and technologies, PT = {(corn, dry grind), (corn stover, digestion), (corn stover, incineration), (wood chips, incineration), (MSW, incineration), (manure, incineration), (manure, digestion), (timber, sawing)} PIP = pairs of intermediate products and their produced products, PIP = {(corn, bioethanol), (corn, DDGS), (corn stover, electricity), (corn stover, heat), (corn stover, digestate), (wood chips, electricity), (wood chips, heat), (MSW, electricity), (MSW, heat), (manure, electricity), (manure, heat), (manure, digestate), (timber, boards)}

(B.6a)

j=1

(B6.b)

j=2

(B6.c)

j=3

(B6.d)

j=4

(B6.e)

j=5

(B6.f)

j=6

(B6.g)

j=7

NOMENCLATURE FOR CASE STUDY PROBLEM 2

Acronyms

(B.5)

j=0

AUTHOR INFORMATION

The authors declare no competing financial interest.

( −1)mod(j ,2)z1 + ( −1)mod(⌊j /2⌋,2)z 2 + ( −1)mod(⌊j /4⌋,2)z 3

which corresponds to z1 + z 2 + z 3 ≤ 0,

ASSOCIATED CONTENT

Notes

(B.3)

(B.4)

j = 0, ..., 7

260 − − 36 10 20 − 6000

*Tel.: +1 613 533 6582. Fax: +1 613 533 6637. E-mail: xiang. [email protected].

For example, if n = 3, the 1-norm bound B.1 can be transformed into eight linear constraints in the form of inequality B.4 as

≤ 0,

230 210 150 − − 3 340 −

Corresponding Author

j = 0, ..., 2n − 1

i=1

sugar beets

500 150 238 170 − − 2.5 300 −

Values of parameters in case study problem 2. This material is available free of charge via the Internet at http://pubs.acs.org.

(B.2)

which can be equivalently transformed into the following 2n linear inequalities n

corn

S Supporting Information *

where mod(x,y) stands for the modulo operation that finds the remainder of division of x by y. ⌊·⌋ is a function returns the largest integer that is smaller than its argument. Therefore, inequality B.1 becomes n

wheat

(B6.h) n

Note that one does not always need 2 linear inequalities for the 1-norm bound. For example, if all elements in z in the preceding example are known to be nonnegative, then inequality B.1 is equivalent to inequality B.6a; that is, bound B.1 can be reformulated into one linear inequality.

■

APPENDIX C. PARAMETERS IN CASE STUDY PROBLEM 1 See Table C.1 for the problem 1 parameters. 5732



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qL1,L2,UP = capacity of the preprocessing centers for raw pi material pi, t/year qL3,UP = capacity of technology t, t/year t

PIPT = groups of intermediate products, their produced products and related technology, PIPT = {(corn, bioethanol, dry grind), (corn, DDGS, dry grind), (corn stover, digestate, digestion), (corn stover, electricity, incineration), (corn stover, heat, incineration), (MSW, electricity, incineration), (MSW, heat, incineration), (manure, digestate, digestion), (manure, electricity, incineration), (manure, heat, incineration), (timber, boards, sawing)}

Continuous Variables

cinv = total investment costs, €/year cop = total operating costs, €/year ctr = total transportation costs, €/year PB = profit before taxes, €/year qL1,L2 i,m,pi = rate of material entering the preprocessing center, t/ year qL1 i,pi = rate of raw material harvesting, t/year qL2,L3 m,n,pi = rate of material exiting the preprocessing center, t/ year qL3,L4 n,j,pp = rate of product sent to demand location, t/year qL2,L3 n,pi,pp,t = rate of product production, t/year qL2,L3 n,pi,t = rate of material sent to technology option t, t/year

Superscripts

conv = conversion fix = fixed part of investment costs inv = investment costs L1 = harvesting layer L2 = preprocessing layer L3 = main processing layer L4 = demand layer LO = lower bound op = operating costs price = price of products road = road conditions tr = transportation UP = upper bound var = variable part of investment costs

Binary Variables

■

yL2 m = selection or rejection of preprocessing centers yL3 n,t = selection or rejection of main plants

REFERENCES

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Indices

i = supply zones j = demand locations je = export level jo = local level m = preprocessing centers n = main processing plants p = products pi = raw materials pp = produced products t = technology options Parameters

Ai,pi = available area of material pi, km2 cop,L2 = operating costs of material pi in the preprocessing pi center, €/t cop,L3 pi,t = operating costs of material pi in the main plant, €/t cfix,inv,L2 = fixed investment costs of the preprocessing center, €/year ctr,La,Lb = cost coefficient for transportation of product p from p layer a to layer b, €/(t·km) cpi = cost of raw material pi, €/t cprice pp = price of produced product, €/t or €/MWh or €/MJ cfix,inv,L3 = fixed investment costs of main plant technology, t €/year ctvar,inv,L3 = variable investment costs of main plant technology, €/t La,Lb Dx,y = distance between object x in layer a and object y in layer b, km DemLO j,pp = lower bound of product demand, t/year or MWh/ year or MJ/year DemUP j,pp = upper bound of product demand, t/year or MWh/ year or MJ/year f conv,L2 = conversion factor through the preprocessing center pi f conv,L3 pi,pp,t = conversion factor through the main plant f road,La,Lb = road condition factor of object x in layer a and x,y object y in layer b HYpi = yield of raw material pi, t/(t·km) qL2,UP = overall capacity of the preprocessing centers, t/year 5733



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Robust Scenario Formulations for Strategic Supply Chain Optimization

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