MILP reformulation of the multi-echelon stochastic inventory system ...

MILP reformulation of the multi-echelonstochastic inventory system with uncertaindemandsAxel NybergÅbo Akademi UniversityIgnacio E. Grossmann ∗Dept. of Chemical Engineering, Carnegie Mellon University, Pittsburgh,USATapio WesterlundÅbo Akademi UniversityJuly 13, 20121 IntroductionIn this article we present an effective mixed-integer linear programming (MILP)formulation for design of a multi-echelon stochastic inventory system with uncertaincustomer demands. In You and Grossmann [2010] a three-echelon supplychain with inventories under uncertainty is presented. In that supply chain, thelocation of the plants and the customer demand zones (CDZ) are known. Potentialdistribution centers (DC) are given and the objective is to decide whichDCs to install in order to minimize total costs, which include transportationcost, installation cost as well as inventory holding costs. The formulation alsodetermines the service times for each DC, and what the size of the safety stockshould be at all DCs and CDZs. This model uses single sourcing, which is oftenthe case for supply chains in industrial gases or specialty chemicals. That meansthat all DCs are served by only one plant and each CDZ is served by only oneDC. The detailed model can be found in You and Grossmann [2010].In this short note we first reformulate the mixed-integer nonlinear programming(MINLP) model presented in You and Grossmann [2010] and You andGrossmann [2011], using an alternative linearization scheme leading to a modelthat is significantly smaller in size, and tighter. We also present a successivepiecewise linear approximation with which we can solve the model with a sufficientlysmall optimality gap without the need of a global optimization solver.∗ Author to whom correspondence should be addresed: grossmann@cmu.edu1

Figure 1: Network of multiple plants, DCs and CDZs2 Model FormulationIn You and Grossmann [2010] the original MINLP model (P0) is stated as:Min : ∑ f j Y j + ∑ ∑ ∑A ijk X ij Z jk + ∑ ∑B jk Z jk (1)j∈J i∈I j∈J k∈K j∈J k∈K+ ∑ √∑q1 j N j σk 2Z jk + ∑ √q2 k Lk (2)j∈Jk∈K k∈Ks.t.N j ≥ ∑ i∈I(SI i + t1 ij )X ij − S j , ∀j, (3)L k ≥ ∑ j∈J(S j + t2 jk )Z jk − R k , ∀k, (4)∑X ij = Y j , ∀j, (5)i∈I∑Z jk = 1, ∀k, (6)j∈JZ jk ≤ Y j , ∀j, k, (7)X ij , Y j , Z jk ∈ {0, 1}, ∀i, j, k, (8)S j ≥ 0, N j ≥ 0, ∀j, (9)L k ≥ 0, ∀k, (10)with the following parameters(see section 9 for complete nomenclature):A ijk = (c1 ij χ + θ1 j t1 ij ) · µ k ,B jk = (g j χ + c2 jk χ + θ2 k t2 jk ) · µ k ,q1 j = λ1 j · h1 j ,q2 k = λ2 k · h2 k · σ k .In order to decrease the number of nonlinear terms, the authors linearize ina similar way as in Glover [1975], the bilinear terms in (P0) to obtain a newformulation, (P1), with fewer nonlinear constraints. In the objective function,2

whereA ij = (c1 ij χ + θ1 j t1 ij ), (25)is a new constant independent of k. According to Eq. (5) at most one of the X ijvariables in ∑ i A ijX ij can be nonzero. Thus, this term can now be linearizedusing (|I| + 1) · j new continuous variables, XZ ij , instead of 2 · |I| · |J| · |K| asin Eq. (2). Hence, for the linearization, the objective function is written as:∑ ∑A ij XZ ij , (26)j∈J i∈Iand the new constraints (instead of Eqs. (11) to (14)) as,XZ 0j + ∑ iXZ ij = ∑ kµ k Z jk ∀j, (27)XZ ij ≤ ∑ kµ k X ij ∀i, j, (28)XZ 0j ≤ ∑ kµ k (1 − Y j ) ∀j, (29)where the last variable Y j comes from Eq. (5).Since the objective is to minimize, and all variables are nonnegative we cansubstitute the N j in the objective function, with the right hand side ( ∑ i∈I (SI i+t1 ij )X ij −S j ) of Eq. (3). Instead of having bilinear terms between the continuousvariables N j and the binary variables Z jk , we can now linearize the expressionbelow as follows:( ∑ (SI i + t1 ij )X ij − S j ) · ( ∑ Z jk σk), 2 (30)i∈Ik∈Kwhich can be written as:( ∑ (SI i + t1 ij )X ij ) · ( ∑ Z jk σk) 2 − S j · ( ∑ Z jk σk). 2 (31)i∈Ik∈Kk∈KAgain, it can be noted that according to Eq. (5) at most one of the binaryX ij variables in the above equation can be nonzero. The first part of thisterm can therefore be linearized analogously as in Eqs. (27) to (29). Notethat the variables are exactly the same in the bilinear terms, and therefore thesame linearizations would work in both cases if it were not for the constantspreceding the variables. However, we choose to write these linearizations withnew continuous variables ZX ij as well since this formulation is tighter than theone presented in You and Grossmann [2010].ZX 0j + ∑ iZX ij = ∑ kσ 2 kZ jk ∀j, (32)ZX ij ≤ ∑ kσ 2 kx ij ∀i, j, (33)ZX 0j ≤ ∑ kσ 2 k(1 − Y j ) ∀j, (34)4

In the second half of Eq. (31), the bilinear terms in S j · ( ∑ k∈K Z jkσk 2 ) are againexactly the same as the ones already linearized in Eqs. (15) to (17). Thereforethe bilinear terms are already defined earlier and we can write the expressionunder the first square root term as:∑S1 ij · ZX ij − ∑ σk 2 · SZ jk (35)k∈Ki∈I4 Reformulated modelBased on the linearizations in the previous section, the reformulated nonlinearmodel (R1) is as follows:∑Min : ∑ f j Y j + ∑ A ij XZ ij + ∑j∈J j∈J i∈I j∈J+ ∑ j∈Jq1 j√ ∑i∈IS1 ij · ZX ij − ∑ k∈K∑B jk Z jkk∈Kσ 2 k · SZ jk + ∑ k∈Kq2 k√Lk(36)s.t.L k ≥ ∑ j∈JSZ jk + t2 jk · Z jk − R k , ∀k, (37)∑X ij = Y j , ∀j, (38)i∈I∑Z jk = 1, ∀k, (39)j∈JZ jk ≤ Y j , ∀j, k, (40)XZ 0j + ∑ iXZ ij = ∑ kµ k Z jk ∀j, (41)XZ ij ≤ ∑ kµ k x ij ∀i, j, (42)XZ 0j ≤ ∑ kµ k (1 − Y j ) ∀j, (43)ZX 0j + ∑ iZX ij = ∑ kσ 2 kZ jk ∀j, (44)ZX ij ≤ ∑ kσ 2 kx ij ∀i, j, (45)ZX 0j ≤ ∑ kσ 2 k(1 − Y j ) ∀j, (46)SZ jk + SZ1 jk = S j , ∀j, k, (47)SZ1 jk ≤ (1 − Z jk ) · S U j , ∀j, k, (48)where all varialbes are ≥ 0. As shown later on in Table 3 this formulation issignificantly smaller in size as well as tighter than P1.5

5 Concave nonlinear termsIn the same manner as in You and Grossmann [2010] the only remaining nonlinearterms in R1 can be underestimated to obtain an MILP formulation (R2)of the model, which provides a good starting solution as well as a lower boundfor R1. Since we use the same variables and constraints, every feasible solutionof the MILP R2 is also a feasible solution to the MINLP R1. Since weunderestimate R1, a valid lower bound for R2 is also a lower bound for R1.In You and Grossmann [2010] (P1) is relaxed in order to obtain the MILP P2,which is solved with CPLEX to obtain a good starting solution for the MINLP.You and Grossmann [2010] underestimate the nonlinear square root terms √ L kand √ NZV j with their respective secants√L kL UBkand√ NZV jNZV UBjas proposedby Soland [1971]. However, since one of the binary Z jk variables in Eq. (4) isnonzero, the lower bound for L k , when R k is a parameter becomes:L LBk= minj∈J {t2 jk − R k } ∀k. (49)In order to obtain a tighter underestimator, we use the following function, whichis the secant in the feasible region of L k :√ √L LBk− L UBk)((L LBk) − LUB k) L k +√L UBk√ √− ( L LBk− L UBk)((L LBk) − LUB k) LUBk (50)In Fig. 2 the difference is shown between underestimating the square root withthe provided lower bounds versus underestimating with the lower bound 0. Thisis not the case when responsiveness R k is defined as a variable as in You andGrossmann [2011].√L kSecantEq. (50)√ L kL LBkL kL UBkFigure 2: Underestimators with lower bounds and with lower bound set to zero5.1 Piecewise linear approximationIn order to tighten the underestimations for the convex square root terms, weuse a sequential piecewise linear approximation approach. Since the MILP R2formulation gives a relatively small gap, even on larger instances, we introduce asuccessive piecewise linear approximation scheme in order to obtain sufficiently6

√ Lk√˜Lk√ L kL LBkL ∗ k 2L ∗ k 3L UBkL kFigure 3: Intervals with the sequential approach after a few iterationssmall gaps. We use the δ−formulation presented in Dantzig [1963] and whosetightness has been studied by Padberg [2000]. Since there are |J| + |K| squareroot terms in the model, it is not a good idea to add too many intervals persquare root term. Therefore, we solve the underestimating MILP several times,and only partition the terms that are greater than zero in the underestimation.In the equations below, we show the piecewise linearization used for the term√Lk . In the model we also partition the second square root term analogously.L k = L LBk +√ √˜Lk = L LBk+N∑n=1N∑δ n , (51)n=1√L ∗ k n+1− √ L ∗ k nL ∗ k n+1− L ∗ k nδ n , (52)(L ∗ k n+1− L ∗ k n+)w n ≤ δ n ≤ (L ∗ k n+1− L ∗ k n)w n−1 , n = 2, . . . , N − 1, (53)(L ∗ k 2− L ∗ k 1)w 1 ≤ δ n ≤ L ∗ k 2− L ∗ k 1, (54)δ n ≤ (L UBk − L ∗ k N)w N−1 , (55)w n ∈ {0, 1}, n = 1, . . . , N − 1. (56)In the sequential approach, we start by solving the MILP model R2, then weadd a grid point for each square root term that is larger than its lower bound.The MILP will therefore be exact in the solution found. We then solve the modelagain with the new grid points and if we find a better solution we again add gridpoints at the new solution. Therefore, N, which is the number of grid points,will be different for all square root terms. Most of the terms will stay at zero, andtherefore no new grid points or variables will be needed for those terms. Afterno improvement is found, the final MILP with the added grid points is solved toa predefined optimality gap. Since the piecewise underestimator is exact in thegrid points, this approach can also be used to close the gap completely. This isa similar approach as the branch-and-refine method presented in Leyffer et al.[2008] and You et al. [2011].7

6 Computational ResultsIn this section, we present computational results on the same size instances as inYou and Grossmann [2010]. All computations in this paper are conducted on anAsus UX31E ultra book with a 2.8 GHz quad-core Intel processor and 4 Gb ofram. As a MILP solver we used CPLEX 12.3 with the default parameters. Allconstant values in the model are generated randomly with uniform distributionwithin predefined values. These values can be found in section 9. In order to beable to reproduce the runs, the default random seed in GAMS 23.7.3 was used.The difficulty of these problems are strongly dependent on the values assignedto the different costs.P2R2|I| |J| |K| Bin. Vars. Con. Vars. Const. Bin. Vars. Cont. Vars. Const.2 20 20 460 2,480 5,300 460 980 1,8405 30 50 1,680 13,640 33,190 1,680 3470 658010 50 100 5,550 70,250 185,350 5,550 11,200 21,50020 50 100 6,050 120,250 335,350 6,050 12,200 22,4003 50 150 7,700 52,800 120,450 7,700 15,650 30,90015 100 200 21,600 300,300 1,040,700 21,600 43,400 83,800Table 1: Sizes of the old and reformulated modelsIn Table 1 the model sizes of P2 and R2 are compared. As can be seen, thedifference is very large, especially for the larger instances. The root node valuesof the relaxed LP for the different formulations is shown in Table 2. As can beseen from Table 1 and Table 2, R2 is both smaller in size and at least as tightas P2.P2 R2|I| |J| |K| LB LB2 20 20 1,024,834 1,026,0545 30 50 2,087,756 2,087,75610 50 100 3,550,280 3,550,28020 50 100 3,234,617 3,234,6173 50 150 4,779,460 4,779,46015 100 200 6,468,441 6,468,441Table 2: Comparison of the tightness in the root node of the relaxed MILPs P2and R2In Table 3 the computational comparisons of the reformulated MILP (R2)and the original formulation (P2) are presented. As can be observed, P2 canonly be solved for the smallest instance within the time limit of 10 minutes. Onthe other hand for the new formulation R2, all instances are solved in a fewseconds. This comparison is made with the same secant function for the squareroot terms in both cases (i.e. zero lower bound in Eq. (50)). If the squareroot term in R2 is relaxed with the tighter underestimator shown in section 5,then the optimal values of the MILP will be higher and therefore closer to theoptimal value of R1. However the solution times for R2 would still be as fast asin Table 3.Table 4 shows the global optimum of the MILP R2, when underestimatingthe square root terms with the tighter lower bound in Eq. (50) and the correspondingcalculated feasible solution of the problem when evaluating the squareroot terms. Note that the optimal solutions for R2 are higher in Table 4 than inTable 3 for the same problems, while the solution times are roughly the same.8

P2R2|I| |J| |K| UB LB Gap Time(s) UB LB Gap Time(s)2 20 20 1,584,557 1,584,557 0.00% 1.27 1,584,557 1,584,557 0.00% 0.095 30 50 3,248,076 2,988,151 7.80% 600 3,169,880 3,169,880 0.00% 0.5810 50 100 6,893,324 3,569,929 48.20% 600 5,173,888 5,173,888 0.00% 4.1820 50 100 5,969,190 3,245,514 45.63% 600 4,602,706 4,602,706 0.00% 6.153 50 150 10,852,632 5,009,423 53.84% 600 7,151,942 7,151,942 0.00% 2.9215 100 200 54,755,897 6,407,108 88.30% 600 8,816,596 8,816,596 0.00% 23.4Table 3: Comparison of the performance of the MILPs P2 and R2, with a timelimit of 600 seconds|I| |J| |K| MILP (R2) MINLP (R1) Gap Time(s)2 20 20 1,708,930 1,783,540 4.18% 0.095 30 50 3,538,980 3,770,404 6.14% 0.6710 50 100 5,862,307 6,215,596 5.68% 4.7420 50 100 5,294,980 5,396,136 1.87% 6.863 50 150 8,161,606 8,479,354 3.75% 1.6515 100 200 10,318,630 10,727,860 3.81% 28.2Table 4: Gap between the optimal solution of R2 and the calculated feasibleMINLP solutionIn Table 5 solution times for the sequential piecewise approach presented insection 5.1 are shown. Even the larger instances can be solved within reasonablecomputational time. However, the solution times for different instances with thesame size, can be completely different. This formulation is strongly dependenton the values of the parameters. This is also why the solution time is longerfor the third instance in Table 5 than for the fourth, even though the fourthinstance is larger in size.|I| |J| |K| Solution LB Gap Time(s)2 20 20 1,776,969 1,769,882 0.40% 0.945 30 50 3,728,020 3,704,300 0.64 % 24910 50 100 6,182,142 6,120,321 0.99% 9,04220 50 100 5,396,821 5,368,078 0.53% 1,0423 50 150 8,489,428 8,409,329 0.94% 41515 100 200 10,704,022 10,597,052 0.99% 10,840Table 5: Solutions for different size instances with the sequential piecewise approach7 ConclusionsIn this short note, we have showed that by reformulating the three-stage multiecheloninventory system with specific exact linearizations, we can solve largerproblems directly with an MILP solver without the need of decomposing theproblem. The new formulation is significantly smaller in size, both in the numberof variables as well as in the number of constraints. An MILP underestimation ofthe problem can be solved very quickly in order to obtain a good feasible solutionfor the MINLP. The paper shows a simple sequential piecewise approximationscheme that can be used to solve the problem within a desired optimality gap.9

8 AcknowledgmentsThe financial supports from the Academy of Finland project 127992 as well asthe Foundation of Åbo Akademi University, as part of the grant for the Centerof Excellence in Optimization and Systems Engineering, are gratefully acknowledged.Financial support from the Center of Advanced Process Decision-making(CAPD) is also acknowledged. Axel Nyberg is grateful for his stay at the CAPD.9 NotationModelsP0, original MINLPP1, original reformulated MINLPP2, original underestimating MILPR1, reformulated P1R2, reformulated P2SetsI = set of suppliersJ = set of candidate DC locationsK = set of CDZsParametersBelow are the input parameters for the instances. All parameters are generatedrandomly with a uniform distribution. The data files used are available fromthe authors, upon request.c1 ij = t1 ij × U[0.05, 0.1], unit transportation cost between supplier and DCc2 jk = t2 jk × U[0.05, 0.1], unit transportation cost from DC to CDZf j = U[150, 000, 160, 000], fixed cost for installing a DC at location jg j = U[0.01, 0.1], annual variable cost coefficient for installing DC at location jh1 j = U[0.1, 1], unit inventory holding cost at DCh2 k = U[0.1, 1], unit inventory holding cost at CDZR k = 0, maximum guaranteed service time to customers at CDZ kSI i = U[1, 5](integers), guaranteed service time of plant it1 ij = U[1, 7](integers), order processing time of DC j if it is served by supplier it2 jk = U[1, 3](integers), order processing time of CDZ k if it is served by DC jµ k = U[75, 150], daily mean demand at CDZ kσ 2 k = U[0, 50], daily variance of demand at CDZ kχ = 365, days per yearθ1 ij = U[0.1, 1], annual unit cost of pipeline inventory from plant i to DC jθ2 jk = U[0.1, 1], annual unit cost of pipeline inventory from DC j to CDZ kλ1 j = 1.96, safety stock factor of DC jλ2 k = 1.96, safety stock factor of CDZ kBinary VariablesX ij = 1 if DC j is served by supplier iY j = 1 if a DC is installed at location j10

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