A LAGRANGIAN RELAXATION APPROACH FOR SUPPLY CHAIN

A LAGRANGIAN RELAXATION APPROACH FOR SUPPLY CHAIN PLANNING WITH ORDER/SETUP COSTS AND CAPACITY CONSTRAINTS Haoxun CHEN

Chengbin CHU

Industrial System Optimization Laboratory Technology University of Troyes, France [email protected]

[email protected]

Abstract A heuristic approach is developed for supply chain planning modeled as multi-item multi-level capacitated lot sizing problems. The heuristic combines Lagrangian relaxation (LR) with local search. Different from existing LR approaches that relax capacity constraints and/or inventory balance constraints, our approach only relaxes the technical constraints that each 0-1 setup variable must take value 1 if its corresponding continuous variable is positive. The relaxed problem is approximately solved by using the simplex algorithm for linear programming, while Lagrange multipliers are updated by using a surrogate subgradient method that ensures the convergence of the dual problem in case of the approximate resolution of the relaxed problem. At each iteration, a feasible solution of the original problem is constructed from the solution of the relaxed problem. The feasible solution is further improved by a local search that changes the values of two setup variables at each time. By taking the advantages of a special structure of the lot-sizing problem, the local search can be implemented by using a modified simplex algorithm, which significantly reduces its computation time. Numerical experiments show that our approach can find very good solutions for problems of realistic sizes in a short computation time and is more effective than an existing commercial optimization code. Keywords: Supply chain planning, lot sizing, lagrangian relaxation, local search, linear programming, simplex algorithm

1. Introduction Driven by fierce global competition, many companies have recognized that effective management of their supply chains can result in reduced costs and increased profits. Supply chain management (SCM) encompasses all management activities that aim at satisfying ISSN 1004-3756/03/1201/98 CN11-2983/N ©JSSSE 2003

customer needs at minimal costs for all companies involved in the production and delivery of products and services to customers. One important issue in SCM is supply chain planning. According to a definition of AMR Research, advanced supply chain planning is the process of balancing materials and plant resources to best meet customer demands while JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING Vol. 12, No.1, pp98-110 March, 2003

Haoxun CHEN, Chengbin CHU

achieving the business goal for reducing costs. There are two principle concepts for supply chain planning. The first one is recently applied in SCM software packages commonly known as Advanced Planning and Scheduling Systems (APS). APS covers various capabilities such as "finite capacity scheduling" or "constraint-based scheduling" at the plant floor level, through advanced long time horizon supply chain planning and coordination. APS at planning level is often LP-based, which formulates a planning problem as a deterministic linear programming or mixed integer linear programming problem (model) and then solves and applies it in a rolling schedule context (Silver et al. 1998). In the model, customer demands are estimated based on demand forecast. At each planning cycle, a mathematical program is solved, either to optimality by using an exact algorithm or a heuristics is applied. Only the production or distribution decisions for the current period are really implemented in factories or warehouses, while the others will be updated in the next planning cycle based on updated demand forecast and resource availability. The second concept explicitly considers demand uncertainty in its planning model. It is mostly developed around the multi-echelon inventory system theory (Federgruen 1993, Silver et al. 1998). Following this concept, supply chains are modeled as multi-echelon inventory systems with stochastic demand, and local (decentralized) stock policies or base-stock (echelon stock) polices are used to control the inventory replenishment of the systems. However, because of stochastic nature, for most multi-echelon inventory models, their optimal

inventory policies are either unknown or very difficult to calculate even if capacity constraints are not considered in those models. In this paper, we adopt the first concept for its relative simplicity. We consider supply chain planning in the medium term, where the problem is to establish quantities for purchasing, production, and distribution over a time horizon of several months so that the total cost of inventory holding and order/setup is minimized while satisfying capacity constraints. This problem can be formulated as a multi-level capacitated lot-sizing problem (MLCLSP) (Secuk et al. 1999). Mathematically, it is a NP-hard mixed integer programming (MIP) problem with 0-1 and continuous variables. Because real planning problems usually involve thousands of 0-1 variables and tens thousand of constraints, exact methods are impractical for their unbearable long computation time. Many researchers have therefore been seeking for effective approximate or heuristic methods. Existing methods in this category can be classified into four classes: decomposition approaches, meta-heuristic search approaches, LP-based approaches, and Lagrangean based approaches (Katok et al. 1998). The decomposition approaches ignore the multi-level structure of the MLCLSP and solves a sequence of single level capacitated lot sizing problems (Tempelmeier and Helber 1994). The meta-heuristic search approaches include simulated annealing, tabu search, genetic algorithms, and evolution strategies (Salomon 1991). The LP-based approaches involve LP relaxation of a mixed integer programming formulation of the MLCLSP (Harrison and Lewis 1995, Katok et al. 1998). The Lagrang-

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A Lagrangian Relaxation Approach for Supply Chain Planning with Order/Setup Costs and Capacity Constraints

ian based approaches use a branch-and-bound strategy with Lagrangian relaxation to solve sub-problems followed by a smoothing procedure to eliminate infeasibilities (Billington et al. 1986, Tempelmeier and Derstroff 1996). Tempelmeier and Derstroff (1996) present such an approach to the MLCLSP. In the approach, both the inventory balancing constraints between two consecutive levels and the capacity constraints at each level are relaxed, and Lagrange multipliers are updated using sub-gradient optimization to compute lower bounds. Feasible solutions are then constructed by a heuristic finite scheduling procedure that shifts production away from overloaded periods. A recent overview of the lot-sizing literature is given by Drexl and Kimms (1997). In this paper, we consider supply chain planning in the setting of MLCLSP and present a heuristics that combines Lagrangian relaxation (LR) with local search. Different from existing LR approaches that relax capacity constraints and/or inventory balance constraints, our approach only relaxes the technical constraints that each 0-1 setup variable must take value 1 if its corresponding continuous variable is positive. The relaxed problem is approximately solved by using the simplex algorithm for linear programming, while Lagrange multipliers are updated by using a surrogate sub-gradient method that ensures the convergence of the dual problem in case of the approximate resolution of the relaxed problem. At each iteration, a feasible solution of the original problem is constructed from the solution of the relaxed problem. The feasible solution is further improved by a local search that changes the values of two setup variables at each time. By taking the advantages

100

of a special structure of the MLCLSP problem, the local search can be implemented by using a modified simplex algorithm that significantly reduces its computation time. The performance of our approach is compared with a commercial code, IBM Optimization System Library (OSL) 3.0, on two sets of testing problems on a Pentium III PC with 700 MB CPU. One is a set of small-size randomly generated problems that are optimally solved by OSL. The other is a set of medium-size random problems that are solved to best by OSL with a time limitation of 10000 seconds. For the former, our approach finds solutions within 1% derivation of the optimal cost. For the latter, our approach finds solutions significantly better than those found by the time constrained OSL in a much shorter computation time.

2. Problem Formulation We consider a supply chain with multiple facilities consisting of manufacturing factories or distribution warehouses. Each facility has a finite capacity and involves multiple intermediate or finished products. The production and distribution of the products share the capacities. The supply chain planning is carried out over a time horizon of several periods. Each intermediate or finished product incurs an inventory holding cost if it holds inventory, and incurs order/setup cost if a purchasing order is placed or a production order is released to produce the product. To produce an intermediate or finished product, it may require multiple units of several other intermediate products. That is defined by the bill-of-materials (BOM) of the product. The production of each



product from intermediate products of its immediate upstream level has a lead time that represents the minimum time from the release of a production order to the time that the product is available for usage in other products or for delivery to customers. If production capacity is not available when the production order is released, the real lead time for producing the product may be longer. A similar lead time exists for an distribution operation. Customer demands for finished products are assumed to be known but may change over time. There is no external demand for intermediate products. The demand for an intermediate product is only internally generated by the production of intermediate or finished products at its downstream level. For simplicity, setup times are not considered in this paper. The notations to be used for problem formulation are described in the following: Indices i = 1,...,N

index of intermediate and finished products, index of planning periods, index of facilities.

t = 1,...,T k = 1,...,K Parameters hi: inventory holding cost rate of item i ($ per unit per period), si: set-up cost of item i ($ per set-up), li: minimum lead time for item i aij: number of units of item i required for the production of one unit of item j, dit: demand for item i in period t, bik: capacity utilization of item i at facility k (capacity units per unit of item i), ckt: capacity of facility k at time t (units of capacity). Decision variables xit: production/purchasing quantity of item

i initialized in period t, δit: setup/order variable, δit = 1 if item i is produced or purchased in period t, 0 otherwise, yit: inventory of item i at the end of period t. With these notations, the supply chain planning problem can be formulated as the following mixed integer programming: MLCLSP: N T

Min C = ∑ ∑ (hi y it + s i δ it ) i =1 t =1

(1)

subject to N

yi,t-1 + x i ,t −li – yit − ∑ a ij x jt = dit j =1

i = 1,...,N; t = 1,...,T (2a) N

∑ bik x it ≤ ckt, k = 1, ...,K; t = 1, ...,T

(2b)

i =1

xit(1 – δit) = 0, i = 1,...,N; t = 1,...,T (2c) xit ≥ 0, yit ≥ 0, (2d) δit = 0 or 1. (2e) where equations (2a) are inventory balance constraints, inequalities (2b) are capacity constraints, equations (2c) are technical constraints. Let J = N×T and M = (N+K)×T, the problem can be reformulated in a standard form as: P: Min C = c1x + c2δ

(3)

subject to Ax + u = b xj(1 – δj) = 0, j = 1,..., J

(4a) (4b)

xj ≥ 0,

j = 1,..., J

(4c)

ui ≥ 0,

i = 1,..., M

(4d)

δj = 0 or 1, j = 1,..., J

(4e)

T

where x = (x1, x2,...,xJ) is a vector representing


101


all production variables xit, δ = (δ1, δ2,..., δJ)T is a vector representing all setup variables, u = (u1, u2,..., uM)T is a vector of slack variables, c1 ∈ RJ (R is the set of real numbers), c2 ∈ RJ, A ∈ RM×J, b ∈ RM.

3. Lagrangian Relaxation Problem MLCLSP has been proved to be NP-hard. As a result, exact methods may not be appropriate for practical use for their exponentially increasing computation time since real supply chain planning problems usually involve thousands of 0-1 variables and tens thousand of constraints. Lagrangian relaxation (LR) has been used to solve this problem in the literature. Existing LR approaches usually relax the capacity constraints and/or the inventory balance constraints. This leads to an easy-to-solve relaxed problem, but the excessive relaxation of important constraints makes the construction of a feasible solution difficult and causes the quality of the solution far from optimality sometimes. In this section, we will develop a LR approach that only relaxes the technical constraints that each 0-1 setup variable must take value 1 if its corresponding continuous variable is positive. The relaxed problem is approximately solved by using the simplex algorithm for linear programming, while Lagrange multipliers are updated by using a surrogate subgradient method that ensures the convergence of the dual problem in case of the approximate resolution of the relaxed problem.

3.1 Relaxation Framework By introducing Lagrange multipliers { λj, j = 1, …, J} to relax constraints (4b), problem P becomes:

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RP: Min L((x, δ ), λ ) J

J

j =1

j =1

= ∑ (c1j x j + c 2j δ j ) + ∑ λ j x j (1 − δ j )

(5)

J

= ∑ [(c1j + λ j ) x j + c 2j δ j − λ j x jδ j ] j =1

subject to (4a), (4c), (4d), and (4e). Let D(λ) denote the optimal value of RP for a given multiplier vector λ = { λj, j = 1,..., J}, the Lagrangian dual problem of RP is: DP: Max D(λ), λ

(6)

where D(λ) = Min {L((x,δ), λ) | s.t. (4a), (4c), x ,δ

(4d), and (4e) }.

3.2 Approximate Resolution of the Relaxed Problem The relaxed problem RP can not be decomposed into subproblems and cannot be easily solved to optimality because of the coupling terms λjxjδj in its objective function. However, it can be approximately solved by using a Gauss-Seidel iteration-like method. That is, the problem is solved for a given {δj} to obtain an optimal {xj} at the first iteration, it is then solved for the {xj} given by last iteration to obtain an optimal {δj}, and so on, until the same {xj} and {δj} are obtained at two consecutive iterations or a given number of iterations is reached. When {δj} is given as { δˆ j }, problem RP becomes: RP1: Min L J

= ∑ [(c1j + λ j − λ jδˆ j ) x j + c 2j δˆ j ]

(7)

j =1



subject to (4a), (4c), and (4d). This is a linear programming problem which can be solved by using the simplex algorithm. When {xj} is given as { xˆ j }, problem RP becomes:

defined as ~

J

L(( x, δ ), λ ) ≡ ∑ (c 1j x j + c 2j δ j ) j =1

(10)

J

+ ∑ λ j x j (1 − δ j ) j =1

2

RP : MinL J

= ∑ [(c1j + λ j ) xˆ j + (c 2j − λ j xˆ j )δ j ] j =1

(8)

for some (x, δ) satisfying constraints (4a), (4c), (4d), and (4e), and the corresponding surrogate sub-gradient is defined as g~ ((x,δ)) ≡ { g~ ((x,δ))} = {x (1 – δ )} (11) j

j

subject to (4e).

j

Note that (x, δ) is not required to be an optimal solution of the relaxed problem RP and

2

The optimal solution of RP is:

can take different values.

δj = 0, if c 2j − λj xˆ j > 0, δj = 1, if c 2j − λj xˆ j ≤ 0.

In our application, (x,

δ) is taken as the approximate solution of RP obtained by using the method presented in last (9)

subsection. Given

the

multiplier

vector

λk

and

3.3 Dual Maximization by Surrogate Subgradient Method

approximate solution (x ,δ ) of the relaxed

A Lagrangian dual problem is usually solved by using the subgradient (SG) method, but the application of SG requires optimal resolution of each relaxed problem. Since the relaxed problem we consider here is only approximated solved, the method is no longer applicable. Fortunately, there is a recently developed subgradient-like method called surrogate subgradient (SSG) method (Zhao et al. 1999), which can be used to solve our dual problem in case of the approximate resolution of the relaxed problem. In the following, we briefly introduce the method in the context of our problem. SSG is similar to SG except for the definition of subgradient and the step sizing scheme for the update of multipliers. For our problem, as an extension of the dual in (6), the surrogate dual is

updates the multiplier vector according to λk+1 = λk + sk g~ k , (12) k ~ where g is the surrogate subgradient given by

k

k

problem at the kth iteration, the SSG method

g~ k = g~ ((xk,δk)) = { x kj (1 – δ kj ) }

with stepsize sk satisfying ~ 0 < sk < ( D * − L k ) g~ k

2

(13)

(14)

Here D* = max D(λ) is the optimal value of λ

~ ~ dual problem DP, Lk = Lk ((xk,δk), λk) is the

surrogate dual at the kth iteration. The conditions for ensuring that SSG converges towards the optimal solution of the dual problem are: 1) Initially at multiplier vector λ0, the solution {x0, δ 0} of the relaxed problem satisfies


103


L((x0, δ 0), λ0) < D*

(15)

2) At each iteration with multiplier vector λ (k ≥ 1), the solution {xk, δk} of the relaxed problem satisfies (16) L((xk, δ k), λk) < L((xk−1, δ k−1), λk) where (xk−1, δk−1) is the solution of the relaxed problem obtained at last iteration with multiplier vector λk−1. In our implementation of SSG, the multiplier vector λ is initiated at zero, i.e., λ0 = 0. In this case, the first convergence condition holds since the coupling terms λjxjδj in the objective function of RP at the first iteration disappear and the solution of RP obtained by using the method presented in last subsection is in fact optimal. In most cases, the second condition also holds after problems RP1 and RP2 are solved by one time at each iteration. This is because at the iteration with multiplier vector λk, xk is an optimal solution of RP with δ = δ k−1, and δ k is an optimal solution of RP with x = xk, so we have L((xk, δk), λk) ≤ L((xk, δk−1), λk) ≤ L((xk−1, δk−1), λk). This implies that L((xk, δ k), λk) ≤ L((xk−1, δ k−1 ), λk) When L((xk, δ k), λk) = L((xk−1, δ k−1), λk), RP1 and RP2 can be solved once again with new data to improve the current solution (xk, δk) until L((xk, δk), λk) < L((xk−1,δk−1),λk) holds. One important factor affecting the performance of the surrogate sub-gradient method is step sizing. Since the surrogate dual defined by (10) is not a dual in the common sense, its value may exceed the optimal (minimum) value of the original problem. For this reason, an adaptive step sizing scheme which estimates the optimal value based on the surrogate dual rather than the best primal obtained so far is adopted in our implementation of the method. With the scheme, the step size sk k

104

is set according to the formula: 2 ~ sk = β ( Dˆ * − L k ) g~ k ,

(17)

where β is a parameter with 0 < β < 1, ~ Dˆ * = (1 + ω / θ ρ ) × L[ k ] is an estimate of the ~ optimal dual D*, L[ k ] is the best surrogate dual obtained prior to iteration k. Parameters ω and ρ are chosen as ω ∈ [0.1, 1.0] and ρ ∈ [1.1, 1.5], respectively. Parameter θ is adaptively adjusted ~ ~ with θ = max (1, θ −1) if L k > L[ k ] , and θ = θ + 1 otherwise.

4. Local Search Improvement At each iteration of the LR approach when subproblem RP1 is solved, a feasible solution to the original lot sizing problem is obtained by setting each setup variable δi corresponding to positive xi of the solution of RP1 to 1 and all other setup variables to 0. This feasible solution can be further improved by a local search procedure using a modified simplex algorithm. In order to do so, consider a linear programming problem derived from problem P by extracting its linear part, i.e., the part relating to variables {xj} and {ui} as described in the following: PL: Min C = c1x

(18)

subject to Ax + u = b (19a) xj ≥ 0, j = 1,..., J (19b) (19c) ui ≥ 0, i = 1,..., M Suppose that the optimal solution of RP1 is x , its corresponding solution to the original problem P is ( x , δ ), where δ i = 1 if xi > 0 and δ i = 0 if xi = 0. The solution x is a feasible basic solution (FBS) to problem PL or an extreme point to the constraint set of PL since PL and RP1 have the same set of constraints.



Starting from x , we want to find a new FBS x ' of PL, which is adjacent to x , such that its corresponding solution ( x ' , δ ' ) to problem P is better than ( x , δ ), i.e., C( x ' , δ ' ) < C( x , δ ), where δ i' = 1 if xi' > 0 and δ i' = 0 if xi' = 0. Since x ' is an extreme point of PL adjacent to x , x ' can be obtained from x by a pivot operation that exchanges the places of a nonbasic variable and a basic variable in x , i.e., a nonbasic variable is changed to basic, and a basic variable is changed to nonbasic. The problem is how to identify a nonbasic variable and a basic variable to exchange so that C( x ' , δ ' ) < C( x , δ ). Suppose that a non-basic variable xj is selected to change to basic. According to linear programming theory, a basic variable that is first driven to negative as variable xj increases from 0, denoted by xi j , will be selected as the basic variable to be exchanged with the non-basic variable.

Suppose

that

the

reduced

cost

corresponding to non-basic variable xj in the simplex tableau of problem PL is c 1j and that the

maximum

possible

increment

of

the

non-basic variable before a basic variable is driven to negative is v j . On the one hand, since only two set-up variables δ j and δ i j have

different values in ( x ' , δ ' ) and ( x , δ ), the

difference of the setup costs of the two solutions is c 2j − c i2j . On the other hand, the difference of the holding costs of the two solutions is c 1j v j . Thus, the difference of the objective

values of problem P at solutions ( x ' , δ ' ) and ( x , δ ) is c 1j v j + c 2j − ci2j . The condition for

selecting a non-basic variable xj such that C( x ' , δ ' ) < C( x , δ ) can then be formulated as:

c 1j v j + c 2j − ci2j < 0

(20)

Note that if a slack variable ui is selected as a nonbasic variable or a basic variable, we should take c 2j = 0 or ci2j = 0. In the local search procedure, each time a nonbasic variable satisfying condition (20) is selected to change to basic by a pivot operation. This process continues until no nonbasic variable satisfying (20) can be found. The computational complexity for such a selection is O(J*M), where J and M are the number of production variables and the number of constraints of PL, respectively.

5. Computational Results In this section, we evaluate the performance of our approach, with compared to IBM OSL version 3, on a Pentium III PC with 700 MB CPU. OSL is one of the most powerful commercial codes for solving linear and mixed integer programming problems, so a comparison of our algorithm with OSL provides good benchmarks. First, we test the algorithm on a set of small-size random problems to see how good its solutions are compared with the optimal solutions obtained by OSL. We then compare our algorithm with OSL on a set of medium-size random problems to evaluate its performance for problems of more realistic sizes.

5.1 Small-size Random Instances This set of 10 problems is generated from supply chains with a BOM structure as shown in Figure 1. In each instance, 8 intermediate products (numbered from 1 to 8) and 2 finished products (numbered 8 and 9) are produced by 3 facilities, where the finished products are manufactured and assembled from the


105


intermediate products. The set of products produced by the three facilities are {1, 2, 3, 4}, {5, 6} and {7, 8, 9, 10}, respectively. The planning horizon is 6 periods. 1

3 5

2

7

9

8

10

6

4

Figure 1 BOM Structure for Small Problems Other data of the problems are randomly generated in the following way: the minimum lead time for producing each product from intermediate products of its immediate upstream level is simply assumed zero. For each product, its inventory holding cost rate and setup cost are assumed to be constant over time. They are randomly generated from uniform distribution U[1, 10] and U[500, 1500], respectively. The

demands of the finished products are generated from normal distribution N(100,10).There is no external demand for intermediate products. The capacity utilization (in capacity unit) of producing one unit of each product by a facility is randomly generated from uniform distribution U[1, 5]. To set facility capacity, we first calculate the average workload per period for each facility (in capacity unit), and then set the capacity of the facility in each period to be the workload multiplied by a facility utilization ratio randomly generated from uniform distribution U[0.7, 0.9]. The initial inventory of each product is set to be its average lead-time demand plus a safety stock with z-value 1.0. For this set of problems, we compare the solutions obtained by our algorithm with the optimal solutions obtained by OSL. The step size parameter β of our algorithm is taken as 0.9 and the algorithm is terminated after 200

Table 1 Computational Results for Small Problems Prob.

CLRLS

COPT

PD

CPULRLS

CPUOSL

1

31098

31057

0.13

1.30

202.8

2

17798

17798

0.00

1.41

55.3

3

57239

56689

0.97

0.69

11323.3

4

12736

12736

0.00

1.31

1082.6

5

52494

52494

0.00

0.75

14079.1

6

33794

33794

0.00

1.11

13902.3

7

11524

11524

0.00

0.98

1704.4

8

59214

59214

0.00

0.96

12396.2

9

47716

47612

0.22

0.55

9514.4

10

22235

22235

0.00

0.70

890.3

No.

* PD = (C LRLS − C OPT ) C OPT × 100

106



iterations. The computational results are given in Table 1, where CLRLS and COPT are the cost obtained by our algorithm and the optimal cost obtained by OSL respectively, PD is the percentage difference between CLRLS and COPT, CUPLRLS and CPUOSL are the computation times (in seconds) of our algorithm and OSL respectively. From this table, we can see that our algorithm finds an optimal solution for 7 of the 10 problems and finds a near-optimal solution within 1% derivation of the optimal cost for other 3 problems. The computation time required by our algorithm is around 1 second, much shorter than the time required by OSL. For some problems, OSL runs more than 10000 seconds to locate an optimal solution.

These results show that our algorithm is very effective for small problems.

5.2 Medium-size Random Instances This set of 10 problems is generated from supply chains with a BOM structure as shown in Figure 2. In each instance, 54 intermediate products and 6 finished products (numbered from 1 to 6) are produced by 10 facilities, where the finished products are manufactured and assembled from the intermediate products. The BOM as shown in Figure 2 has 10 levels, each level corresponds to a facility which produces all intermediate or finished products at the level. The planning horizon is 12 periods.

Table 2 Computational Results for Medium Problems Prob.

CLRLS

COSL

PD

CPULRLS

1

180293

211682

−14.83

274.2

2

628326

688459

−6.28

307.1

3

509178

565038

−9.88

287.3

4

433164

490835

−11.75

264.2

5

463583

515723

−10.11

242.6

6

791100

874423

−9.53

217.2

7

647865

721564

−10.21

220.0

8

773182

814285

−5.05

326.4

9

636511

702263

−9.36

252.8

10

407575

498464

−18.23

277.9

No.

PD = (C LRLS − C OSL ) C OSL × 100 Other data of the medium problems are randomly generated in the same way as for the small problems except for the setting of facility capacities and initial inventories. Now, the

capacity of each facility is set to be its average workload per period multiplied by a facility utilization ratio randomly generated from uniform distribution U[0.5, 0.7], and the initial


107


inventory of each product is set to be its average lead-time demand plus a safety stock with z-value 2.0. The decrease of facility utilization ratio and the increase of safety stock z-value compared with those of the small problems are only for ensuring that the randomly generated problems with more BOM levels are feasible. For this set of problems, we compare the performance of our algorithm with a time-limited (time-truncated) OSL branch and bound algorithm which is terminated after the CPU time of 10000 seconds. The step size parameter β and the number of iterations of our algorithm are taken the same as those for the small problems. The computational results are given in Table 2, where COSL is the best cost of a problem obtained by OSL with the time limitation, PD is the percentage difference between the cost of our algorithm and the OSL best cost.

obtained by our algorithm is 10.5% lower than that obtained by OSL on average and the computation time required by our algorithm is only a small fraction of the time required by OSL. These results show that our algorithm is more effective than OSL for the medium-size problems. In summary, our approach can find very good solutions for small to medium-size problems in a short computation time and is more effective than the commercial optimization code OSL.

6. Conclusions In this paper, a heuristic approach that combines Lagrangian relaxation with local search improvement has been developed for supply chain planning modeled as multi-item multi-level capacitated lot sizing problems. Numerical experiments show that our approach can find very good solutions for problems of realistic sizes in a short computation time and is more effective than an existing commercial optimization code. Further work is to extend the approach to supply chain planning with demand uncertainty.

References

1

2

3

4

5

6

Figure 2 BOM Structure for Medium Problems From this table, we can see that the cost

108

[1] Billington, P. J., McClain, J. O. and Thomas, L. J., “Heuristics for Multilevel Lot-Sizing with a Bottleneck”, Management Science, Vol. 32, No. 8, pp989-1006, 1986. [2] Federgruen, A., “Centralized Planning Models for Multi-Echelon Inventory Systems under Uncertainty”, Graves, S. C., Rinnooy Kan, A. H. G., and Zipkin, P. H.,



Logistics of Production and Inventory, North-Holland, pp133-173, 1993. [3] Drexl, A., Kimms, A., “Lot Sizing and Scheduling

Survey

and

Extensions”,

European Journal of Operational Research, Vol. 99, pp221-235, 1997. [4] Harrison, T. P. and Lewis, H. S., “Lot Sizing in Serial Assembly Systems with Multiple Constrained

Resources”,

[5] Katok E., Lewis, H. S., and Harrison T. P., “Lot Sizing in General Assembly Systems with Setup Costs, Setup Times and Multiple Resources”,

Management

Science, Vol. 44, No. 6, 1998. [6] Salomon, M., Determining Lotsizing Models for Production Planning, in Lecture Notes in Economics and Mathematical Systems, Vol. 355, Springer Verlag, Heidelberg, 1991. [7] Secuk E. S., Simpson, N. C. and Vakharia, A. J., “Integrated Production/Distribution Planning in Supply Chains: An Invited Review”, European Journal of Operational Research, Vol. 115, pp219-236, 1999. [8] Silver, E. A., Pyke D. F. and Peterson, R., Inventory Management and Production Planning and Scheduling, Wiley, New York, 1998. [9] Tempelmeier H. and Derstroff, M., A “Lagrangean-Based Heuristics for Dynamic Multi-Level

Multi-Item

Structure”, European Journal of Operational Research, Vol. 75, pp296-311, 1994. [11]Zhao, X., Luh, P. B. and Wang, J., “TheSurrogate Gradient Algorithm for Lagrangian Relaxation Method”, Journal of Optimization Theory and Applications, Vol. 100, No. 3, p699-71, 1999.

Management

Science, Vol. 41, No.11, 1995.

Constrained

Capacitated Lotsizing for General Product

Constrained

Lotsizing with Setup Times”, Management Science, Vol. 42, pp738-757, 1996. [10] Tempelmeier, H. and Helber, S., “A Heuristtic for Dynamic Multi-Item Multi-Level

Haoxun Chen received his BSc degree in applied mathematics from Fudan University, China in 1984, his Master and Ph.D. degrees in systems engineering from Xi’an Jiaotong University, China in 1997 and 1990, respectively. Since 1990, he had been with Xi’an Jiaotong University, where he was an Associate Professor from 1993 to 1996. He visited INRIA-Lorraine, France as a Visiting Professor in 1994, the University of Magdeburg, Germany as a Research Fellow of the Alexander von Humboldt Foundation in 1997 and 1998, and the University of Connecticut as a Research Assistant Professor in 1999 and 2000. His research interests include discrete event system control, production planning and scheduling, and supply chain management. He has published more than 60 papers in technical journals and conference proceedings and received the Best Transactions Paper Award from IEEE Robotics and Automation Society in 1998. He is currently an Associate Professor at the Technology University of Troyes, France. Chengbin Chu received his BSc degree from Hefei University of Technology, China in Electrical Engineering in 1985 and his PhD degree from Metz University, France in


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Computer Science in 1990. He worked at INRIA, France, from 1987 to 1996. He has been a Professor at Troyes University of Technology, France since 1996, where he is also the Director of the Industrial Systems Optimization Laboratory. He is interested in research areas related to operations research and modeling, analysis and optimization of production systems and supply chains. He is author or co-author of one book and more than thirty articles in

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journals such as Operations Research, SIAM Journal on Computing, and IEEE Transactions on Robotics and Automation. He also published more than fifty papers in conference proceedings. He received the First Prize of Robert Faure Award in 1996, and received the Best Transactions Paper Award from IEEE Robotics and Automation Society in 1998. He is an Associate Editor of IEEE Transactions on Robotics and Automation.
