Lagrange Relaxation for the Capacitated Multi-item Lot-Sizing Problem

1. Introduction

The capacitated multi-item lot-sizing problem referred to as CLSP is often encountered in industry production settings where multiple products are produced and a bottleneck resource exists [1,2,3]. The problem is to determine the production lot sizes of products and timing over a finite number of periods so that the total cost consisting of production costs, set-up costs, and inventory costs is minimal and meets the production demand of each product in each period. Mathematically, the CLSP can be formulated as follows.

$$\begin{array}{l}（\mathrm{CLSP}) \\ \mathrm{minimize} z={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^I(p_{it}{x}_{it}+h_{it}{I}_{it}+s_{it}{y}_{it})}}\end{array}$$

(1)

$$\begin{array}{l}s.t.\\ I_{it-1}+x_{it}-I_{it}=d_{it},i\in I,t\in T\end{array}$$

(2)

$$\begin{array}{l}{\displaystyle \sum _{i\in I}{a}_i{x}_{it}\le C_t,t\in T}\end{array}$$

(3)

$$\begin{array}{l}{x}_{it}\le M{y}_{it},i\in I,t\in T\end{array}$$

(4)

$$\begin{array}{l}{x}_{it,}{I}_{it}\ge 0,i\in I,t\in T\end{array}$$

(5)

$$\begin{array}{l}{I}_{i0},I_{iT}=0,i\in I\end{array}$$

(6)

$$\begin{array}{l}{y}_{it}\in \left\{0,1\right\},i\in I,t\in T \left(7\right)\end{array}$$

(7)

where the meanings of the notations are as follows.

Indices and sets:

i product

t period

I set of products

T set of periods

Parameters:

p_it unit linear production cost for the product i in period t

h_it unit inventory holding cost for one unit of product i between periods t and t+1

s_it setup cost for the product i in period t

d_it demand of product i in period t

a_i unit capacity requirement for the product i

C_t available capacity in period t

M a large positive number

Variables:

x_it amount of product i produced in period t

I_it inventory of product i at the end of period t

y_it binary variables, =1, if x_it >0; =0, x_it =0

In the CLSP, the objective function (1) minimizes the total production, inventory holding and setup costs. Constraint (2) represents the inventory balance, where for any product i the production in period t plus the remaining inventory at the end of period t-1 minus the demand in period t is equal to the inventory at the end of period t. Constraint (3) is the capacity constraint for the bottleneck resource. The capacity consumption of all products in each period t cannot exceed the available amount in that period. Constraint (4) is the associated relation of variables x and y, where a product i is produced in a period t if and only if it has been set up in that period. Constraints (5) - (7) define the decision variables. Note that inventories are non-negative meaning no backlogs allowed. Note also that when production costs do not change over time, the first cost in the objective function is constant and can be removed. The problem often encountered in many important industry application areas, such as production planning and scheduling, logistics management and quality control, etc., is NP-hard [4], thus the solution of the CLSP has continuously been a hot research area in both industry and academic communities [5,6,7].

2. Literature Review

Many publications about the subject of “lot-sizing or lot sizes” can be found in the literature [8,9]. Lot sizing problems can be traced back to the EOQ problem [10], the economic ordering quantity model. EOQ is a trade-off of ordering/setup cost and inventory holding cost to achieve the optimal economic lot sizes. EOQ supposes that the problem is for a single product, stationary, invariable demand, and unlimited production capacity. EOQ can be solved with differential methods. While some complex characteristics are added to the “lot sizes” problem, the problems will be very difficult to solve. For example, when product demand changes over time, more products are produced, etc. However, one of the most essential changes for problem-solving is the existence of production capacity constraints. Consequently, the complexity of a lot-sizing problem can be characterized in terms of the length of the horizon, number of items, and capacity constraints. Based on these characteristics, the lot-sizing problems can be classified into the uncapacitated single-item lot-sizing problem (SILSP), the capacitated single-item lot-sizing problem (CSILSP), the uncapacitated multi-item lot-sizing problem (ULSP), and the capacitated single-level multi-item lot-sizing problem (CLSP). Among them, CLSP is the most common and difficult problem, which has become the core of lot-sizing problem research. However, the SILSP is the simplest lot sizing problem specific to a single product, finite planning horizon of T periods, and no capacity constraints. The SILSPs are solvable in O (T²), except for some variants with some essential extensions, such as setup time and (or) cost, backlogs, time windows, etc., which are NP-hard. The ULSPs are the extension of the SILSPs with multiple items and thus are also solvable in O (T²). The CSILSPs are the extension of the SILSPs with capacity constraints and NP-hard [4]. The CLSP is the extension of CSILSP with multiple items, thus NP-hard. Because of the difficulties of the CLSP, most of the existing solution approaches are heuristic and the exact solution approaches are only for the small-sized problems. In general, the solution approaches to CLSP can be divided into three classes: common heuristics, meta-heuristics, and mathematic programming-based approaches.

Heuristics is sometimes the unique choice when the capacity constraints are very tight in a CLSP. The well-known common heuristics include lot-for-lot, least unit cost (LUC) [11], part-period balancing (PPB) [12,13], Silver-Meal (SM) [14], Lambrecht-Vanderveken (LV) [15], and Dixon-Silver (DS) [16] heuristics. Lot-for-lot heuristic directly converts the demand matrix to a lot-size matrix to obtain an initial solution. This approach maximizes the setup costs and minimizes the inventory holding costs. It is suitable for problems with looser capacity constraints and can quickly obtain an initial feasible solution. LUC heuristic tracks the least cost per unit for production, it can be expressed mathematically as:

$$\underset{t}{\arg \min } k(t)=\frac{s+h{\displaystyle \sum _{\tau =1}^t(\tau -1)d_{\tau }}}{{\displaystyle \sum _{\tau =1}^t{d}_{\tau }}}, t=1,\mathrm{...},T$$

(8)

Here and after, s indicates the setup cost, h, unit inventory holding cost. The PPB heuristic aims at seeking the maximum production over the demand of several periods when the total holding cost does not exceed a single setup cost. Expressed by the formula:

$$\underset{t}{\arg \max } h{\displaystyle \sum _{\tau =1}^t(\tau -1)d_{\tau }-s}\le 0, t=1,\mathrm{...},T$$

(9)

The SM heuristic attempts to minimize the average cost per unit time for each lot. Mathematically, it can be expressed as:

$$\underset{t}{\arg \min }k(t)=\frac{s+h{\displaystyle \sum _{\tau =1}^t(\tau -1)d_{\tau }}}{t}, t=1,\mathrm{...},T$$

(10)

The LV heuristic is an extended Eisenhut heuristic [17]. LV’s idea is analogous to Eisenhut's heuristic, a period-by-period heuristic. Because we always may suppose that the current period is 1, the LV then computes the maximum order up to the period:

$$\underset{t}{\min }{B}_t={\displaystyle \sum _{\tau =2}^t({\displaystyle \sum _{i=1}^I{d}_{i\tau }-C_{\tau })>0,t=2,\mathrm{...},T}}$$

(11)

Suppose that$B_{t^{*}}>0$, this means that from period 2 to period t^*, the cumulative demands exceed the cumulative capacity by$B_{t^{*}}$units that must be backward shifted to period 1. Repeat the above process until the current period is T and a feasible solution is obtained. DS heuristic can be regarded as an extension of the SM heuristic for multiple items. Because the single-item SM heuristic for the multiple-item lot-sizing problems is not necessarily optimal even feasible if the capacity limitations exist. Consequently, for a multiple items problem, the DS would be to increase the item which results in the largest decrease in average costs per unit time per unit capacity.

$$\underset{u_i}{\arg \max }{u}_i=\frac{A{C}_i(T_i)-A{C}_i(T_i+1)}{d_{i,T_i+1}}>0, i=1,\mathrm{...},I$$

(12)

$\mathrm{Where}, A{C}_i(T_i)=(s_i+h_i{\displaystyle \sum _{\tau =1}^{T_i}(\tau -1)d_{i\tau }})/T_i,T_i=1,\mathrm{...},T$, denotes the average cost per unit time of a lot of item i which will satisfy T_i periods’ requirements. Therefore, u_i indicates the marginal decrease in average costs per unit of capacity absorbed under increasing the production of item i from T_i to T_i+1.

Mathematic programming-based approaches are suited for lot-sizing problems with complex constraints and multiple items. The well-known W-W [18] dynamic programming method is for the uncapacitated single-item lot-sizing problem. It solves the USLP in time O (T²). Barany et al. [19] proposed a linear Programming (LP) based approach for CLSP. This approach embeds valid inequalities into a branch and bound framework to solve the linear programming relaxation of the CLSP. Up to 20-item×13-period CLSPs are solved optimally with this approach. Lasdon and Terjung [20] used column generation and generalized upper bound for solving CLSP. Up to 393-item × 6-period CLSPs are solved with their approach with good approximation. Bahl [21] proposed a column generation heuristic for CLSP, where the time-consuming W-W dynamic programming algorithm is replaced by the PPB heuristic in computing shadow prices thus reducing CPU time with a slight loss in optimality. Hindi [22] proposed a variable redefinition approach for the CLSP. The approach uses computationally efficient approaches, such as shortest-path and minimum-cost network flow for obtaining low bounds and upper bounds in a branch and bound framework. Results show the frequent generation of good upper bounds. Thizy and Wassenhove [23] proposed a Lagrangian relaxation approach for CLSP. The Lagrangian relaxation problem consists of |I| uncapacitated single-item problems and each of them is solved with the W-W dynamic programming in O (T²). Feasible solutions are obtained from the resulting transportation problems with specified production periods. The Lagrange multipliers are updated by a subgradient procedure [24]. Trigeiro et al. [25] proposed a Lagrangian relaxation for CLSP with setup times. They relax the capacity constraints to the objective function and solve these decomposed single-item problems with the W-W dynamic programming. Finding feasible solutions is by a heuristic called smooth procedure. Diaby et al. [26] proposed a Lagrangian relaxation for CLSP in which Lagrangian relaxation is used within a framework of branch and bound algorithm to generate bounds. Initial upper bounds are generated by three heuristics: (1) DS heuristic (2) a linear programming-based relaxation and (3) an extension of the solution procedure of Thizy and Van Wassenhove [23] with setup times and multiple resources. Practical-sized problems are tested and the results are very well.

Meta-heuristics have also been used to solve CLSP for many applications. Xie and Dong [27] proposed a heuristic genetic algorithm (GA) for CLSP. They design a domain-specific encoding scheme for lot sizes and provide a heuristic shifting procedure for decoding. Computational results show that the algorithm converges a solution within 200 generations for most of the small-scale examples (N ≤10) and within 500 generations for most of the modest examples (N ≤20). Gaafar [28] applied a genetic algorithm (GA) to solve dynamic lot-sizing with batch ordering and backorders. The approach uses a “012” coding scheme specific to the problem. Computational results show that the proposed genetic algorithm outperforms the modified SM heuristic in terms of optimal rate and percentage deviation. Gaafar et al. [29] proposed a simulated annealing (SA) algorithm for lot-sizing problems. In their algorithm, the “012” coding scheme and 16 mutation-based moves are used for solving dynamic lot-sizing problems and the computational results show the performance of the SA outperforms the compared GA and the modified SM heuristic. Hindi [30] proposed a tabu search (TS) heuristic for CLSP. In his implementation, firstly, a reformulation of the original problem is executed by variable redefinition. The new problem has tighter bounds but more variables and thus is solved by column generation. The resulting feasible solution is further improved by solving a minimum-cost network flow problem, and then the improved solution is used as a starting point for a tabu search procedure. Computational results show the effectiveness of the TS. More solutions for CLSP with metaheuristics are referred to [31].

In this paper, we propose a Lagrange relaxation approach for the CLSP, the work is closely related to that of [23]. However, the difference is clear in the methods for finding the low bounds and upper bounds. The contributions of this paper are as follows.

(1)

A Lagrange relaxation approach for CLSP is implemented.

(2)

A stepping-stone algorithm is developed for solving the resulting transportation problem.

(3)

A fix-up heuristic is proposed for obtaining feasible solutions.

(4)

A local neighborhood search heuristic is used for further searching for high-quality solutions, which increases the chance of finding the optimal solution.

3. Solution Approach

3.1. Lagrange Relaxation

We propose a Lagrange relaxation approach for CLSP [32,33]. Let u_t, t =1, ..., T, be nonnegative Lagrange multipliers for capacity constraints (3), and then a Lagrange relaxation problem (LR (u)) is obtained as follows.

$$\begin{array}{l}LR (u): z_u=min{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^I(p_{it}{x}_{it}+h_{it}{I}_{it}+s_i{y}_{it})}}+{\displaystyle \sum _{t\in T}{u}_t(} {\displaystyle \sum _i{a}_i}{x}_{it}-C_t)\\ ={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in I}((p_{it}+} u_t{a}_i)x_{it}+h_{it}{I}_{it}+s_i{y}_{it})-{\displaystyle \sum _{t\in T}{u}_t{C}_t}}\\ s.t.\\ (2),(4)-(6), u_t\ge 0,t\in T.\end{array}$$

The LR (u) problem can be then decomposed into |I| independent uncapacitated single-item lot-sizing problems, each of which is solvable in O(T²) [18], any of the previously mentioned single-item problem methods, for example, lot-for-lot, LUC, W-W dynamic programming, etc., can be used for this aim. In this paper, the W-W dynamic programming is chosen. Note that for any given u, the solution of the LR(u) will give a lower bound z_LB for the original problem (CLSP).

3.2. Obtaining Feasible Solutions

Let (y_u, x_u, I_u) be the optimal solution of z_u for a given u, note that when the production periods are specified by fixing y_it at 0 or 1 for all i = 1, …, I, t = 1, …, T, the problem can be transformed into a transportation problem with |T| origins and |T|×|I| destinations. Also, note that some of the arcs of the transportation problem are forbidden due to both preventing backlogging or production not starting in that period.

The transportation problem (TP) is formulated as follows.

$$TP(y):z_y=min{\displaystyle \sum _{i\in I}{\displaystyle \sum _{t\in T}{\displaystyle \sum _{\tau \in \mathrm{T}}{\delta }_{it\tau }{\xi }_{it\tau }}}}$$

(13)

s.t.

$${\displaystyle \sum _{t\in T}{\xi }_{it\tau }} \ge a_i{d}_{i\tau },i\in I,t\in T$$

(14)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{\tau \in T}{\xi }_{it\tau }}}\le C_t,t\in T$$

(15)

$${\xi }_{it\tau }=0, \mathrm{if} y_{it}=0 \mathrm{or}t>\tau ;i\in I,t\in T,\tau \in T$$

(16)

$${\xi }_{it\tau }\ge 0,i\in I,t\in T,t\in T$$

(17)

$${\delta }_{it\tau }=\{\begin{array}{c}(p_{it}+{\displaystyle \sum _{q=t}^{\tau -1}{h}_{iq}})/a_i,t\le \tau \\ +\infty ,otherwise\end{array}, i\in I,t\in T,\tau \in T$$

(18)

${\xi }_{it\tau }$: the amount of product i produced in the period t help satisfy the demand in the future period$\tau$.

To solve the resulting transportation problem TP (y), we developed a stepping-stone algorithm [34,35]. The algorithm is shown in Algorithm 1.

Algorithm 1: Stepping-stone algorithm

Get the initial basic feasible solution with the matrix minimum rule Calculate dual variables: uit, vt for the current basis solution Find a non-basic variable with a negative reduced cost which will enter the basis to improve the current basic solution; if such a variable is not found, then stop and the optimal solution has been obtained, else go to step 4; Find the closed loop including the newly found non-basic variable Update values of elements along the found closed loop and obtain a new basic Return to step 2;

Note that the solution to the transportation problem will give an upper bound z_UB of the CLSP if it is feasible. As an illustrative example from the literature [23], here, a 3-item×4-period CLSP with specified y is considered. The problem and data are shown in Table 1, Table 2 and Table 3. The iteration process of the algorithm is shown in Table 4, Table 5, Table 6, Table 7 and Table 8.

In Table 1, a number with a square indicates that the production in the period occurs. In Table 3, the resulting transportation problem is formed with y specified in Table 1. The marks X and X indicate the corresponding arcs are forbidden either because y_it = 0 or because$\tau <t$, respectively. The initial basis is listed in Table 4. The superscripts are the basis variable labels. The initial objective value is 385.3(no setups included). The optimal solution is obtained with 4 iterations and the objective values are gradually reduced to 285.3, 193.3, 162.3, and 146 (optimal). The four closed-loops found are (14-1-0), (10-12-13-1-0-14-15), (13-1-0-14-15-10-4), and (13-1-0), marked with purple dotted boxes.

3.3. Fix-up Heuristic

For a given y_u, from the LR(u), the resulting transportation problem TP(y) often is infeasible. To obtain a feasible solution, we introduce a fix-up policy to make it feasible. The fix-up policy shifts lots among periods using lot techniques. The fix-up heuristic is shown in Algorithm 2.

Algorithm 2: Fix-up heuristic

Check infeasible period cp, compute the remaining capacity RCj, $R{C}_j={\displaystyle \sum _{i\in I}{a}_i{d}_{ij}-L_j}$, for each period j=1,…,T. If (cp < T) go to forward pass; else go to backward pass. Forward pass: for each product check inventory period by period, forward move products lots into periods cp+1, cp+2,…, T, with a cost-effective fashion until no sufficient capacity is available. Go to 4 Backward pass: if the following periods no excessive capacities exist, backward move products lots into periods cp-1, cp-2,…, with cost-effective fashions. Iteratively execute 3 and 4, until a feasible solution is obtained.

$R{C}_j={\displaystyle \sum _{i\in I}{a}_i{d}_{ij}-L_j}$Check infeasible period cp, compute the remaining capacity RC_j,

, for each period j=1,…,T.

If (cp < T) go to forward pass; else go to backward pass.

Forward pass: for each product check inventory period by period, forward move products lots into periods cp+1, cp+2,…, T, with a cost-effective fashion until no sufficient capacity is available. Go to 4

Backward pass: if the following periods no excessive capacities exist, backward move products lots into periods cp-1, cp-2,…, with cost-effective fashions.

Iteratively execute 3 and 4, until a feasible solution is obtained.

3.4. Local Neighborhood Search Algorithm

to find more high-quality solutions, we proposed a local neighborhood search algorithm for the current feasible solution. The neighborhood structures are defined as follows.

(1)

The neighborhood of the lot-move

The first neighborhood structure is for shifting lots between two periods for a single item. Assume there exists an item i, and any two periods t₁ and t₂, two lots$x_{i{t}_1}$> 0,$x_{i{t}_2}$> 0, in that corresponding periods, and again assume there are no positive lots among t₁ and t₂. The neighborhood of lot-move is then defined for all possible moves between t₁ and t₂ as above stated. As an illustrative example, let$\Delta$> 0 be the lot from period t₁ to period t₂, then have

$$\{\begin{array}{c}{x}_{i{t}_1}=x_{i{t}_1}-\Delta \\ x_{i{t}_2}=x_{i{t}_2}+\Delta \end{array},$$

(14)

Note that the lot move may be bidirectional. The forward move reduces inventories, while the backward move may reduce setups. The cost change can be computed from formula (15), (16).

$$\cos \mathrm{t}=-\lambda h_i\cdot \Delta -s_i,\mathrm{for}\mathrm{forward}\mathrm{moves}$$

(20)

$$\cos \mathrm{t}=\lambda h_i\cdot \Delta -s_i,\mathrm{for}\mathrm{backward}\mathrm{moves}$$

(21)

When the inventory cost h_i changes from period to period, the formula needs to adjust correspondingly. The cardinal of the lot-move neighborhood is O(n³).

(2)

The neighborhood of the 2-opt lot-exchanges

The neighborhood is for the 2-opt lot-exchanges which change lots for two items in two continuous periods. Assume there are two items i₁, i₂, and two continuous periods, t₁ and t₂, the related lots are$x_{i_1{t}_1},x_{i_1{t}_2},x_{i_2{t}_1}, x_{i_2{t}_2}$, respectively. Now we swap two lots for two items i₁ and i₂ and two continuous periods t₁ and t₂, and assume that${\Delta }_1\le \min (x_{i_1{t}_2},x_{i_2{t}_1})$,${\Delta }_2\le \min (x_{i_1{t}_1},x_{i_2{t}_2})$, then have $\{\begin{array}{c}{x}_{i_1{t}_1}=x_{i_1{t}_1}+{\Delta }_1\\ x_{i_1{t}_2}=x_{i_1{t}_2}-{\Delta }_1\\ x_{i_2{t}_1}=x_{i_2{t}_1}-{\Delta }_1\\ x_{i_2{t}_2}=x_{i_2{t}_2}+{\Delta }_1\end{array}$, for forward exchange,$\{\begin{array}{c}{x}_{i_1{t}_1}=x_{i_1{t}_1}-{\Delta }_2\\ x_{i_1{t}_2}=x_{i_1{t}_2}+{\Delta }_2\\ x_{i_2{t}_1}=x_{i_2{t}_1}+{\Delta }_2\\ x_{i_2{t}_2}=x_{i_2{t}_2}-{\Delta }_2\end{array}$, for backward exchange.

The cardinal of the neighborhood is O(n⁴).

As an illustrative example, below a 3-item×4-period lot-sizing problem is considered [15]. The data is referred to in the literature. The 2-opt lot-exchange local neighborhood search results are shown in Figure 1.

The initial solution is obtained from the DS heuristic, and the objective function is 1415.7. Two 2-opt lot exchanges were performed and two improved solutions were obtained with values of 1414 and 1412.6, optimal.

3.5. Subgradient Algorithm

For a given u, the solution of the TP(y) gives an upper bound of the original problem. To obtain the maximum upper bound, the traditional subgradient algorithm [31] is selected in this paper. By subgradient theories, ${\displaystyle \sum _{i\in I}{a}_i{x}_{it}}-C_t$ is a subgradient of function z_u at point u. The Lagrange multipliers u are updated using the formula.

$$u_t^{k+1}=\max \left\{0,u_t^k+s^k({\displaystyle \sum _{i\in I}{a}_i{x}_{it}^k}-C_t)\right\} ,s^k=\frac{{\eta }^k(z^{best}-z_u^k)}{{\displaystyle \sum _{t\in T}{({\displaystyle \sum _{i\in I}{a}_i{x}_{it}^k}-C_t)}^2}}$$

(17)

where, $x_{it}^k$ is the solution of LR (u^k), z^best is the best upper bound found, η⁰= 2. The subgradient algorithm is shown in Algorithm 3.

Algorithm 3: Subgradient algorithm

1: given a Lagrangian multiplier uk, k = 0; 2: loop 3: Arbitrarily select a subgradient from ∂(zu), If any of the termination criteria are met then stop; else uk+1 = max {0, uk + θksk}; 4: k=k+1; 5: if k > N then stop; 6: end loop

Notes: 1. selection of θ_k:

${\displaystyle \sum _{k=1}^{\infty }{\theta }_k=} \infty ,\theta \to 0,k\to \infty$;

• θ_k = θ₀ρ^k, 0<ρ<1;
• ${\theta }_k=\frac{z_{UB}^k-z_{LB}^k}{\Vert s^k\Vert }{\eta }_k$;

3

Iterations: N,

4

s^k = 0 or$\Vert s^k\Vert \le \epsilon$,

5

|z_UB^k - z_LB^k| < ε

6

u^k or z(u^k) does not change within a given number of iterations (e.g., 7).

3.6. Overview of the Lagrange Relaxation Algorithm

Based on the description of the components of the algorithm, we give the complete algorithm framework as shown in Algorithm 4.

Algorithm 4: Lagrangian relaxation main procedure

// Initialization 1: k =0 2: LB = -1E+10 // a sufficiently small lower bound 3: UB = 1E+10 // a sufficiently large upper bound 4: Gap = 1E-2 // dual gap percentage 5: K = 5000 // the maximum number of iterations 6: ηk = 2.0 // step-size 7: uk = 0 // Lagrangian multipliers 8: loop // main cycle beginning 9: Solve the Lagrangian relaxation LR (uk) by the W-W dynamic programming, and calculate the current lower bound ZLB (uk) 10: Solve the resulting transportation problem by the stepping-stone algorithm for the y from the solution of the current LR to obtain a ZUB (uk) if it is feasible else { Execute fix-up heuristic Solve the corresponding transportation problem using the stepping-stone algorithm } 11: Execute the local neighborhood search algorithm for raising solution quality 12: if ZLB (uk) < ZLB, ZLB = ZLB (uk) // update low bounds 13: if ZUB (uk) >ZUB, ZUB = ZUB (uk) // update upper bounds 14: if [ (ZUBk – ZLBk) / ZUBk] ≤ Gap, then stop 15: k++, if k >= K then stop; 16: Update Lagrangian multipliers with the subgradient algorithm 17: ηk =1/2*ηk; // update step-size 18: end loop // main cycle ending

4. Computational Results

4.1. Comparison Study

The solution approach proposed in this paper has been implemented on a personal computer using VC++ 6.0 programming language. Computational experiments for the benchmarks and the randomly generated large-sized CLSPs are performed. The benchmarks are from the literature [23]. The computational results are listed in Table 10. The data is listed in Table 9.

The computational results are listed in Table 10. Four approaches, LRFN (the proposed approach), LR (without the local neighborhood search), DS, and LV are compared, wherein the results of DS and LV are from the literature [23], and the optimal solutions are from CPLEX 7.0. We can see that the LRFN proposed in this paper is better than others. For the problems with looser capacity, where the capacity utilization ratios are below 73% of TVW3 and TVW4, the LRFN obtains the optimal solution. For the problems with tighter capacity, TVW2, where the capacity utilization ratio is 91%, LRFN obtains the optimal solution, others obtain the near-optimal solutions, especially, and LV has a larger error, about 11.3%. For the tightest capacity problem, TVW1, where the capacity utilization ratio is 93%, LRFN obtains the best near optimum, and the error is below 1%, but the errors of DS and LV are 3.3% and 6.4%, respectively.

4.2. Large-Sized Problems

The benchmarks we previously computed had been regarded as large problems in that time (1980s). However, today computer ability has made essential progress. In addition, the difficulty of lot-sizing problems can be evaluated with the following criteria:

(1)

The number of time periods, |T| and the number of items, |I|.

(2)

Tightness of the capacity utilization: $\frac{{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^I{d}_{it}}}}{{\displaystyle \sum _{t=1}^T{C}_t}}$.

Thus, we compute the larger problems at different difficulty levels. The data of these problems are randomly generated and adhere to the following agreements.

(1)

the maximal number of items is set to 512;

(2)

the number of periods is set to 48;

(3)

the maximal utilization ratio is set to 93%.

The other problem-specific data are properly randomly generated, for example, C_t: rand (800, 2000), d_it: rand (0,100), s_i: rand (100, 2000), p_i:(1,8), h_i:(1,4), and a_i=1, and so on. Computational results for these problems are listed in Table 11.

In Table 11, the column “Iteration found” represents that the best solution is found in a certain iteration. The meanings of other columns are clear. As shown in Table 11, the average dual gap is within 2% (exactly 1.76) except for some extreme examples, like problem No. 15 the dual gap is 9.54. The reason for that result may be the excessively tight capacity utilization. Usually, the looser the capacity utilization ratio, the smaller the dual gap is; and vice versa.

5. Conclusions

In this paper, we propose a Lagrange relaxation approach for solving the capacitated multi-item lot-sizing problem, called CLSP. The CLSP is often encountered in industry production settings and studying its solution approaches is significant. Our work mainly includes: (1) developing a stepping-stone algorithm for solving the resulting transportation problem to obtain upper bounds, (2) developing a fix-up procedure for obtaining feasible solutions, (3) developing a local search heuristic for obtaining high-quality solutions. Computational experiments from the benchmarks and the randomly generated large-sized examples verify the effectiveness and efficiency of the proposed solution approach.