The stochastic uncapacitated lot-sizing problems with incremental quantity discount have been studied in this paper. First, a multistage stochastic mixed integer model is established by the scenario analysis approach and an equivalent reformulation is obtained through proper relaxation under the decreasing unit order price assumption. The proposed reformulation allows us to extend the production-path property to this framework, and furthermore we provide a more accurate characterization of the optimal solution. Then, a backward dynamic programming algorithm is developed to obtain the optimal solution and considering its exponential computation complexity in term of time stages, we design a new rolling horizon heuristic based on the proposed property. Comparisons with the commercial solver CPLEX and other heuristics indicate better performance of our proposed algorithms in both quality of solution and run time.
The lot-sizing problems have been the subject of intensive research in the last few decades. The basic definition of single-item lot-sizing problems can be stated as follows: the order (production), inventory and backlog quantities in each period should be determined to meet the deterministic but dynamic demand over a finite time horizon. The objective is to minimize the total costs, which consist of the fixed setup cost, order cost and inventory cost. Different quantity discount policies, such as all-units quantity discount and incremental quantity discount, have been widely executed in practice and thus have also been introduced into the lot-sizing problems.
Although the deterministic planning and scheduling models have been intensively studied, in practice there are many different sources of uncertainties, such as customer demand, production lead-time, and product price, which make information that will be needed in subsequent decision stages unavailable to the decision maker. In such cases, the solution provided by a deterministic model may be of little value in terms of applicability of the model’s recommendations, see Beraldi et al. [
Wagner and Whitin [
For the stochastic version problem, although Ahmed et al. [
Besides the algorithms based on the extended W-W properties, Lulli and Sen [
To the best of our knowledge, little research has been reported on the stochastic lot-sizing problems with incremental quantity discount (SLP-IQD). However as it is reported in the survey by Munson and Rosenblatt [
The remainder of the paper is organized as follows. In Section
First, we will establish an mathematical model for the deterministic uncapacitated lot-sizing problems with incremental quantity discount (DULS-IQD). Considering a planning horizon of
The incremental quantity discount cost structure is given as follows:
Thus, the DULS-IQD can be formulated as
The objective function (
In this subsection, a stochastic model is established by scenario analysis approach. We assume that the problem parameters follow discrete time stochastic processes with a finite probability space and evolve as a multistage stochastic scenario tree. Let node
Here using the above notation, the deterministic model in the above subsection can be extended to the stochastic environments by replacing the stage index
An equivalent mixed integer programming formulation can be easily obtained by introducing auxiliary order quantity variables and corresponding Boolean variables. A group of variables are assigned for each order quantity interval as follows:
For model brevity, we introduce the following notations:
Formulation ESP1 is equivalent to the original formulation SP.
Suppose that
For each optimal solution of SP,
Further, by relaxing the constraints on Boolean variables and order quantity variables in (
Under the decreasing unit order price assumption, formulation ESP2 is equivalent to ESP1 and SP.
Because the feasible set of the relaxed formulation ESP2 contains the feasible set of ESP1, the optimal solutions of ESP2 must be proven feasible for ESP1. Let
Second, it is asserted that the relaxed constraints
From the above analysis, a better solution can always be obtained by setting
Third, we prove that the relaxed constraints
In this section, we explore the property of the SULS-IQD and design-efficient algorithms. It is necessary to highlight the differences between the deterministic problems and the stochastic problems. First, Ahmed et al. [
Formulation ESP2 presented in Section
For any instance of SULS-IDQ, there exists an optimal solution
Note that under assumption that all lead time is equal to 1 and by similar arguments, the second optimal condition can be regarded as an extension of the Semi-Wagner-Whitin Property in Huang and Küçükyavuz [
For any optimal solution of ESP2,
First, it is asserted that there exists at least an optimal solution by Weierstrass’ theorem since the feasible set is compact (note that
We scan the optimal solution
The objective cost for
Since
Since
Thus, the optimal solution holding the proposed property can always be constructed after finite steps.
To recursively calculate the optimal solution, the following functions are introduced as in Guan and Miller [
From Proposition
To obtain the exact optimal solution of SP, it is not necessary to completely characterize the value function
Without loss of generality, we assume
For each stage For each node For each possible initial inventory (if step 1: Calculate step 2: Calculate otherwise step 3: Calculate End For Iteration ( End For Iteration ( End For Iteration (
In dynamic lot-sizing and scheduling problems with a large planning horizon, rolling horizon heuristics have been developed to decompose the original large-scale problem into a set of smaller subproblems. See, for example, [
At each iteration for
At step 1 for given node
where
At step 2 for given node
where BF depends on the distribution of demand. We will set proper
In the above strategies, the major modification comes from the iteration for
For each node
This lemma is proven by induction from nodes at stage
Now consider the possible initial inventory at node
Combine the above two cases, the conclusion holds for node
Figure
Example of RHH with
By summarizing the above analysis, the rolling horizon heuristic is given as in Algorithm
For each stage For each node For each initial inventory Set step 1: If by ( step 2: If calculate step 3: Calculate End For Iteration ( End For Iteration ( End For Iteration (
Next the computation complexity of RHH is analyzed. For given node
For any instance of SULS-IDQ, the rolling horizon heuristic with parameter
The above analysis can be applied to the dynamic programming algorithm, thus the total run time for DP is given by
In this section, the computational results on both DP and RHH are reported. In the computational analysis, we first concentrate on identifying proper settings of parameters FT and BT for RHH by comparison of its relative error gap and run time with DP’s. Then, DP and RHH are compared with the CPLEX solver and other heuristics for the lot-sizing problems with fixed charge.
In order to evaluate performance of the proposed DP and RHH, and explore the proper parameters settings of
Problem instances.
Instance |
|
|
|
Int. | Cont. |
---|---|---|---|---|---|
|
2 | 10 | 1023 | 3069 | 7161 |
|
2 | 11 | 2047 | 6141 | 8188 |
|
2 | 12 | 4095 | 12285 | 16380 |
|
3 | 7 | 1093 | 3279 | 4372 |
|
3 | 8 | 3280 | 9840 | 13120 |
|
3 | 9 | 9841 | 29523 | 39564 |
|
4 | 6 | 1365 | 4095 | 5460 |
|
4 | 7 | 5461 | 16383 | 21844 |
Problem parameters.
Parameter | Distribution |
---|---|
|
Uniform |
|
Uniform |
|
Uniform |
|
Uniform |
|
Truncated normal |
In order to evaluate the performance of the proposed RHH method with different parameters, we define two different implementations of RHH and the problem parameter
Performance of RHH with different (FT, BT).
Instance | cv |
|
|
DP | |||
---|---|---|---|---|---|---|---|
RE (%) | CPU (sec) | RE (%) | CPU (sec) | Value | CPU (sec) | ||
|
0.05 | 0.289 | 0.016 | 0.0 | 0.094 | 12968.28 | 0.100 |
|
0.15 | 0.168 | 0.015 | 0.0 | 0.110 | 12991.93 | 0.985 |
|
0.25 | 0.135 | 0.015 | 0.0 | 0.110 | 13082.23 | 0.969 |
|
0.05 | 0.039 | 0.079 | 0.0 | 0.219 | 25694.23 | 4.672 |
|
0.15 | 0.015 | 0.079 | 0.0 | 0.219 | 26111.22 | 4.672 |
|
0.25 | 0.005 | 0.078 | 0.0 | 0.218 | 26447.11 | 4.703 |
|
0.05 | 0.009 | 0.172 | 0.0 | 1.109 | 51470.76 | 22.047 |
|
0.15 | 0.008 | 0.172 | 0.0 | 1.109 | 51722.50 | 22.109 |
|
0.25 | 0.006 | 0.156 | 0.0 | 1.110 | 52772.90 | 22.172 |
|
0.05 | 2.431 | 0.047 | 0.509 | 0.156 | 8322.42 | 0.656 |
|
0.15 | 1.529 | 0.031 | 0.195 | 0.125 | 8998.02 | 0.641 |
|
0.25 | 1.391 | 0.047 | 0.158 | 0.125 | 9126.63 | 0.640 |
|
0.05 | 2.877 | 0.140 | 0.635 | 0.516 | 24898.40 | 7.344 |
|
0.15 | 1.869 | 0.125 | 0.409 | 0.516 | 26456.16 | 7.359 |
|
0.25 | 1.212 | 0.141 | 0.190 | 0.500 | 27716.36 | 7.375 |
|
0.05 | 2.950 | 0.438 | 0.214 | 5.937 | 74120.22 | 81.484 |
|
0.15 | 1.879 | 0.453 | 0.064 | 5.969 | 79096.31 | 81.563 |
|
0.25 | 1.434 | 0.453 | 0.054 | 6.047 | 82641.24 | 81.844 |
|
0.05 | 7.894 | 0.031 | 1.270 | 0.093 | 7814.61 | 0.828 |
|
0.15 | 5.590 | 0.016 | 0.783 | 0.093 | 8551.74 | 0.813 |
|
0.25 | 4.350 | 0.015 | 0.435 | 0.094 | 9024.62 | 0.812 |
|
0.05 | 1.697 | 0.437 | 0.247 | 2.219 | 30614.77 | 16.969 |
|
0.15 | 1.001 | 0.453 | 0.075 | 2.234 | 33982.89 | 16.938 |
|
0.25 | 0.597 | 0.437 | 0.053 | 2.234 | 36099.06 | 17.032 |
Then in the second experiment, we concentrate on the comparison of solution quality and computation time with standard MIP solver CPLEX (version 11.1) and another heuristic dynamic slope scaling procedure (DSSP). DSSP proposed by Kim and Pardalos [
Comparison with DSSP and CPLEX.
|
RHH | DSSP | CPLEX | DP | ||||
---|---|---|---|---|---|---|---|---|
RE (%) | CPU (sec) | RE (%) | CPU (sec) | RE (%) | CPU (sec) | Value | CPU (sec) | |
|
0.0 | 0.094 | 2.225 | 1.062 | 11.881 | 5.110 | 11452.16 | 0.985 |
|
0.0 | 0.109 | 2.299 | 0.907 | 10.441 | 5.063 | 11653.62 | 0.984 |
|
0.0 | 0.094 | 2.581 | 1.062 | 9.555 | 5.015 | 11685.286 | 0.984 |
|
0.0 | 0.516 | 1.986 | 2.172 | 0.619 | 25.750 | 22602.25 | 4.719 |
|
0.0 | 0.516 | 2.583 | 2.438 | 3.800 | 24.093 | 23408.59 | 4.719 |
|
0.0 | 0.500 | 3.040 | 2.094 | 6.103 | 24.110 | 23259.73 | 4.719 |
|
0.0 | 1.140 | 1.947 | 4.578 | 0.645 | 111.891 | 45820.49 | 22.187 |
|
0.0 | 1.125 | 2.845 | 4.250 | 1.162 | 114.657 | 46665.20 | 22.172 |
|
0.0 | 1.125 | 3.104 | 4.406 | 0.858 | 113.641 | 47492.82 | 22.218 |
|
0.704 | 0.140 | 0.721 | 1.015 | 3.744 | 3.391 | 7285.71 | 0.656 |
|
0.110 | 0.140 | 0.893 | 0.828 | 15.302 | 3.375 | 7916.02 | 0.656 |
|
0.035 | 0.140 | 1.089 | 0.735 | 13.440 | 3.297 | 8279.95 | 0.641 |
|
0.867 | 0.515 | 0.484 | 3.313 | 3.334 | 37.516 | 21619.82 | 7.453 |
|
0.174 | 0.500 | 0.632 | 2.796 | 1.199 | 38.265 | 23767.47 | 7.422 |
|
0.198 | 0.516 | 1.163 | 3.110 | 0.814 | 37.703 | 24543.10 | 7.438 |
|
0.202 | 6.063 | 0.401 | 8.781 | 0.800 | 414.046 | 65133.43 | 82.640 |
|
0.068 | 6.109 | 0.747 | 11.453 | 1.879 | 412.860 | 70243.82 | 82.375 |
|
0.022 | 6.110 | 1.034 | 11.157 | 0.843 | 419.937 | 74203.32 | 82.563 |
|
0.201 | 0.360 | 0.716 | 1.078 | 4.300 | 4.250 | 6681.52 | 0.828 |
|
0.0 | 0.359 | 0.889 | 0.859 | 1.742 | 4.172 | 7480.93 | 0.812 |
|
0.0 | 0.344 | 1.368 | 1.188 | 1.188 | 4.2347 | 7980.60 | 0.828 |
|
0.367 | 2.204 | 0.429 | 4.578 | 3.918 | 86.125 | 26885.65 | 17.265 |
|
0.082 | 2.172 | 0.579 | 4.812 | 0.741 | 87.125 | 30213.60 | 17.234 |
|
0.0 | 2.171 | 1.060 | 5.594 | 1.111 | 89.468 | 32503.12 | 17.157 |
In summary, the proposed DP can solve the SULP-IQD efficiently compared with the standard CPLEX solver and by properly setting the parameters, we obtain effective RHH which outperforms the DSSP heuristic for the tested instances. The computational results also show that RHH performs better for problem instances with a larger number of stages and high coefficient of variation.
In this paper, we study the stochastic uncapacitated lot-sizing problems with incremental quantity discount where the uncertain parameters are supposed to evolve as discrete time stochastic processes. First, we establish the original stochastic formulation by scenario analysis approach. Another two formulations are built by introducing auxiliary variables and relaxing certain constraints. Then, it is proven that under the decreasing unit order price assumption, the relaxed formulation is equivalent to the original one. Based on this reformulation, the extended production-path property is presented for the SULP-IQD and it enhances the ability to further refine the desired optimal solution by providing a more accurate characterization.
To obtain the exact optimal solution, a dynamic programming algorithm is developed. Although the dynamic programming algorithm has the polynomial-time computational complexity in terms of the number of nodes, it runs exponentially in terms of the number of stages. Thus, a new rolling horizon heuristic is further designed which contains three types of strategies to reduce the computational time. The nonproduction strategy is designed based on the accurate characterization of the optimal solution, and the look-forward and look-backward strategy is developed to overcome the complete enumeration calculations in the production and nonproduction value function. Numerical experiments are carried out to identify proper parameters settings of the proposed RHH and to evaluate the performance of the proposed algorithms by comparison with the CPLEX solver and DSSP heuristic. The computational results of a large group of problem instances with different parameters setting suggest that DP outperforms the CPLEX solver in run time required for obtaining optimal solution and the proposed RHH demonstrates satisfactory run time and solution quality compared with DSSP heuristic; moreover, as the computational complexity analysis suggests, the performance of RHH is better for problems with a greater number of stages.
Since the main difficulties for the stochastic lot-sizing problems stem from the setup cost and uncertain parameters, it will be an area of future research to analyze the properties of the problem and present effective algorithms for the stochastic lot-sizing problems with complex setup requirements, such as setup carryovers by Buschkühl et al. [
The paper is supported by the National Nature Science Foundation of China (NSFC no. 60874071 and 60834004) and Research Fund for the Doctoral Program of Higher Education of China (RFDP no. 20090002110035).