Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7209303, 18 pages

https://doi.org/10.1155/2017/7209303

## Fix-and-Optimize and Variable Neighborhood Search Approaches for Stochastic Multi-Item Capacitated Lot-Sizing Problems

^{1}Department of Automation, Tsinghua University, Beijing 100084, China^{2}Department of Basic Science, Military Transportation University, Tianjin 300161, China

Correspondence should be addressed to Shiji Song

Received 19 October 2016; Revised 18 January 2017; Accepted 1 March 2017; Published 12 April 2017

Academic Editor: Honglei Xu

Copyright © 2017 Liuxi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We discuss stochastic multi-item capacitated lot-sizing problems with and without setup carryovers (also known as link lot size), S-MICLSP and S-MICLSP-L. The two models are motivated from a real-world steel enterprise. To overcome the nonlinearity of the models, a piecewise linear approximation method is proposed. We develop a new fix-and-optimize (FO) approach to solve the approximated models. Compared with the existing FO approach(es), our FO is based on the concept of “-degree-connection” for decomposing the problems. Furthermore, we also propose an integrative approach combining our FO and variable neighborhood search (FO-VNS), which can improve the solution quality of our FO approach by diversifying the search space. Numerical experiments are performed on the instances following the nature of realistic steel products. Our approximation method is shown to be efficient. The results also show that the proposed FO and FO-VNS approaches significantly outperform the recent FO approaches, and the FO-VNS approaches can be more outstanding on the solution quality with moderate computational effort.

#### 1. Introduction

The stochastic multi-item capacitated lot-sizing problem (S-MICLSP) and its setup carryover extension (also known as linked lot size extension, in [1], abbreviated to “-L” ), S-MICLSP-L, are designed to map an industrial optimization problem in a realistic steel enterprise. The problem setting is as follows: there are several types of steel products. These products differ in various attributes: chemical composition (mixture), width, thickness, shape (bar, rod, tube, pipe, plate, sheet, etc.), microstructure (ferritic, pearlitic, martensitic, etc.), physical strength, and other attributes. Usually, each type of steel products should be used for only one particular purpose, and each purpose can be satisfied by one or several steel products. Hence, in this steel enterprise, a typical production schedule is made based on the need of one particular purpose, rather than the need of one particular customer. There are two categories of production scheduling, solid scheduling and flexible scheduling, applied in different factories in this steel enterprise. The solid one schedules all types of products simultaneously with a high frequency of restarting production states, while the flexible one schedules parts of the types simultaneously with a low frequency of restarting production states. All of the production schedules are made before the whole planning horizon. The problems with these settings can be suitably mapped to S-MICLSP and S-MICLSP-L.

Both S-MICLSP and S-MICLSP-L are stochastic generalizations of the capacitated lot-sizing problems (CLSP, see [2]) and they consider backlogging and setup carryovers jointly. The deterministic CLSPs with backlogging or setup carryovers individually have been tackled by various models in the literature. We refer the interested readers to [3, 4] for the most recent review on CLSPs. For the lot-sizing problems considering backlogging and setup carryovers jointly, models were treated in [5–7]. Their problem formulations were similar to [8], who first solved problems with setup carryovers. All of the above studies focused on how to solve the lot-sizing problems by designing heuristics. As the authors highlighted, although there is a significant amount of research literature on CLSPs, the literature on problems that consider backlogging and setup carryover jointly is rather scarce.

In this paper, we assume that demand is continuously stochastic which can cover majority of demand environment. Due to the model uncertainty, approximation methods are applied to reformulate the lot-sizing models for performing deterministic mixed integer programming (MIP) in the literature. Haugen et al. [9] generated subproblems for each scenario solved heuristically to capture the nature of demand uncertainty and specify a reasonable number of representative scenarios. Brandimarte [10] modeled the demand uncertainty through generating scenario trees. They made the generated scenario trees match the first, second, third, and fourth moments of the given distribution. The scenario method or scenario-generated method can also be found in [11–14]. Almost all approximation methods for lot-sizing problems are scenario methods. Nevertheless, Mietzner and Reger [15] stated the advantages and disadvantages of scenario methods. One of the crucial disadvantages is as follows: to capture more properties of the uncertainty, the approximated models should ensure an adequate number of scenarios, but the practice of scenario methods can be very time-consuming. This leads to the contradiction between computational time and approximation accuracy. In the following part, we will propose our approximation method to overcome this drawback.

Since the approximated models can perform deterministic MIP, the methods used for deterministic CLSP and its extensions can be also applied to the approximated models. Historically, exact methods (branch & bound technique, Lagrangian relaxation, cut-generation technique, etc.) and metaheuristics (genetic algorithm, particle swarm optimization, tabu search, etc.) are adopted in the deterministic lot-sizing models. We refer interested readers to [16] for further review. Recently, MIP-based heuristics are developed to solve lot-sizing models since they combine the advantages of exact methods and (meta-)heuristics. An MIP-based heuristic shown to be outstanding is called fix-and-optimize (FO) approach, which is proposed by Sahling et al. [17]. The authors presented three types of decomposition method: product decomposition, resource decomposition, and time periods decomposition. Based on the work of [17], variants of FO are developed by [18–20]. However, all of the variants follow the decomposition framework of [17].

Although FO exhibits its efficiency and effectiveness in the literature, it follows a prespecified trajectory and hence it is a local search method. This may result in low solution quality. To enhance the search space of FO approach, one can apply variable neighborhood search (VNS) proposed by [21]. VNS is a metaheuristic which involves two key steps. The first key step is using a local search method to obtain local optimum and the second is systematically changing the neighborhood structure of each local search. Unlike other metaheuristics, VNS does not follow a prespecified trajectory but explores increasingly distant neighborhoods of the current incumbent solution. Since VNS can enhance the search space, many integrative frameworks with VNS are proposed to solve lot-sizing problems. Hindi et al. [22] proposed an integrative Lagrangian relaxation- (LR-) VNS framework for the CLSP with setup times and got good feasible solutions. Zhao et al. [23] and Seeanner et al. [24] developed another type of VNS, the so-called variable neighborhood decomposition search (VNDS) to solve multilevel lot-sizing problems, and provided promising computational results. All of the above studies throw light upon solving lot-sizing problems by combining VNS.

Newly, Chen [25] proposed an excellent integrative framework combining FO and VNS for deterministic lot-sizing problems. Since our models have* “many-to-one” demand structure*, his framework cannot be applied to our models. However, motivated from his work, we propose our FO and integrative FO-VNS for our stochastic lot-sizing problems. Compared with the work of [25], our proposed FO allows capacity-infeasible (overtime cost is not zero) solutions and can be applied to* “many-to-one” demand structure*, while he prohibited capacity-infeasible solutions and his framework was only valid for one-to-one demand structure. Thus, we apply the integrative framework to models without setup carryovers, S-MICLSP, and successfully extend it to our setup carryovers version, S-MICLSP-L, while Chen [25] only applied his framework to models without setup carryovers.

In this paper, we follow a similar analytical procedure of solving stochastic lot-sizing problems to the reviewed literature. However, despite the above, our paper demonstrates other unique characteristics which distinguish from the existing related literature as follows:(1)Derived from realistic industrial problems, we formulate S-MICLSP and S-MICLSP-L models considering backlogging, production overtime, and initial inventory at the same time, which is much more complicated than the existing models in the literature.(2)We propose a piecewise linear method to approximate S-MICLSP and S-MICLSP-L models. This method is simple and easy-to-implement, providing a good trade-off between computational time and approximation accuracy. This method overcomes the drawback of scenarios generating on the computational end.(3)A new FO approach is proposed for our approximated models. Differing from the decomposition framework of [17], this approach decomposes the main problem based on the combined information of products, resources, demands, and time periods.(4)An integrative VNS heuristic which uses FO as the local search engine is proposed to solve our approximated models. This combined approach is running on a specially designed neighborhood structure.

The outline of this paper is structured as follows: we formulate our S-MICLSP and S-MICLSP-L models and propose our piecewise linear approximation method in Section 2. Our proposed FO approach and combined method (FO-VNS) are described in Sections 3 and 4. Numerical experiments of the two approaches on instances generated from a realistic case are presented in Section 5. In Section 6, the concluding remarks as well as discussions on future research are provided. The generating method is lengthy and is relegated to Appendix.

#### 2. Models Formulation and Approximation

In this section, we first formulate S-MICLSP and S-MICLSP-L models. To overcome the nonlinearity and intractability of the models, we then propose a piecewise linear approximation method to reformulate the models. These approximated models are deterministic and hence can be tractably solved by our following proposed algorithms.

##### 2.1. Model Description

In our models, demands have no* one-to-one correspondence* to products. Demands can be satisfied by multiple products and categorized into different classes by the purposes. We can use the term “demand class” to describe one demand for purpose. The term “demand class” can help readers recognize the unique structure of demands in our models. But to avoid ambiguity, we equate the term “demand” to the term “demand class” and use “demand” mostly in the context. For detailed description, we make additional assumptions as follows:(i)General capacitated lot-sizing problems assumptions:(a) lot-sizing for multiple products(b) finite time of planning horizon(c) initial inventories(d) capacitated production resource(e) decision before planning horizon(ii)Demands assumptions:(a) continuously randomized on a known distribution with a finite support, independent, and identically distributed from period to period for each demand (class)(b) *many-to-one structure*: each product can only satisfy one demand (class), while each demand (class) can be satisfied by multiple products(iii) Big-bucket assumption (see [26]):(a) permit the production of multiple products during a single period(iv) Linked lot sizes assumption (see [1]):(a) the setup state of a resource to be carried over from the current period to the next period(v) Other assumptions:(a) overtime production and backlogging setting are allowed, with high penalty costs.(b) no lead times(c) expected cost minimization objective(d)continuous variables for lot sizes

Note that the overtime production is allowed since the requirement of flexibility. This assumption is often used in practice if no feasible production plan could be found otherwise the following two facts: one is the production capacity limits are frequently “soft” as machines could run longer than the planned daily operating time, the other one is the total volume of production could be increased slightly if machines could run below their technical limits by default.

Using the symbols given in Notations, the S-MICLSP can be formulated as given below.*subject to* constraintsThe objective function (1) to be minimized is the sum of setup costs, production costs, inventory holding costs, backlogging penalty costs, and overtime costs. Constraints (2) and (3) are the inventory-balanced equations that each demand (class) can be satisfied by multiple products. Additionally, constraints (4) imply that the quantity of one product used to fulfill one demand (class) should not exceed the sum of inventory and production quantity. Constraints (5) give the capacity constraint of each resource in each period with overtime. Constraints (6) are the coupling constraints linking each production variable with its corresponding setup variables , where the choice of each large positive number must not limit any feasible production quantity of product in period . The coupling constraints imply that if for all and . The nonnegative real or binary nature of each variable in the model is indicated by constraints (7) and (8).

The S-MICLSP-L allows the setup state of each resource to be carried over from the current period to the next period. To formulate the S-MICLSP-L, additional binary variables indicating setup carryovers and additional constraints linking the setup state variables with the setup carryover variables are required. We adopt the formulation of [27] with overtime. Additional variables can also be found in Notations. The S-MICLSP-L can be formulated as given below.*subject to* constraints (2)(4), (6)(8), and constraintsConstraints (10) are similar to constraints (5). Constraints (11) imply that, in each period, the setup carryover of a resource is possible only for at most one product. Constraints (12) indicate that the setup carryover of a resource for product occurs in period only if the resource is set up for the item in both periods and . Constraints (13) indicate multiperiod setup carryovers. Constraints (14) specify the binary nature of setup carryover variables.

##### 2.2. Piecewise Linear Approximation

Both and in S-MICLSP and S-MICLSP-L models are nonlinear stochastic functions. We have specified the continuous nature of demand uncertainty. Thus it is intractable to solve these models. We could apply scenario method to approximate the models. However, we have discussed in Section 1 that the computational efforts can be unacceptable and the precision of approximation can be low. Fortunately, it is possible to replace the functions of and by suitably chosen piecewise linear functions. The functions of and can be approximated as follows. Let denote the total amount available to fill the cumulated demand from period to period (cumulated quantity produced up to period plus initial inventory in period 1). Let denote the cumulated demand from period up to period and let denote the associated density function. Denote the expected physical inventory on hand at the end of period for demand corresponding to . Then consequently is equal towhere is the well-known expected loss function or the failure function of the random variable with respect to the quantity . Figure 1 illustrates the function .