Abstract
We discuss stochastic multiitem capacitated lotsizing problems with and without setup carryovers (also known as link lot size), SMICLSP and SMICLSPL. The two models are motivated from a realworld steel enterprise. To overcome the nonlinearity of the models, a piecewise linear approximation method is proposed. We develop a new fixandoptimize (FO) approach to solve the approximated models. Compared with the existing FO approach(es), our FO is based on the concept of “degreeconnection” for decomposing the problems. Furthermore, we also propose an integrative approach combining our FO and variable neighborhood search (FOVNS), 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 FOVNS approaches significantly outperform the recent FO approaches, and the FOVNS approaches can be more outstanding on the solution quality with moderate computational effort.
1. Introduction
The stochastic multiitem capacitated lotsizing problem (SMICLSP) and its setup carryover extension (also known as linked lot size extension, in [1], abbreviated to “L” ), SMICLSPL, 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 SMICLSP and SMICLSPL.
Both SMICLSP and SMICLSPL are stochastic generalizations of the capacitated lotsizing 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 lotsizing 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 lotsizing 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 lotsizing 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 scenariogenerated method can also be found in [11–14]. Almost all approximation methods for lotsizing 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 timeconsuming. 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, cutgeneration technique, etc.) and metaheuristics (genetic algorithm, particle swarm optimization, tabu search, etc.) are adopted in the deterministic lotsizing models. We refer interested readers to [16] for further review. Recently, MIPbased heuristics are developed to solve lotsizing models since they combine the advantages of exact methods and (meta)heuristics. An MIPbased heuristic shown to be outstanding is called fixandoptimize (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 lotsizing 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 socalled variable neighborhood decomposition search (VNDS) to solve multilevel lotsizing problems, and provided promising computational results. All of the above studies throw light upon solving lotsizing problems by combining VNS.
Newly, Chen [25] proposed an excellent integrative framework combining FO and VNS for deterministic lotsizing problems. Since our models have “manytoone” demand structure, his framework cannot be applied to our models. However, motivated from his work, we propose our FO and integrative FOVNS for our stochastic lotsizing problems. Compared with the work of [25], our proposed FO allows capacityinfeasible (overtime cost is not zero) solutions and can be applied to “manytoone” demand structure, while he prohibited capacityinfeasible solutions and his framework was only valid for onetoone demand structure. Thus, we apply the integrative framework to models without setup carryovers, SMICLSP, and successfully extend it to our setup carryovers version, SMICLSPL, while Chen [25] only applied his framework to models without setup carryovers.
In this paper, we follow a similar analytical procedure of solving stochastic lotsizing 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 SMICLSP and SMICLSPL 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 SMICLSP and SMICLSPL models. This method is simple and easytoimplement, providing a good tradeoff 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 SMICLSP and SMICLSPL models and propose our piecewise linear approximation method in Section 2. Our proposed FO approach and combined method (FOVNS) 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 SMICLSP and SMICLSPL 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 onetoone 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 lotsizing problems assumptions:(a) lotsizing 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) manytoone structure: each product can only satisfy one demand (class), while each demand (class) can be satisfied by multiple products(iii) Bigbucket 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 SMICLSP 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 inventorybalanced 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 SMICLSPL allows the setup state of each resource to be carried over from the current period to the next period. To formulate the SMICLSPL, 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 SMICLSPL 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 SMICLSP and SMICLSPL 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 wellknown expected loss function or the failure function of the random variable with respect to the quantity . Figure 1 illustrates the function .
A backlog of demand occurs at the end of period , if the cumulated demand up to period , , is greater than the cumulated production quantity up to period , . Hence the expected backlog of demand at the end of period , denoted by , isConsider the backlog just after production but before demand occurrence in period . This backlog cannot be affected by demand since demand does not occur. Hence the expected backlog just after production but before demand occurrence of demand corresponding to in period , denoted by , isThe expected backlog number of demand in period can be expressed as the difference between the backlog at the end of period and the expected backlog just after production but before demand occurrence of period .Define line segments with interval limits for demand that mark the cumulated production up to period . Let be the lower limit of the relevant region for demand . Accordingly, the slope of the inventory on hand function for the line segment isSimilar to the above calculation, the nonlinear function of backlogging in period can be approximated, whereby the slope can be calculated asBoth and are calculated from the function when segments are defined (see Figure 1). Once slope values of expected inventories and backlogging can be found from the distribution of the random variable , we can approximate the original model in the following. Let be the production quantity of product for demand in period associated with interval . As is convex, should be satisfied. Also, is the cumulated production quantity of product for demand up to period and (see Figure 1). Thus constraints (2)(4) can be rewritten. We introduce the slope values into the model. Let be the expected inventory and be the expected backlogging at point . In that sense, the physical inventory of product in period can be approximated as , and the backlog number can be approximated as . All the additional symbols in this section are listed in Notations. The following linear approximated SMICLSP is obtained:subject to constraints (8) and constraintsConstraints (22)(24) are approximated constraints of (2)(4) by piecewise linear approximation. Constraints (25)(27) are similar to constraints (5)(7).
The linear approximated SMICLSPL is as follows:subject to constraints (22)(24), (26)(27), (8), and (11)(14) and constraintsConstraints (29) are similar to constraints (25).
Models (21) and (28) are both deterministic models; hence these models can be tractably solved by our next proposed algorithms.
3. New FixandOptimize (FO) Approach
In the FO approach of [17], a series of MIP subproblems is solved in each of which most of the binary setup variables are tentatively fixed to or . Only a subset of binary variables of the original model is treated as decision variables and “optimized” by a run of an MIP solver. MIP subproblems are generated using three types of basic decompositions, product decomposition, resource decomposition, and time periods decomposition. The authors also proposed three more combined decomposition methods: (1) product decomposition first and then resource decomposition, (2) product decomposition first and then time periods decomposition, and (3) product decomposition first, then resource decomposition, and finally time periods decomposition. Figure 2 exemplifies the FO approach of product decomposition with 4 products. In this section, we will propose our FO approach, which differs from [17]. In the following, we first define the socalled “degreeconnection” to combine the decision of resources, products, demands, and time periods. Then the subproblems of the fixandoptimize approach can be redefined based on the concept “degreeconnection” (as we have discussed before, our models have “manytoone” demand structure; we need to state that, in the work of [25], a similar concept “Interrelatedness” is defined; however, his concept is for onetoone demand structure and cannot be applied to our models). Finally, we present our FO approach for both the SMICLSP and SMICLSPL models.
3.1. Definition of “DegreeConnection”
In SMICLSP, the binary setup variables are closely connected to other decision variables. We can infer from constraints (26) that if the setup variable is set to be zero, no production can be planned in this period. If the setup variable is set to be one, the corresponding production can be made. If the value of changes, the value of and may also change due to constraints (24). The change of may also cause the change of , whereby , due to constraints (23). Similarly, the change of may also cause the change of , whereby , due to constraints (25).
First we define “degreeconnection.” Let denote the set of all binary setup variables. We say two setup variables and have “degreeconnection” or is “degreeconnected” to if one of the following conditions holds:(1)Period time and period time are consecutive; that is, and ;(2)Product and both satisfy demand ; that is, and ;(3)Product and are produced by the same resource ; that is, and ;then we can define the set of binary setup variables that are “degreeconnected” to , denoted by , as follows: For any , whereby is a subset of all binary setup variables , the set of binary setup variables that are “degreeconnected” to , denoted by , is given byNow we can continue to define “degreeconnection” based on “degreeconnection.” We say two binary setup variables and have “degreeconnection” or is “degreeconnected” to if there exists a setup variable such that both and are “degreeconnected” to . Based on the previous definitions, “degreeconnection,” “degreeconnection”,…, “degreeconnection” can be defined by induction. Then the sets , can be defined.
Similar to “degreeconnection”, we can define the set of binary setup variables that are “degreeconnected” to , denoted by , as follows: Without ambiguity, we define “degreeconnection” for completeness as follows, if two binary setup variables and are “degreeconnected” or have “degreeconnection” if and only if and . Hence, the definition of “degreeconnection” is reflexive, transitive, and symmetric.
3.2. New FO Approach for SMICLSP
Note that our FO approach solves a series of subproblems iteratively. The key to defining subproblems of our FO approach is to clarify which should be reoptimized and which should be fixed in each iteration. In the following, we apply the concept of “degreeconnection” to decompose SMICLSP and define subproblems of our FO approach. However, we also need some complementary definitions of “degreeconnection.” Recall the set of all binary setup variables, . For any binary setup variable, , we define the complement set of denoted by , which is the set of binary setup variables that are not “degreeconnected” to .
The subproblems of level associated with , denoted by , are simple and are defined in the following: is fixed and is reoptimized. In this definition, we need to point out is a control parameter and the bigger the level is, the more binary setup variables are reoptimized in the corresponding subproblem . Due to this reason, we limit the maximum number of to for each subproblem in the following numerical experiments of Section 5.
To describe our FO approach, we denote a setup plan or a setup solution of the model. Also, is called the values of all setup variables at a solution of the model. Note that our FO approach allows the capacityinfeasible solutions in each loop of solving subproblems, while Chen [25] only selected capacityfeasible solutions. We present the pseudocode of our FO approach in Algorithm 1.

Note that, from the pseudocode of Algorithm 1, we only choose a pair from in each iteration. Hence, the number of possible subproblems is , where is the number of the items and is the number of periods. The number of iterations, , is another control parameter of the approach. Since may be very big and it may be too timeconsuming for the approach to terminate after iterations, we need to take an appropriate that is equal to or smaller than in the numerical simulation.
3.3. New FO Approach for SMICLSPL
Before we propose the new FO approach for SMICLSPL, we need to define the “degreeconnection” of the setup carryover variables in a similar way and extend the scope of “degreeconnection” of . If changes, it may cause the change of the setup carryover variable , whereby . The change of can also cause the change of the consecutive setup carryover variables and by the consecutive setup carryover constraints. From the resource constraints, we can infer that if changes, may change, whereby .
An observation shows that is restricted by . Conversely, if changes, may change according to the constraints of the model. The change of can also influence and whereby or . Now we can define (the subscript indicates the approach for SMICLSPL), the set of binary variables that are “degreeconnected” to , or the set of binary variables that have “degreeconnection” with as follows: The definition of “degreeconnection” for , is similar. We say that and have “connection” or is “connected” to if there is a finite integer such that they are “degreeconnected.”
Respectively, we need to redefine (we use another denotation for SMICLSPL), the set of binary variables that is “degreeconnected” to , or the set of binary variables that have “degreeconnection” with as follows: With the “degreeconnection” of defined above, we can similarly derive the new definition of “degreeconnection” for as previously described. Based on the new definition of “degreeconnection” for , we can define the subproblems for SMICLSPL, with defining the fix binary variables set and the reoptimized binary variables set , for any binary setup variable or pair . The fixandoptimize approach for SMICLSPL is the same as SMICLSP, except that their definitions of and are different.
4. Integrative FO and Variable Neighborhood Search (VNS) Approach
We have stated that FO is a local method in Section 1. Since the structure of the feasiblesolution set defined by the concept “degreeconnection” can be relatively large, the solution searched by the FO approach can only be a local optimum in most cases. In order to find a global optimum, or a solution close to the global optimum, Chen [25] proposed an excellent framework which integrates FO and VNS. In his paper, the integrative framework emphasized great performances comparing to his FO approach. In this section, we partly adopt the framework and propose our integrative FOVNS approach for the SMICLSP and SMICLSPL. The main novelty in contrast to [25] is that our FOVNS can be extended to models with setup carryovers.
4.1. Integrative FOVNS for SMICLSP
To describe our FOVNS approach, we will use symbols as shown in Notations. To integrate VNS with our FO approach, we need to predefine a finite set of neighborhood structures , , with , where is the neighborhood index. can be defined as (35), where is an integer associated with neighborhood index .We also define model SMICLSP, which is a linear relaxation model allowing derived from the original model by adding constraint (35) to it, where is the subproblem level described in Section 3. In the FOVNS approach for SMICLSP, we obtain the local optimum of SMICLSP by applying our proposed FO approach in Section 3 as local search engine.
To enhance the search space, we need to shake the starting solution generated by each local search loop. In the shaking procedure, we apply our proposed FO approach as the swapping generator. We define the swapping initial solution , which can possibly change from to , and use to define the subproblems of the FO generator. We also use a tabu list to keep the diversity realized by all previous shaking. Our shaking procedure is similar to [25] except for the swapping generator. With the above descriptions and notations, our FOVNS approach for the model can be presented in pseudocode (see Algorithm 2).

4.2. Integrative FOVNS for SMICLSPL
Due to the sophisticated multilevel structure, Chen [25] did not present the integrative framework for models with setup carryovers. However, in this paper, our model is singlelevel and hence we can extend our integrative FOVNS to SMICLSPL. Similar to FOVNS for SMICLSP, the effective extension of FOVNS for SMICLSPL also has two main parts, local search and shaking. Using symbols given in Notations, we now propose the FOVNS for SMICLSPL.
We denote the setup and carryover plan. To construct a local searching engine for SMICLSPL, we first define the neighborhood structure of asthen the linear relaxation SMICLSPL can be defined by adding the following constraint to the SMICLSPL model where , and . We predefine that for all .
The swapping of the current setup and carryover plan means that its value is possibly changed from and to and . The tabu list contains the setup and carryover plans which will be prevented from being selected by any future swap.
With the above statements, our FOVNS approach for SMICLSPL is similar as Algorithm 2 for SMICLSP and we neglect the pseudocode.
5. Numerical Experiments
In this section, we evaluate the performances of our proposed FO and FOVNS approaches.
5.1. Experimental Design
We generate problem instances based on the attributes of realworld steel products. The developed instance generator is documented in Appendix, as well as the instance settings. The experimental design structure is as follows: we generate instances with products, demands (or demand classes), resources, and periods. Both SMICLSP and SMICLSPL are tested on the same instances.
All algorithms are coded in C++ in the environment of Microsoft Visual Studio 2012, and all instances are tested on a PC with Intel Corei5 GHz CPU, GB RAM. We compare our approaches with the fixandoptimize approach of [17]. All LP and MIP subproblems involved are solved by calling the MIP solver of ILOG CPLEX 12.7. All problems and subproblems use a relative MIP gap tolerance of ; the time for ILOG CPLEX 12.7 to solve each subproblem is limited to 2 s for SMICLSP and 4 s for SMICLSPL.
As the results depend on the number of line segments used in the approximated models (see (21) and (28) in Section 2.2) as well as the computational times, we solved each of the problem instances with 5, 10, and 15 line segments. The numerical experiment shows that 10 line segments provide a good compromise between accuracy and computational times. Hence we will choose 10 as the segments number in all other tests and comparisons.
Table 1 lists the algorithm variants compared in the computational experiments. We describe Table 1 in the following four aspects. (1) FO1 is referred to the fixandoptimize approach proposed by [17], in which they presented three decomposition methods. They defined the subproblems, respectively, by product, resource, and time periodoriented decomposition (we refer to Ptype, Rtype, and Ttype decomposition for short in the context). Further, they combined the three decomposition types and presented three more variants: Ptype first and then Rtype, Ptype first and then Ttype, and Ptype first and then Rtype ending with Ttype. (2) FO2 is referred to our newly proposed FO approach, in which each variant is entitled with FO2L. Recall that is the control parameter of FO2 introduced in Section 3, and FO2 terminates if subproblems are consecutively solved without improvement; that is, at most subproblems are solved in each iteration. Otherwise, the solution of one subproblem is better than the current best solution of the main problem, and FO2 will proceed to a new iteration after the update of the best solution. The decomposition level is another control parameter described in Section 3. Recall that a bigger implies that more binary setup variables are reoptimized in one subproblem. Because considering subproblems of level larger than is too timeconsuming, we only test FO2 with subproblems of level (see Section 3.2). (3) FOVNS is referred to our integrative FO and VNS method. We vary the decomposition level and obtain three variants of FOVNS. Overall, absolute time limit for the algorithms is set to 10 minutes for SMICLSP and 20 minutes for SMICLSPL, which can be slightly exceeded by finalizing code. (4) and are referred to standard software (CPLEX, Branch & Cut) as comparisons. The absolute time limit for is set to be same as FOVNS, while the time limit of is more than by 30 times.
5.2. Results and Interpretation
Table 2 describes the notations used to measure the solution quality.
5.2.1. Computational Results on SMICLSP
Table 3 illustrates the computational results on SMICLSP model. From this table, we can see that FO2 outperforms FO1 in terms of the solution quality. Among the six variants of FO1, FO1PRT obtains the lowest . It can be observed from Table 3 that the variants of FO2 with bigger and bigger consume more computational time than FO1. However, FO2 can reduce significantly compared with FO1 in general. 7 out of 9 FO2 variants perform a better with respect to FO1 on average, meanwhile 5 out of 9 FO2 variants are better with respect to FO1PRT. Also, another observation from this table is that FO2 is more efficient than FO1 in general. Taking FO2L2 as an example, this FO2 variant can obtain both reduced and compared either with FO1 on average or the best FO1 variant. Among all the variants of FO2, it seems that FO2L2 makes the best tradeoff between and since it can provide better solutions with lower computational time.
For other measurements of the solution quality, FO2 still performs better than FO1 under most circumstances. Considering , which describes the proportion of periods with overtime, all of the FO1 variants are not able to provide solutions completely without overtime. However, when , FO2 can make at a zero level, which implies lower overtime cost than FO1. For other FO2 variants, they can also obtain competitive results against FO1. Considering , only 2 out of 6 FO1 variants obtain 100 percent service level while 5 out of 9 FO2 variants can achieve this goal. We find that, for bigger and , FO2 tends to achieve a high service level from the column of Table 3. Consider , which indicates the solution quality by combining and in a statistical sense. The results of FO2L () imply that there exist solutions without overtime and backlogging for all tested instances, while FO1 obtain solutions at a relative lower proportion. We can also observe that for a lower , FO2L2 performs better than FO1P and FO1R, while other FO2L () perform worse than all the FO1 variants.
The three rows entitled “FOVNSL” of Table 3 report the solution quality of FOVNS variants with different . From the results, the three variants of FOVNS obtain around 4.95%~5.99% better solutions compared with FO1 on the average, and around 3.55%~4.60% compared with the best FO1 significantly, while the computational time can be considerably larger by 172.32%~173.06% and 156.81%~157.50% compared with the average FO1 and the best FO1. However, the SMICLSP (also SMICLSPL) considered in this paper is usually solved weekly or monthly in a tactical decision for a factory of one steel enterprise. It is thus worth spending more but reasonable time to obtain a significantly better production plan by our FOVNS.
The last two rows of Table 3 provide comparisons between CPLEX and our proposed algorithms. runs the same time limit as FOVNS but it performs worse than all other algorithms. Running a much longer time limit, performs much better. However compared to FOVNS, the solution quality of is slightly improved with an extremely increased computational effort.
5.2.2. Computational Results on SMICLSPL
The computational results on the SMICLSPL model are given in Table 4. By the similar observations to Table 3, FO2 still outperforms FO1 in terms of the solution quality. Particularly, FO2L1 and FO2L2 outperform all the FO1 variants when considering and , which implies they make the best tradeoff between and in all FO2 variants. Similar to Table 3, FOVNS gives quite outstanding results compared with FO1 and FO2. FOVNS performs a better by reducing 5.17%~6.16% against FO1PRT. All of the FOVNS variants provide of , yet FO1 and FO2 fail to do so. The results of the last two rows in Table 4 are similar to Table 3. reports its drawbacks in terms of solution quality and computational effort, while reports its better solution quality with much longer computational time.
According to Tables 3 and 4, it can be confirmed that the advantages of FO2 and FOVNS against FO1 for SMICLSPL grow compared with SMICLSP. For example, FO2L3 has −3.93% of while it has −3.20% in the SMICLSP case from Table 3. Also, this FO2 variant can obtain −2.50% of while it can obtain 1.77% in the SMICLSP case from Table 3. This observation can also be found for all variants of FOVNS.
5.2.3. Comparison of Various Cases of Parameter Settings
To check the effectiveness and efficiency of our proposed approaches under various cases of parameter settings, we implement FO1PRT, FO2L3, and FOVNSL3 (best variants of FO1, FO2, and FOVNS tested ever) again using test cases of different TBO (time between orders) and ST (setup times). TBO can be computationally defined as the ratio between setup costs and inventory costs (see [28]). Varying different parameters, the results of Table 5 with measurements of and are illustrated by groups of combinations of TBOs (low with TBO , medium with TBO , and high with TBO) and STs (low with ST , and high with ST ).
We compare the results in terms of FO1PRT for the benchmark. From Table 5, we can see that for high TBO cases, FO2L3 and FOVNSL3 significantly outperform FO1PRT in terms of solution quality. These outstanding performances of FO2L3 and FOVNSL3 can also be found in cases of high ST, compared to cases of low ST. Considering cases of high or medium TBO, FO2L3 and FOVNSL3 obtain lower in SMICLSPL than that in SMICLSP. It implies that for high or medium TBO cases, the advantages of FO2 and FOVNS in SMICLSPL grow compared to that in SMICLSP, which is similar to the previous observations in Section 5.2.2. All of the above results imply that FO2 and FOVNS are much more effective than FO1 in cases of high or medium TBO and high ST.
5.2.4. Comparison with Scenario Methods
Typically, because of the continuous nature of demand uncertainty and the dynamic structure of the problems, multistage stochastic modeling methods can be applied and hence scenario methods are used to approximate the models. This methodology is distinguished from ours and here we numerically compare the difference between them. The methodology using scenario method is denoted by SCNBC, where Branch & Cut approach is used to solve the models approximated by scenario method. We generate realizations of random demand for each period and hence the number of scenarios is , where is the number of periods. We choose for small test cases. Our methodologies are denoted by PWLFO2 and PWLFOVNS, whereby FO2L3 and FOVNSL3 are, respectively, applied.
We compare the results in terms of the methodology of PWLFO2 for the benchmark. From Table 6, we can generally see that the methodology of PWLFOVNS is competitive with SCN5BC and SCN10BC in terms of . The entry “” indicates that there are no computational results. Since the number of scenarios is growing in size exponentially, it is impossible to compute finalized results in reasonable time even when is relatively small. Both PWLFO2 and PWLFOVNS are more splendid than SCN5BC in terms of solution quality. SCN10BC can obtain better solutions when ; however the computational time is unacceptable when . Our proposed PWLFOVNS obtains competitive solution quality against SCN10BC while the computational efforts are much less. All of the results imply that our proposed approximation method and solving approaches are efficient.
6. Conclusion
The key contributions and findings of this paper are as follows:(1)We formulate dynamic SMICLSP and SMICLSPL models mapped to a realistic problem in steel production. We also propose a piecewise linear approximation method to reformulate models that can be solved tractably. This method is novel and can balance the approximated accuracy and computational times. This method can also be extended to other cases such as the lotsizing problems with substitutions (see [18]) and safety stocks (see [29]).(2)We present a new fixandoptimize (FO) approach for both SMICLSP and SMICLSPL which possesses a novel way of decomposing. Our FO decomposes the problems based on the concept of “degreeconnection” described in Section 3. This decomposition method combines all the information about products, resources, demands, and time periods. Hence, our presented approach is more effective in each iteration. The computational experiments show that our proposed approach outperforms the recent one.(3)We develop an integrative FOVNS approach, based on diversifying search space by VNS. The FOVNS explores more promising regions in each iteration. This approach extends the scope of [25] and can be applied to models with setup carryovers. From numerical results, FOVNS can obtain solutions with quite high quality by consuming reasonable time in a tactical planning decision, especially in the testing on SMICLSPL model.
This paper can explore several avenues in future research; for example:(i)Extend the SMICLSP and SMICLSPL models to the multilevel versions.(ii)Find a more effective way to construct decomposition frameworks in each iteration.(iii)Develop a more efficient local search engine for the integrative metaheuristics.
Appendix
Generating Problem Instances
To solve the real SMICLSP and SMICLSPL applications, we need to gather data in steel production. However, due to the reasons of confidentiality, we have to generate sufficient problem instances, in which structure is as realistic as possible and capacities are rather tight. To ensure that the optimal ordering decisions are nontrivial, we propose a method for generating problem instances based on the realistic attributes of products. The symbols used for describing the instance generator are given in Notations. To generate instances, we also delineate the settings and assumptions of our method which are contained in Table 7.
Each product can be described by values of several attributes . We assume that all attribute values are integers and ⩾ 0. Suppose that we have known the number of elements in demands (or demand classes) , products , and resources , the next question is how to determine and . From the context, it is obvious that , the number of elements in , is not greater than , which is the number of elements in . Then we can use clustering algorithms such as Kmeans clustering and hierarchical clustering to cluster the elements into classes after the attribute values are realized (see [30] for more details about clustering algorithms). Thus is determined. As for , we are building it by assigning the resources randomly to each product.
The following two groups of attributes distinguished by realworld steel production can help us generate the instances further.
(1) Property. Attributes that are in products themselves. The products are significantly different from other products by this kind of attributes. These attributes are also the key factors for generating the setup parameters such as and . The set containing this kind of attributes is denoted by .
(2) Measurement. Attributes for specification of some kinds of standards, such as width and length. The variation of these values can also fluctuate the marginal parameters such as , . The set containing this kind of attributes is denoted by .
Table 8 lists an example of product attributes in steel production. The underlying motivation of this classification of the attributes is the fact that the occupied resource of one product is mainly dependent on its “Property” attribute. Hence, the idea for generating the resource parameters, and , is as follows: we assume that both and of one product are linear functions of its attributes value . Particularly, for , we assume that it is linear function of only whose attributes are “Property.” For all attributes, the weights and (which could also be negative) are multiplied with the attribute value . We also assume the two base values, and , for generating and , where and (with values cutoff). Thus, we can generate and as follows:
An observation from production viewpoint is that the unit production cost is strongly correlated to while the setup cost is strongly correlated to . In order to generate and , we first sample random variable from a normal distribution . Using these random numbers, both and are calculated using same random value, so that they are correlated. We also introduce and as multipliers for physical unit standardization since we assume that is dimensionless:
The procedure for generating and is as follows: we first calculate the minimum amount of the resource capacity required for producing sufficient quantities of each product to fulfill the expected demand. We denote this amount .The corresponding capacity of resource , , is estimated by the unit capacity consumption and the setup capacity , while the overtime cost, , can be also estimated by the unit product cost and the setup cost . Both and are affected by and so they can be correlated with each other.
Note that the concept of the instance generator can be extended to the aspects of other products in the production management. This methodology can make the structure of the generating instances as realistic as possible since all the parameters are based on the attributes of the products.
Notations
Notations Used in Model FormulationsSet of resources ()  
Set of products ()  
Set of demands (or demand classes) ()  
Set of periods ()  
If and only if product can fulfill demand (class)  
Set of products that can fulfill demand (class)  
Set of products that produced by resource  
Set of demands (or demand classes) whose demand can be fulfilled by product  
(only one element under manytoone structure assumption). 
Incurred setup cost when production for product is ready  
Unit production cost of product  
Holding cost for storing product to next period per unit and period  
Backlogging penalty cost for demand (class) per unit and period  
Overtime cost of resource per unit of overtime  
Capacity required for manufacturing one unit of product  
Setup time capacity required for manufacturing product  
Available capacity of resource  
Large number, required for setup forcing constraint  
Number of units of product that satisfies one unit of demand (class) for any . 
Backlog number of demand (class) during period  
Physical inventory level of product at the end of period 
Demand (class) in period . 
Quantity of product used to fulfill demand (class) at period  
Production quantity of product at period  
Binary variable that indicates whether production of product occurs at period  
Binary setup carryover variable for item at the beginning of period  
Additional capacity of resource at period . 
Set of segments (). 
Cumulated demand of demand from period up to period  
Physical inventory level of demand at the end of period  
Backlog number of demand after production at period , but before demand occurrence  
Backlog number of demand at the end of period . 
Slope value of the on hand inventory for demand in period associated with segment  
Slope value of the backlog for demand in period associated with segment  
Expected loss function or the failure function of the random variable with respect to the quantity . 
Cumulated production quantity of product for demand in period associated with interval  
Production quantity of product for demand in period . 
Setup plan for the current solution  
Setup plan for the incumbent (the current best solution)  
Computation time so far  
Maximum computation time allowed  
Index of neighborhood structure  
Maximum number of neighborhood considered. 
Carryover plan for the current solution  
Carryover plan for the incumbent (the current best solution)  
Carryover plan for linear relaxation model  
Setup and carryover plan for the current solution  
Setup and carryover plan for the incumbent (the current best solution)  
Index of neighborhood structure  
Maximum number of neighborhood considered  
A finite set of neighborhood structures, with  
Maximum distance between two setup and carryover plans, where both of them are in the same neighborhood structure . 
Number of attributes. 
Attributes  
Set of “Property” attributes  
Set of “Measurement” attributes. 
Production capacity weight for attribute  
Setup capacity weight for attribute  
Dimensional multiplier for calculating unit production cost  
Dimensional multiplier for calculating setup cost  
,  Dimensional multipliers for calculating parameters of resource capacity 
,  Dimensionless multipliers for calculating parameters of resource capacity. 
Value for attribute of product  
Base value of production capacity for attribute of product  
Base value of setup capacity for attribute of product  
Random variable, used as a multiplier (dimensionless). 
Total resource capacities required for producing the expected demand for all products. 
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research has been supported by the National Natural Science Foundation of China under Grants 61273233 and U1660202.