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

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 “k-degreeconnection” 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.


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][6][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][12][13][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 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 lotsizing 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][19][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 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 "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.

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.

Model Description.
In our models, demands have no oneto-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) 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.
The objective function (1) to be minimized is the sum of setup costs, production costs, inventory holding costs, backlogging Mathematical Problems in Engineering 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   = 0 if   = 0 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. min subject to constraints (2)∼( 4), ( 6)∼( 8), and constraints Constraints (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 −1 and .Constraints (13) where    (  ) 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    (  ).
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 Consider 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 The 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 .

𝐸 [𝐵
Define  line segments with interval limits    for demand  that mark the cumulated production up to period .Let  0  be the lower limit of the relevant region for demand .Accordingly, the slope of the inventory on hand function for the line segment  is Similar to the above calculation, the nonlinear function of backlogging in period  can be approximated, whereby the slope can be calculated as Both     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   ,  ∈ D,  ∈ T, we can approximate the original model in the following.Let    be the production quantity of product  for demand  in period  associated with interval .
=1    is the cumulated production quantity of product  for demand  up to period  and   = q − q,−1 (see Figure 1).Thus constraints (2)∼( 4) can be rewritten.We introduce the slope values into the model.Let  0   be the expected inventory and  0   be the expected backlogging at point  0  .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 S-MICLSP is obtained: subject to constraints (8) and constraints Mathematical Problems in Engineering ,    ⩾ 0,  ∈ D,  ∈ P  ,  = 1, . . ., ,  ∈ R,  = 1, . . ., .

New Fix-and-Optimize (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 0 or 1.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 so-called "-degree-connection" to combine the decision of resources, products, demands, and time periods.Then the subproblems of the fix-and-optimize approach can be redefined based on the concept "-degree-connection" (as we have discussed before, our models have "many-to-one" demand structure; we need to state that, in the work of [25], a similar concept "Interrelatedness" is defined; however, his concept is for one-to-one demand structure and cannot be applied to our models).Finally, we present our FO approach for both the S-MICLSP and S-MICLSP-L models.

Definition of "𝑘-Degree-Connection".
In S-MICLSP, 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  ,,−1 and  ,,+1 may also change due to constraints (24).The change of   may also cause the change of    ,, , whereby D   = D  = , due to constraints (23).Similarly, the change of   may also cause the change of    ,, , whereby ,   ∈ P  ,  ∈ R, due to constraints (25).
First we define "1-degree-connection." Let Ω = {  |  ∈ P,  ∈ T} denote the set of all binary setup variables.We say two setup variables   ∈ Ω and      ∈ Ω have "1-degreeconnection" or   is "1-degree-connected" to      if one of the following conditions holds: (1) Period time  and period time   are consecutive; that is,   =  and   ∈ { − 1,  + 1}; (2) Product  and   both satisfy demand ; that is, D   = D  =  and   = ; (3) Product  and   are produced by the same resource ; that is, ,   ∈ P  and   = ; then we can define the set of binary setup variables that are "1-degree-connected" to   , denoted by DC(  ), as follows: For any   ∈ S ⊆ Ω, whereby S is a subset of all binary setup variables Ω, the set of binary setup variables that are "1-degree-connected" to S, denoted by DC(S), is given by  Now we can continue to define "2-degree-connection" based on "1-degree-connection." We say two binary setup variables   and      have "2-degree-connection" or   is "2degree-connected" to      if there exists a setup variable   1 , 1 ∈ Ω such that both   and      are "1-degreeconnected" to   1 , 1 .Based on the previous definitions, "3degree-connection," "4-degree-connection",. .., "-degreeconnection" can be defined by induction.Then the sets DC 2 (  ), DC 3 (  ), . . ., DC  (  ) can be defined.
Similar to "1-degree-connection", we can define the set of binary setup variables that are "-degree-connected" to S, denoted by DC(S), as follows: Without ambiguity, we define "0-degree-connection" for completeness as follows, if two binary setup variables   and      are "0-degree-connected" or have "0-degreeconnection" if and only if   =  and   = .Hence, the definition of "-degree-connection" is reflexive, transitive, and symmetric.

New FO Approach for S-MICLSP.
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 "-degree-connection" to decompose S-MICLSP and define subproblems of our FO approach.However, we also need some complementary definitions of "-degree-connection." Recall the set of all binary setup variables, Ω = {  |  ∈ P,  ∈ T}.For any binary setup variable,   ∈ Ω, we define the complement set of DC  (  ) denoted by DC  (  ) = Ω \ DC  (  ), which is the set of binary setup variables that are not "-degree-connected" to   .The subproblems of level  associated with   , denoted by SP   , are simple and are defined in the following: DC  (  ) is fixed and DC  (  ) 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 SP   .Due to this reason, we limit the maximum number of  to 3 for each subproblem SP   in the following numerical experiments of Section 5.
To describe our FO approach, we denote  = {  ,  ∈ P,  ∈ T} 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 capacity-feasible 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 N × T 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 time-consuming for the approach to terminate after  ×  iterations, we need to take an appropriate  that is equal to or smaller than  ×  in the numerical simulation.

New FO Approach for S-MICLSP-L.
Before we propose the new FO approach for S-MICLSP-L, we need to define the "-degree-connection" of the setup carryover variables   in a similar way and extend the scope of "-degree-connection" of   .If   changes, it may cause the change of the setup carryover variable     , whereby D   = D  = .The change of   can also cause the change of the consecutive setup carryover variables  −1, and  +1, by the consecutive setup carryover constraints.From the resource constraints, we can infer that if   changes,     may change, whereby ,   ∈ P  .
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    ,−1 whereby D   = D  =  or ,   ∈ P  .Now we can define DC  (  ) (the subscript  indicates the approach for S-MICLSP-L), the set of binary variables that are "1-degree-connected" to   , or the set of binary variables that have "1-degree-connection" with   as follows: The definition of "-degree-connection" for   , DC   (  ) is similar.We say that   and      have "connection" or   is "connected" to      if there is a finite integer  ⩾ 1 such that they are "-degree-connected." Respectively, we need to redefine DC(  ) (we use another denotation DC  (  ) for S-MICLSP-L), the set of binary variables that is "1-degree-connected" to   , or the set of binary variables that have "1-degree-connection" with   as follows: With the "1-degree-connection" of   defined above, we can similarly derive the new definition of "-degree-connection" for   as previously described.Based on the new definition of "-degree-connection" for   , we can define the subproblems for S-MICLSP-L, with defining the fix binary variables set DC  (  ) and the reoptimized binary variables set DC  (  ), for any binary setup variable   or pair (, ) ∈ N×T.The fix-and-optimize approach for S-MICLSP-L is the same as S-MICLSP, except that their definitions of DC  (  ) and DC  (  ) are different.

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 feasible-solution set defined by the concept "-degree-connection" 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 FO-VNS approach for the S-MICLSP and S-MICLSP-L.The main novelty in contrast to [25] is that our FO-VNS can be extended to models with setup carryovers.

Integrative FO-VNS for S-MICLSP.
To describe our FO-VNS approach, we will use symbols as shown in Notations.
We also define model S-MICLSP(,   , ), which is a linear relaxation model allowing 0 ⩽   ⩽ 1 derived from the original model by adding constraint (35) to it, where  is the subproblem level described in Section 3. In the FO-VNS approach for S-MICLSP, we obtain the local optimum of S-MICLSP(,   , ) 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   = 1 −   , 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 FO-VNS approach for the model can be presented in pseudocode (see Algorithm 2).

Integrative FO-VNS for S-MICLSP-L.
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 single-level and hence we can extend our integrative FO-VNS to S-MICLSP-L.Similar to FO-VNS for S-MICLSP, the effective extension of FO-VNS for S-MICLSP-L also has two main parts, local search and shaking.Using symbols given in Notations, we now propose the FO-VNS for S-MICLSP-L.
The swapping of the current setup and carryover plan   means that its value is possibly changed from   =   and   =   to   = 1 −   and   =    ,−1 (1 −   ).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 FO-VNS approach for S-MICLSP-L is similar as Algorithm 2 for S-MICLSP and we neglect the pseudocode.

Numerical Experiments
In this section, we evaluate the performances of our proposed FO and FO-VNS approaches.

Experimental Design.
We generate problem instances based on the attributes of real-world 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 100 instances with 10 products, 5 demands (or demand classes), 5 resources, and 50 periods.Both S-MICLSP and S-MICLSP-L 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 Core-i5 3.20 GHz CPU, 4 GB RAM.We compare our approaches with the fix-and-optimize 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 10 −4 ; the time for ILOG CPLEX 12.7 to solve each subproblem is limited to 2 s for S-MICLSP and 4 s for S-MICLSP-L.
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 100 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 fix-andoptimize 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 P-type, R-type, and Tdecomposition for short in the context).Further, they combined the three decomposition types and presented three more variants: P-type first and then R-type, P-type first and then T-type, and P-type first and then R-type ending with Ttype.(2) FO2 is referred to our newly proposed FO approach, in which each variant is entitled with FO2--L.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 3 is too time-consuming, we only test FO2 with subproblems of level  = 1, 2, 3 (see Section 3.2).( 3) FO-VNS is referred to our integrative FO and VNS method.We vary the decomposition level  and obtain three variants of FO-VNS.Overall, absolute time limit for the algorithms is set to 10 minutes for S-MICLSP and 20 minutes for S-MICLSP-L, which can be slightly exceeded by finalizing code.( 4) CPX std and CPX 30std are referred to standard software (CPLEX, Branch & Cut) as comparisons.The absolute time limit for CPX std is set to be same as FO-VNS, while the time limit of CPX 30std is more than CPX std by 30 times. 2 describes the notations used to measure the solution quality.3 illustrates the computational results on S-MICLSP model.From this table, we can see that FO2 outperforms FO1 in terms of the solution quality.Among the six variants of FO1, FO1-PRT obtains the lowest Cost.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 Cost significantly compared with FO1 in general.7 out of 9 FO2 variants perform a better Cost with respect to FO1 on average, meanwhile 5 out of 9 FO2 variants are better with respect to FO1-PRT.Also, another observation from this table is that FO2 is more efficient than FO1 in general.Taking FO2- 2 -L2 as an example, this FO2 variant can obtain both reduced Cost and Time compared either with FO1 on average or the best FO1 variant.Among all the variants of FO2, it seems that FO2- 2 -L2 makes the best trade-off between Cost and Time since it can provide better solutions with lower computational time.

Computational Results on S-MICLSP. Table
For other measurements of the solution quality, FO2 still performs better than FO1 under most circumstances.Considering rate ot , which describes the proportion of periods with overtime, all of the FO1 variants are not able to provide The results of FO2- 1 -L ( = 1, 2, 3) imply that there exist solutions without overtime and backlogging for all tested instances, while FO1 obtain solutions at a relative lower rate NOB proportion.We can also observe that for a lower  =  3 , FO2- 3 -L2 performs better than FO1-P and FO1-R, while other FO2- 3 -L ( = 1, 3) perform worse than all the FO1 variants.The three rows entitled "FO-VNS-L" of Table 3 report the solution quality of FO-VNS variants with different .From the results, the three variants of FO-VNS obtain around 4.95%∼5.99%better solutions compared with FO1 However, the S-MICLSP (also S-MICLSP-L) 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 FO-VNS.The last two rows of Table 3 provide comparisons between CPLEX and our proposed algorithms.CPX std runs the same time limit as FO-VNS but it performs worse than all other algorithms.Running a much longer time limit, CPX 30std performs much better.However compared to FO-VNS, the solution quality of CPX 30std is slightly improved with an extremely increased computational effort.

Computational Results on S-MICLSP-L.
The computational results on the S-MICLSP-L model are given in Table 4.By the similar observations to  3. CPX std reports its drawbacks in terms of solution quality and computational effort, while CPX 30std 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 FO-VNS against FO1 for S-MICLSP-L grow compared with S-MICLSP.For example, FO2- 1 -L3 has −3.93% of dev Cost Average while it has −3.20% in the S-MICLSP case from Table 3.Also, this FO2 variant can obtain −2.50% of dev Cost Best while it can obtain 1.77% in the S-MICLSP case from Table 3.This observation can also be found for all variants of FO-VNS.

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 FO1-PRT, FO2- 1 -L3, and FO-VNS-L3 (best variants of FO1, FO2, and FO-VNS 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 Cost and Time are illustrated by groups of combinations of TBOs (low with TBO < 2.0, medium with 2.0 ⩽ TBO < 3.0, and high with 3.0 < TBO) and STs (low with ST < 25, and high with ST ⩾ 25).
We compare the results in terms of FO1-PRT for the benchmark.From Table 5, we can see that for high TBO cases, FO2- 1 -L3 and FO-VNS-L3 significantly outperform FO1-PRT in terms of solution quality.These outstanding performances of FO2- 1 -L3 and FO-VNS-L3 can also be found in cases of high ST, compared to cases of low ST.We compare the results in terms of the methodology of PWL-FO2 for the benchmark.From Table 6, we can generally see that the methodology of PWL-FO-VNS is competitive with SCN5-BC and SCN10-BC in terms of Cost.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 PWL-FO2 and PWL-FO-VNS are more splendid than SCN5-BC in terms of solution quality.SCN10-BC can obtain better solutions when  = 2, 3; however the computational time is unacceptable when  = 6, 10.Our proposed PWL-FO-VNS obtains competitive solution quality against SCN10-BC while the computational efforts are much less.All of the results imply that our proposed approximation method and solving approaches are efficient.

Conclusion
The key contributions and findings of this paper are as follows: (1) We formulate dynamic S-MICLSP and S-MICLSP-L 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 lot-sizing problems with substitutions (see [18]) and safety stocks (see [29]).
(2) We present a new fix-and-optimize (FO) approach for both S-MICLSP and S-MICLSP-L 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 FO-VNS approach, based on diversifying search space by VNS.The FO-VNS 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, FO-VNS can obtain solutions with quite high quality by consuming reasonable time in a tactical planning decision, especially in the testing on S-MICLSP-L model.
(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:

(
] 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 1 to period  (cumulated quantity produced up to period  plus initial inventory in period 1).Let   denote the cumulated demand from period 1 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 to Apply the new FO approach as local search engine to solve S-MICLSP(,   ), with the neighborhood constraint   ().Then   is obtained.Use the new FO approach and the tabu list to generate a new starting solution   from the current solution .If the solution   is better and does not exist in the tabu list, insert it into the list and let  =   .

Table 2 :
Notations for measuring solution quality.
− , 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 Table3.Consider rate NOB , which indicates the solution quality by combining rate ot and  −  in a statistical sense.

Table 3 ,
FO2 still outperforms FO1 in terms of the solution quality.Particularly, FO2- 1 -L1 and FO2- 2 -L2 outperform all the FO1 variants when considering Cost and Time, which implies they make the best trade-off between Cost and Time in all FO2 variants.Similar toTable 3, FO-VNS gives quite outstanding results compared with FO1 and FO2.FO-VNS performs a better Cost by reducing 5.17%∼6.16%against FO1-PRT.All of the FO-VNS variants provide  −  of 100%, yet FO1 and FO2 fail to do so.The results of the last two rows in Table 4 are similar to Table

Table 5 :
Computational results with various parameters settings.Cost in S-MICLSP-L than that in S-MICLSP.It implies that for high or medium TBO cases, the advantages of FO2 and FO-VNS in S-MICLSP-L grow compared to that in S-MICLSP, which is similar to the previous observations in Section 5.2.2.All of the above results imply that FO2 and FO-VNS are much more effective than FO1 in cases of high or medium TBO and high ST.We generate  realizations of random demand for each period and hence the number of scenarios is   , where  is the number of periods.We choose  = 5, 10 for small test cases.Our methodologies are denoted by PWL-FO2 and PWL-FO-VNS, whereby FO2- 1 -L3 and FO-VNS-L3 are, respectively, applied.

Table 6 :
Computational results compared with scenarios methods.

Table 7 :
Instance generator settings.5  , 4  ), with   = (  / ∑ ∈P    )(1/) ∑ ∈T,∈D  [  ]   Demand Normally distributed, stationary across period horizons, but different mean  and variation coefficient / for each demand: ∼ ⩾0,⩽5 (, ) (with values < 0 cut-off), with  ∼  ⩾0 (50, 20) and / ∼  ⩾0 (0.2, 0.1) This paper can explore several avenues in future research; for example: (i) Extend the S-MICLSP and S-MICLSP-L 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.P  : Set of products that can fulfill demand (class)  P  : Set of products that produced by resource  D  : Set of demands (or demand classes) whose demand can be fulfilled by product  (only one element under many-to-one structure assumption).  : Incurred setup cost when production for product  is ready   : Unit production cost of product  ℎ  : Holding cost for storing product  to next period per unit and period   : Backlogging penalty cost for demand (class)  per unit and period oc  : Overtime cost of resource  per unit of overtime   : Capacity required for manufacturing one unit of product  st  : 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 (, ) ∈ A.  during period    : Physical inventory level of product  at the end of 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 .  : Cumulated demand of demand  from period 1 up to period    : Physical inventory level of demand  at the end of period   prod  : Backlog number of demand  after production at period , but before demand occurrence  end  : 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 .Index of neighborhood structure   ()  max : Maximum number of neighborhood considered.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     ,max : Maximum number of neighborhood considered    (  ): A finite set of neighborhood structures, with  1 (  ) ⊆  2 (  ) ⊆ ⋅ ⋅ ⋅ ⊆   ,max (  )    : Maximum distance between two setup and carryover plans, where both of them are in the same neighborhood structure    .Indices and Sets  ∈ F = {1, 2, . . .,   }: Attributes F  ⊆ F: Set of "Property" attributes F  ⊆ F: S e t o f " M e a s u r e m e n t " a t t r i b u t e s . st : Dimensional multipliers for calculating parameters of resource capacity   ,  oc : Dimensionless multipliers for calculating parameters of resource capacity. : Value for attribute  of product   0 : Base value of production capacity for attribute  of product  st 0 : Base value of setup capacity for attribute  of product  : Random variable, used as a multiplier (dimensionless).