Integrated, hierarchical and interactive approaches for the joint resolution of production lot-sizing and scheduling problems

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1 FEDERAL UNIVERSITY OF MINAS GERAIS DEPARTMENT OF PRODUCTION ENGINEERING SCHOOL OF ENGINEERING Integrated, hierarchical and interactive approaches for the joint resolution of production lot-sizing and scheduling problems Fernanda de Freitas Alves Supervisor: Prof. Dr. Martín Gómez Ravetti Belo Horizonte 2017

2 FEDERAL UNIVERSITY OF MINAS GERAIS DEPARTMENT OF PRODUCTION ENGINEERING SCHOOL OF ENGINEERING Integrated, hierarchical and interactive approaches for the joint resolution of production lot-sizing and scheduling problems Fernanda de Freitas Alves Dissertation presented to the Graduate Program in Production Engineering of the Federal University of Minas Gerais as a requirement to obtain a Master s Degree in Production Engineering. Concentration Area: Operational Research and Manufacturing Engineering Research line: Optimization Models and Algorithms for Networked and Production Systems Supervisor: Prof. Dr. Martín Gómez Ravetti Belo Horizonte 2017

3 Abstract To consider production lot-sizing and scheduling problems separately can lead to infeasible production plans, an increase of costs and a poor use of resources. Therefore, this work aims to analyze different forms of association between such problems, comparing the results when integrated, hierarchical and interactive approaches are used to solve them. For the integrated approach, two formulations based on literature models are considered, a continuous, and a timeindexed formulation. We also apply the warm-start technique to the continuous model to find better results since the integrated approach may not find feasible solutions in the limited time. The lower objective function found by the integrated formulations is used for comparison with the other methods. For the hierarchical and interactive approaches, several strategies are proposed to address the problem, analyzing different information shared between the decision levels. All formulations are solved using the CPLEX solver. An analysis of the results is performed considering uniform, normal and Poisson distributions to generate the instance set. Furthermore, three different scenarios related to the relation of the sequence-dependent setup times and the production times are considered. The proposed Interactive Strategy IV (ISIV) presented better results in relation to the other hierarchical and interactive strategies proposed for most of the instances tested. However, results show that all of the proposed strategies are efficient, finding good results in low computational times, and are good alternatives for the connection of production lot-sizing and scheduling problems. Keywords: Production Planning. Lot-sizing. Scheduling. i

4 Acknowledgments First, I would especially like to thank my advisor, Professor Martín Gómez Ravetti, for all the teachings, trust, availability and encouragement. I would also like to thank Professor Maurício Cardoso de Souza for the new ideas and suggestions for the research and the professors Geraldo Robson Mateus and Thiago Henrique Nogueira for the suggestions and availability to be part of the dissertation defense committee. To all my friends, for the friendship, understanding, and affection. To my family, for the love and support during all the moments of my master s degree. Finally, I also would like to thank FAPEMIG for financial support for this research. ii

5 List of Figures 2.1 Approaches to solve production lot-sizing and scheduling problems (Adapted from [50]). The arrows show the flow of information between different decision levels Example of infeasibility. In this case, the capacity of period t = 1 is equal to 112 hours, represented by the dashed line. The makespan obtained by the scheduling problem is equal to 120 hours. Therefore, the sequence obtained is infeasible since MKS > C Flowchart representing the HSI strategy. The arrows show the main information shared between the decision levels. Besides that, all the variables determined in the procedure are transferred to the next period (t = t + 1) when the horizon is rolled Flowchart representing the HSII strategy Example of capacity extension. As MKS < C 1, the procedure may opt to use the idle hours of period t = 1, represented by γ, to produce products of the next periods. However, the extension is limited by the parameter λ. Then, C 2 = C 2 + λc Flowchart representing the interactive strategies ISI, ISII and ISIII, in which feedback information is transferred to the lot-sizing problem. In this work, such information depends on the type of cut added Flowchart representing the HSIV strategy Average computational times. Sample of Scenario 1 Uniform distribution Average deviations observed in relation to the integrated formulations iii

6 List of Tables 2.1 Cited papers that solved lot-sizing and scheduling problems individually or that solved planning problems with the integrated, hierarchical or interactive approaches Literature papers that solved production lot-sizing and scheduling problems with the hierarchical and interactive approaches. The strategies proposed in this work are also presented. The table presents the classification group, the reference to the paper and the information transferred between the levels. The works are shown within the group according to their publication date Relation between the proposed strategies and the papers presented in Table Sets used in the mathematical models Parameters used in the mathematical models Decision variables used in the mathematical models New set and variables used in the IMTIPH mathematical model New parameters used in the hierarchical approach New parameters used in the HSII strategy New sets and parameters used in the interactive strategies New sets, parameters and variables used in the ISIV strategy Distributions used in the instances generation Results obtained for the deviation averages, its respective standard deviation and the resolution CPU time averages iv

7 Contents List of Figures List of Tables iii iv 1 Introduction Objectives Text organization Literature review Integrated approach Non-Integrated approaches Contextualization Hierarchical and interactive approaches for lot-sizing and scheduling problems Relation between the proposed strategies and literature papers Mathematical Modeling and Solution Methods Objectives Solution methods Integrated approach Integrated mixed integer linear programming Model with Continuous Planning Horizon - IMCPH Integrated mixed integer linear programming Model with Time- Indexed Planning Horizon - IMTIPH Integrated mixed integer linear programming Model with Continuous Planning Horizon with the Warm-Start technique - IMCPH - WS Hierarchical approach Hierarchical Strategy I - HSI Hierarchical Strategy II - HSII HSIIa HSIIb Interactive approach Interactive Strategy I - ISI Interactive Strategy II - ISII Interactive Strategy III - ISIII Interactive Strategy IV - ISIV Computational experiments Instances Results and discussions Conclusions and proposals for future works 37 v

8 CONTENTS CONTENTS Bibliography 59 vi

9 Chapter 1 Introduction Production planning consists of a plan that determines resource acquisitions, the size of the production lots, when each lot will be produced, among other decisions to perform the process of converting inputs into final products. It has the objective to meet the demand in the most economical way, seeking the best trade-off between financial goals and customer satisfaction [59]. Production planning is usually divided into three hierarchical decision levels, the strategic, tactical and operational levels, related to the long, medium and short time horizon, respectively. The shorter the planning horizon, the more precise is the demand forecast, and thus the planning is performed with a higher level of detail. See [59] for more information of each hierarchical level. Two decision levels are considered in this work, an upper and a lower level, related to the lot-sizing and the scheduling problems, respectively. The problem solved at the upper level is the master production schedule (MPS), taken into account disaggregated demand, a medium-term planning of months to weeks, and aims to meet demand requirements while satisfying capacity constraints. The MPS minimizes costs, such as inventory, production, and backorder costs. In this work, we will refer to this problem as MPS or lot-sizing problem interchangeably. On the lower level, the problem usually considers a short interval of time of weeks to days, and deals with a detailed shop information, considering various performance measures, such as the minimization of the makespan, minimization of the total completion time and the minimization of the number of late jobs. The MPS consists of defining which products will be produced in each period of the planning horizon, as well as their respective production quantities, considering customer demands and technological limitations of the manufacturing process. The scheduling problem consists in defining the output sequence of the products set to be produced at the lot-sizing level. Decisions related to the lot-sizing are usually taken in the medium term, at the tactical level, while decisions on the scheduling of items consider a short term, at the operational level. Production planning problems connecting lot-sizing and scheduling decisions are being widely studied in the literature since to consider such decisions individually can lead to the infeasibility of the production plan as well as increased costs and poor use of resources. Three different procedures for solving the MPS and scheduling problems are considered: integrated, hierarchical and interactive approaches. In the integrated approach, the lot-sizing and the scheduling problems are simultaneously solved in a single mathematical model. In the hierarchical approach, the problems are solved sequentially with feedback from the scheduling solution. In the interactive case, the lot-sizing problem is firstly solved and transfers the resulting information to address the lower level. If the solution found by the scheduling problem is infeasible, the procedure will loop between the levels searching for feasibility at the tactical level. The main difference between the hierarchical and interactive approaches is that in the former the feedback 1

10 CHAPTER 1. INTRODUCTION 2 from the scheduling is only used to make decisions in the following periods. In the latter, the procedure can loop revisiting the same period until reaching a feasible solution. In this work, two formulations based on literature models are adapted for the integrated approach, one considering continuous time horizon and another one with a discretized time horizon. Given that a rolling horizon technique and a limited CPU time are considered, these formulations can find different solutions. Furthermore, for large instances, the integrated approach may not find feasible solutions due to the complexity of the problem. Since one of the objectives of this work is to compare the proposed strategies to an integrated approach, the continuous integrated model is also solved using a warm-start technique, in which an initial feasible solution is provided. For the hierarchical and interactive methods, various strategies are introduced based on a literature study. To verify the effectiveness of such strategies, they are tested and compared with each other and with the best result obtained by the integrated formulations. The case study for the analysis consists of a capacitated lot-sizing and scheduling problem in a scenario of a single machine with sequence-dependent setup times. It considers multiple products, which must meet demand in each period of the planning horizon. Backorder is allowed and penalized; the factory also has the option to produce in advance, incurring in inventory costs. Therefore, this work considers a trade-off between the backorder and inventory costs, which are the costs to be minimized. 1.1 Objectives This work aims to analyze different forms of association between the production lot-sizing and the scheduling decision problems, comparing the results when integrated, hierarchical and interactive approaches are used to solve them. This dissertation focuses on the analysis of the different types of information shared among both decision levels when addressed with hierarchical and interactive approaches, aiming to propose some strategies to solve production lot-sizing and scheduling problems based on this analysis. Studies that analyze and compare integrated, hierarchical and interactive methodologies were not found in the literature, as well as works with a detailed analysis of the type of information shared between the decision levels. Furthermore, this work still proposes a classification of methods and some strategies to solve production MPS and scheduling problems, presenting new contributions to this research area. 1.2 Text organization The remainder of this dissertation is organized as follows: Chapter 2 introduces a literature review analysis, presenting articles on production planning problems, with a focus on hierarchical and interactive approaches. Chapter 3 presents the case study. Furthermore, the models and methods are described and discussed. The results and general discussions are shown in Chapter 4. Finally, conclusions are drawn in Chapter 5.

11 Chapter 2 Literature review The lot-sizing problem aims to determine in which periods production will occur and the lot sizes of multiple products for each period of the planning horizon [76, 10]. The development and improvement of procedures that solve the lot-sizing problem is of critical importance, given that decisions made at this level affect the performance and productivity of the whole production process [35]. The lot-sizing problem was first approached in 1913, see [32], introducing the classic economic lot size model, the Economic Order Quantity (EOQ). This model considers a single item, infinite planning horizon, stationary demand and aims to determine the lot size so that total costs are minimized, without capacity constraints. In [72] the Economic Production Quantity (EPQ) model is presented, incorporating to the EOQ the production rate. In [62] the use of the Harris model is extended for multiple items with stationary demand produced on the same machine, determining the production schedules. The model proposed in [62] is known as Economic Lot Scheduling Problem (ELSP) and is classified as NP-Hard. In [79] the authors assume a discretized planning horizon considering dynamic demand. The model proposed is known as Wagner-Whitin problem or Dynamic lot size model and, like the EOQ, this model does not consider capacity constraints. The Capacitated Lot Sizing Problem (CLSP) can be regarded as an extension of the Wagner-Whitin problem, including capacity constraints [19]. The CLSP considers multiple items and can be modeled by Mixed Integer Programming (MIP). Other extensions are also known for the lot-sizing problem. More details can be found in [19]. See [3], [83], [40], [1], [84], [13], [73], [15], [12] and [2] for other examples of lot-sizing problems. The scheduling problem determines the optimal allocation of resources, defining the production start and completion dates [41, 70]. This problem is commonly solved to short time intervals like a few days or a few weeks, depending on the specific characteristics of the planning and the organization. Furthermore, the scheduling model should incorporate the specifications of the production process and should know the detailed flow of materials to generate a feasible production plan [70]. Scheduling problems were first analyzed in the 1950s. More complex problems have arisen and, in the 1970s, many of the studied problems were proven to be NP-Hard. Currently, more than 50 years after, techniques and algorithms are still developed to solve scheduling problems, proving its importance and difficult resolution [43]. More information about the scheduling problems can be seen in [41], [8], [58] and [4]. [61], [38], [44], [77], [67], [56] and [48] also discussed several scheduling problems. In [14] the authors present a recent survey about the concurrent resolution of lot-sizing and scheduling problems. The authors introduce the history of different models proposed in the literature over time, also presenting their potential application in industry and directions for further research. A review of supply chain management focusing on the approaches used to connect planning and scheduling problems is performed in [50]. The authors present different 3

12 CHAPTER 2. LITERATURE REVIEW 4 formulations used to model these problems as well as discuss the use of new techniques and existing challenges in this research field. According to the same authors, despite the advances, solving them together still is a challenging task. They classified the methods used to solve planning and scheduling problems in three approaches: full-space methods (or integrated, as called in this work), hierarchical and iterative (or interactive, as called in this work). Figure 2.1 shows an adaptation of [50] considering the specific problems studied in this work, illustrating the three approaches used to solve them. In the hierarchical and interactive methods, the decisions made at the upper level are inputs to the lower level. Considering a single period, if the information flow is only from the upper level to the lower level, the approach is hierarchical. If there is a feedback information from the lower level to the upper level, the approach is interactive. Considering the hierarchical methodology, when no feasible solution is found, the procedure tries to find a feasible schedule, at the lower level. At the contrary, in the interactive approach, the lot-sizing problem may be solved again based on the information returned by the lower level, repeating this procedure until reach a feasible solution [50]. It is important to notice that in the hierarchical method, solutions from scheduling may be used to change the MPS of future periods. Integrated Problem Lot-sizing model Detailed scheduling model Upper level Lot-sizing model Lower level Detailed scheduling model Upper level Lot-sizing model Lower level Detailed scheduling model Feedback a) Integrated approach b) Hierarchical approach c) Iterative approach Figure 2.1: Approaches to solve production lot-sizing and scheduling problems (Adapted from [50]). The arrows show the flow of information between different decision levels. In [78] the authors propose a hierarchical and an integrated model for connecting production aggregate planning level and the MPS level. In the hierarchical approach, the aggregated model belonging to the upper level passes information to the lower level corresponding to the MPS on the number of production lines necessary for each period and target values for the inventory levels, solving the problems sequentially. For the integrated approach, the authors propose a procedure based on the hierarchical method. However, the connection between the two levels is made within a single mathematical model, characterizing the integrated approach. The results showed that the proposed integrated model creates plans with fewer setups, treats seasonality more efficiently and is less susceptible to errors given that coordinates all the decision levels. Besides that, the integrated model found better results when compared to the hierarchical model for all the instances analyzed. Another comparison between the hierarchical and integrated approaches is performed by [65]. The bibliography for the joint resolution of lot-sizing and scheduling problems is very extensive, in which many types of problems and resolution methods are considered. Due to the vast number of works, in the remaining of this chapter, we focus on classical and recent papers for each approach, briefly discussing key features. Lastly, we propose a classification related to the strategy used to solve the problem for all the articles considering hierarchical and interactive approaches, including the strategies proposed in this work.

13 CHAPTER 2. LITERATURE REVIEW Integrated approach The integrated approach, also called monolithic approach, formulates all the studied problems as a single large mathematical model, solving it to optimality [64]. An integrated approach seems to be a good alternative to solve a production lot-sizing jointly to a scheduling problem. However, an integrated model can lead to many decision variables and consequently be able only to deal small instances of the problem [81]. The hierarchical decomposition of MPS and scheduling decisions simplifies the problem but ignores essential constraints of the shop floor when solving the MPS. Therefore, consistency is not guaranteed across all the planning levels. The integrated approach is more complicated, but eliminates connection errors between the levels, as it incorporates all constraints into a single formulation [17]. In [23] the Discrete Lot-sizing and Scheduling Problem (DLSP) is addressed, which consists of determining the quantities to produce and the production sequence of multiple products in a single machine, discretizing the periods considered in the CLSP. The author assumes sequence independent setup costs and develops a method for solving the DLSP based on the Branch-and- Bound technique and on the Lagrangean Relaxation of the capacity constraints. The methods presented satisfactory results for most of the instances tested. In [24], Fleischmann adds to the DLSP sequence-dependent setup costs and time windows, solving the model with a heuristic and Lagrangean Relaxation. In [28] the authors model the DLSP considering sequence-dependent setup costs, adding valid inequalities to the original design. The separation problem is solved with an exact procedure based on a Branch-and-Cut method, reducing the computational time to find optimal solutions. The results proved that the valid inequalities are effective in improving the computational time and strengthen the linear relaxation of the problem. An integrated lot-sizing and scheduling problem in a job-shop environment is studied in [36]. The authors consider flexibility in the machines; they can change their production speeds. Besides that, the sequences generated by the model in each period should satisfy a global fix sequence, considering precedence constraints. The problem is solved with a Particle Swarm Optimization (PSO) algorithm, that has the autonomy to modify its parameters and conditions when performing searches. The problem is also solved with an exact method for comparison purposes, using the best solution found by the PSO as an initial solution to accelerate the convergence to the optimum. However, even using this strategy, the PSO algorithm presented much better results for the allowed computational time, with an acceptable optimality gap. In [85] an integrated lot-sizing and scheduling problem with unrelated parallel machines and capacity constraints in a semiconductor manufacturing industry is presented. Sequencedependent setup times, time windows, machine eligibility and preference constraints are considered. Setup propagation and backlogging are also permitted, using a hybrid heuristic based on Lagrangean decomposition and Simulated Annealing for resolution. The hybrid heuristic is efficient for the instances tested, finding better results than those found by heuristics from the literature. See [31], [20], [21], [75], [71], [66] and [74] for more integrated lot-sizing and scheduling problems in different industrial scenarios. 2.2 Non-Integrated approaches Two non-integrated procedures are here considered, the hierarchical and the interactive approaches. The hierarchical approach consists in the division of the resolution of the problems in several steps. Each one is sequentially solved, in which decisions made at lower levels are based on results from upper levels, reducing the complexity for resolution. These methods are better to be adapted to the organizational structure of the company [78]. The hierarchical approach considers the interdependencies of all planning tasks and tries to coordinate the solution [25].

14 CHAPTER 2. LITERATURE REVIEW 6 Then, the results obtained by this method are suboptimal, given that it does not integrate all the constraints of the MPS and the scheduling problems in a global model [59]. The hierarchical approach can be particularly useful for large instances, since in such cases the integrated approach may not solve the problem due to its complexity [64]. Hierarchical production planning began to be discussed in the 1960s, see [80], but it was only formalized in the 1970s with the works [34], [33] and [7]. In this approach the integrated formulation is replaced by several models that are sequentially solved, in which each step is consistent with the hierarchical decisions that must be made [6]. The authors also add that medium or long-term aggregate decisions are made first at the upper level and set constraints to the operational decision level, that considers detailed information. In [26] this statement is confirmed, showing how hierarchical planning performs throughout the supply chain. At the upper level decisions related to the shop-floor are not considered, making the MPS problem easy to solve. Given the solution of the lot-sizing model, the scheduling problem assigns the production orders to individual resources, defining the operational sequence. The separation of the problem at two planning levels requires that the manufacturing process has an idle capacity or that it is flexible about the given sequence since the solution can be infeasible. This infeasibility may be due to sequence-dependent setup times, loading of resource groups and a lead-time offset of zero time units between two successive operations [70]. In [36] the authors also comment that the consideration of the lot-sizing and the scheduling problems sequentially may result in various corrective actions to make feasible the solution obtained by the scheduling problem. In the hierarchical approach proposed in [34], optimal decisions are taken at the aggregated planning level, setting constraints to the tactical level. The authors consider four decision levels. First, long-term decisions are made based on capacity constraints, allocating products to the plants. At the second level, a production plan is carried out, defining quantities to produce and seasonal inventories. At the next level, the scheduling is set for each family of products, while at the last level the schedule is detailed for each item. The hierarchical method is chosen over the monolithic approach given the limitations of including all the shop-floor details, and to the fact that a single model does not allow the involvement of the company s management in the several stages of the planning [34]. In [52] an unrelated parallel machines problem with sequencedependent setup times is studied. The lot-sizing problem is first solved to optimality considering estimated and aggregate information. Based on the information obtained at the upper level, the scheduling problem is solved with the Greedy Randomized Adaptive Search Procedure (GRASP) heuristic. See [29], [42], [18] and [37] for more works that consider the hierarchical approach. An alternative to the integrated and hierarchical approaches is to use the interactive approach. The interactive approach can be especially applied when the difference between the solution of the planning problem and the solution of the scheduling problem is significant [39]. In this case, the planning problem can be solved again, generating another production plan based on the results of the scheduling level to improve the feasibility of the plan. The authors also state that the complexity to solve production planning problems and scheduling problems together comes from the difficulties of synchronization of both problems. Therefore, an interactive approach is proposed by the authors to coordinate in a better way the inputs and outputs of the decision levels in the semiconductor fabrication. The authors discuss the main information transferred between both problems. A production lot-sizing and scheduling problem in a job-shop environment can be seen in [16]. At the upper level, the lot-sizing problem is solved based on a fixed sequence of jobs considering capacity constraints. At the lower level, scheduling is performed considering fixed batch sizes. Backorder is incorporated into the model to avoid infeasible plans when the two planning levels are considered. The proposed method is solved iteratively sharing information between the two levels to find the best feasible production plan. In [63] the author also uses

15 CHAPTER 2. LITERATURE REVIEW 7 an interactive procedure to solve planning problems, in which seasonal stocks and overtime are defined in the medium term, while in the short term the production lot-sizing and scheduling problems are solved. The base level returns iteratively to the upper level information on the feasibility of the plan. The procedure adjusts the demand defined in the medium term to the demand required in each period of the lot-sizing problem. Also, it performs an approximation of stock levels and individual setup times based on average values defined in the medium term. This approach is implemented through an Artificial Neural Network, comparing the results with other approximation functions. A capacitated parallel machines problem with sequence-dependent setup times and costs is considered in [11], solving it with an interactive approach. If the scheduling problem is infeasible, cuts ensuring the generation of new sequences are added to the lot-sizing problem, which is solved again for the same period. In [86] the authors also study a capacitated lot-sizing and scheduling problem, however, in this case, multiple plants are considered. The lot-sizing level defines the products to produce, their respective quantities and the allocation of raw material to the different plants. Based on this information, a scheduling problem is solved for each plant, considering the values obtained at the upper level as targets to the lower level problem. The procedure is iteratively solved, transferring to the lot-sizing problem the information on the quantities of raw material used, incorporated as a cut to the lot-sizing problem. The constraints enable the definition of new targets to the scheduling problem. In [5] an interactive approach is also proposed, in which the planning problem is solved with a linear programming (LP) model, while a priority rule-based scheduling method is applied to the problem at the lower level. Simulation is then used to verify the feasibility of the resulting production planning. In [69] a lot-sizing and scheduling problem is considered, also using a simulation model, the testing module, to provide information to the lot-sizing problem. The information transferred consists of the sequence generated at the scheduling level, which is also sent to the testing module. The testing module determines the current bottleneck machine and the available capacity and returns this information to the lot-sizing model. The MPS is repeatedly solved for all the lots considered for production. An interactive approach to solve only the scheduling problem can be seen in [47]. [82], [45] and [54] are some examples of works considering the interactive approach to solve production planning and scheduling problems. 2.3 Contextualization This section aims to compare this study to the literature articles cited throughout the Chapter 2. Table 2.1 summarizes articles dealing with the joint or individual approach of lot-sizing and scheduling problems. It is worth noticing that Table 2.1 also includes works on integrated, hierarchical and interactive approaches to different planning problems. The table briefly presents the problem, the reference, the objective function, and the resolution method. In the "Resolution method" column, "MILP" indicates that a Mixed Integer Linear Programming model is used with a commercial solver. As we can observe, most of the objective functions of lot-sizing problems, integrated problems and problems solved at the upper level of hierarchical and interactive approaches address the minimization of costs, such as production, inventory, backorder and setup costs. Additionally, the minimization of makespan is also typical when considering scheduling problems, especially when hierarchical and interactive approaches are employed. We can also observe that there is a great variety of resolution methods used to solve production planning problems. Among the cited works, [65] and [78] are the ones that most resemble the objectives of this dissertation, given that they draw a comparison between hierarchical and integrated approaches. The problems addressed in both papers are different from the problem dealt in this work as well as the resolution methods

16 CHAPTER 2. LITERATURE REVIEW 8 but resembles this one in the fact of comparing different types of approaches to solve production planning problems Hierarchical and interactive approaches for lot-sizing and scheduling problems One of the objectives of this work is to analyze and propose hierarchical and interactive strategies for the lot-sizing and the scheduling problems. We perform a detailed literature review of works verifying the different types of information shared between the decision levels. From this analysis, we propose a classification based on the strategy used by each article to solve the studied problems, dividing these works into five groups, explained next. Table 2.2 shows the proposed classification, presenting the group in which each paper is classified, and the information shared between the different levels. Some of these papers consider other decision levels beyond the lot-sizing and scheduling. In this selection, we consider papers dealing with lot-sizing and scheduling problems in different decision levels.

17 CHAPTER 2. LITERATURE REVIEW 9 Table 2.1: Cited papers that solved lot-sizing and scheduling problems individually or that solved planning problems with the integrated, hierarchical or interactive approaches. Problem Reference Objective function Resolution method Lot-sizing [1] Minimize the production, setup, inventory and lost sales costs Lagrangean relaxation/dynamic programming algorithm/lagrangean heuristics/metaheuristic based on the adaptive large neighborhood search [2] Minimize the fixed and variable production and transportation costs and total holding cost Dynamic programming algorithms [3] Minimize the setup and inventory costs Heuristics based on LP-and-fix and Relax-and-fix [12] Minimize the production, setup, inventory and transfers between plants costs Lagrangean heuristics/integer Programming [13] Minimize the inventory, setup and overtime costs Fix-and-optimize heuristic/variable Neighbourhood Search (VNS) metaheuristic [15] Minimize the purchase cost of the products, the total ordering cost and the total holding cost [32] Minimize the inventory and setup costs EOQ [40] Minimize the total cost minus the penalty received for backlogged supply Heuristic based on Reduce and optimize approach (ROA)/MILP [73] Minimize the setup and inventory holding costs Fix-and-optimize heuristic/column generation heuristic [76] Minimize production, inventory and setup costs Lagrangean relaxation/dynamic programming/heuristic smoothing procedure [79] Minimize setup and inventory costs, charges for filling demand plus the cost of adopting an optimal policy MILP Proposed algorithm [83] Minimize the inventory and setup costs Relax-and-fix heuristic [84] Minimize the inventory, setup and backorder costs Lagrangean relaxation-based heuristic/heuristics from the literature/milp Scheduling [38] Minimize the total weighted completion time/minimization of the number of tardy jobs/minimization of the total weighted tardiness/minimization of maximum lateness [44] Minimize the makespan Dynamic programming algorithms [48] Minimize the absolute deviation from the optimal makespan in the worst-case scenario MILP Local search-based heuristic/simulated annealing-based implementation/exact method from the literature [56] Minimize the makespan MILP/Heuristics from the literature/greedy heuristics [67] Minimize the makespan Proposed exact algorithm/milp [77] Minimize the makespan/minimize fixed processing costs/maximize the profit Parallel Branch-and-Bound algorithm

18 CHAPTER 2. LITERATURE REVIEW 10 Table 2.1: Cited papers that solved lot-sizing and scheduling problems individually or that solved planning problems with the integrated, hierarchical or interactive approaches. Problem Reference Objective function Resolution method Integrated [20] Minimize the total sum of inventory, backorder, machine changeover and tank changeover costs Hierarchical or Interactive [21] Minimize the total sum of inventory, backorder and machine changeover costs Relaxation approach/relax-and-fix heuristic Relax-and-fix heuristic/milp [23] Minimize the setup and inventory holding costs Procedure based on Branch-and-Bound and Lagrangean relaxation/dynamic programming [24] Minimize the sequence-dependent setup costs and inventory holding costs Procedure using Lagrangean relaxation as well as a heuristic [28] Minimize the setup and inventory holding costs Exact method based on Branch-and-Cut/Exact and heuristic algorithms for the separation problem [31] Minimize setup and inventory costs Proposed Branch-and-Bound procedure/milp [36] Minimize the setup, production, inventory and shortage costs MILP/PSO algorithm [66] Maximize the profit MILP [71] Minimize the setup and inventory holding costs/minimize the setup, inventory holding costs for final products and in process inventory costs MILP [74] Minimize the setup, inventory and production costs Branch-and-Cut [75] Minimize overtime and inventory holding costs/minimize overtime, inventory and setup costs MILP/Relax-and-fix heuristic [81] Minimize the setup, inventory and production costs Lagrangean heuristic/constructive heuristic/milp [85] Minimize the inventory, backlog, setup and machine preference costs Hybrid Lagrangean-simulated annealing-based heuristic/lagrangean Relaxation algorithm/simulated Annealing [5] Minimize the production, inventory holding (including work-inprocess inventory) and backorder costs [7] Aggregate level: Minimize the inventory costs, regular hours costs and overtime costs/disaggregated level: Minimize the average annual setup cost and equalize the run-out time of itens of different families of products [9] First level: Minimize inventory costs/second level: Minimize sequence-dependent changeover times/third level: Minimize lost production due to changeover time [11] Minimize the operating, transportation, inventory and changeover costs Linear programming/priority-rule-based scheduling method/discrete-event simulation Linear programming/algorithms from the literature Linear Programming/Network model/tsp Heuristics Hybrid Bilevel-Lagrangean Decomposition

19 CHAPTER 2. LITERATURE REVIEW 11 Table 2.1: Cited papers that solved lot-sizing and scheduling problems individually or that solved planning problems with the integrated, hierarchical or interactive approaches. Problem Reference Objective function Resolution method Hierarchical or Interactive [16] Lot Sizing: Minimize the inventory, backorder, production and setup costs/scheduling: Minimize the makespan Integer programming/heuristics from the literature [18] Minimize the inventory, production and setup costs Algorithms from the literature/branch-and-bound procedure [27] Medium term: Minimize changeover and inventory costs/short term: Meet due dates and minimize the number of changeovers [34] First level: Minimize capital, production and transportation costs/second level: Minimize production and inventory costs/third level: Minimize cycle inventory and changeover costs/fourth level: Maximize the customer service [37] Minimize the loss due to item changeover/minimize the holding cost/minimize the total cost [39] Planning model: Maximize demand satisfaction and minimize the lead time/scheduling: Minimize the gap between the plan and the current status of the scheduling [42] Aggregate level: Minimize changeover costs and maximize throughput efficiency/disaggregated level: Minimize the costs of overutilization of production capacities and deviations from desired inventory levels [45] Planning model: Minimize production, backorder and inventory costs/scheduling: Minimize production costs [46] Minimize the sum of variable manufacturing, setup and inventory costs [47] First level: Maximize the duration of the horizon and products included/second level: Minimize the total number of products included to limit the size and complexity of the scheduling problem/third level: Maximize the production [52] Lot-sizing: Minimize the production, setup, backorder and inventory costs/scheduling: Minimize the weighted sum of jobs completion [53] Minimize the sum of start up and setup costs, inventory and production costs [54] Planning problem: Minimize the penalty of not meeting the supply of products at the reception and the demand of ships, the cost of product allocation in the stockyard and the cost of using routes to transport products/scheduling: Minimize the makespan Adapted Dixon s heuristic/parallel machine scheduling heuristics Linear programming/existing algorithms and systems/development of a new system Heuristic/Neural network approach/simulation Proposed framework using optimization and simulation Heuristics/Goal programming/existing procedures and softwares/linear programming Proposed decomposition framework MILP/Existing algorithms and systems/development of a new system MILP/Decomposition Model MILP/GRASP heuristic Proposed optimization approach/milp/branch-and-bound algorithm MILP/Relax-and-fix heuristic/grasp heuristic/route Capacity Reduction heuristic

20 CHAPTER 2. LITERATURE REVIEW 12 Table 2.1: Cited papers that solved lot-sizing and scheduling problems individually or that solved planning problems with the integrated, hierarchical or interactive approaches. Problem Reference Objective function Resolution method Hierarchical or Interactive [63] Mid-term planning: Minimize overtime and inventory costs/shortterm planning: Minimize the inventory and setup costs as well as the deviation of the overtime in the short-term planning Artificial neural networks/others anticipation functions [68] Minimize the inventory and setup costs Two-stage heuristic/priority rules [69] Meet the due dates considering capacity constraints, minimize the makespan and minimize the inventory holding costs [82] Planning problem: Minimize the raw material cost, inventory cost, backorder cost and operating cost/scheduling: Minimize the slack variables of the required production as well as all the variables minimized at the planning level [86] Planning problem: Maximize the profit/scheduling: Maximize the product throughput Adapted heuristics from the literature/simulation MILP/Heuristic/Interactive framework using a multi-stage stochastic programming formulation Hierarchical [30] Minimize the inventory and setup costs MILP/Proposed heuristic/greedy heuristic & Integrated [51] Minimize setup costs on tanks and on production lines MILP/Two stage optimization heuristic/local search [64] Integrated and scheduling models: Minimize the makespan/balancing model: Balance the workloads [65] Minimize the number of tardy orders/minimize the inventory holding cost/minimize the production line start-ups and part shipments costs [78] Aggregate production planning model: Minimize costs of running, opening and closing production lines and minimize inventory holding costs/mps: Minimize the production, setup, holding inventory, backorders and overtime costs MILP MILP MILP MILP

21 CHAPTER 2. LITERATURE REVIEW 13 Table 2.2: Literature papers that solved production lot-sizing and scheduling problems with the hierarchical and interactive approaches. The strategies proposed in this work are also presented. The table presents the classification group, the reference to the paper and the information transferred between the levels. The works are shown within the group according to their publication date. Group Reference Lot-sizing to scheduling Scheduling to lot-sizing 1 [34] Monthly production plan and the seasonal inventory for each product type at each plant location [27] Master production schedule which states the number of process lines to be used for each diameter in each period [46] Monthly production plan by plant - [42] Lot sizes and line assignments by product and product group and finished inventories by individual products [18] Lot sizes based on setup cost estimates [9] Production quantities and inventory levels [37] Products to be produced and their respective quantities [30] Products to be produced and their respective quantities [51] Assignment of mixture to tanks - [52] Quantities of products produced, stocked and backordered [68] - Products to be produced and the production sequence Proposed HSI Proposed HSII Products to be produced and their respective quantities Products to be produced and their respective quantities 2 [16] Fixed lot sizes Fixed sequence 3 [69] Lot sizes Sequence dependent setup times [5] Disaggregated plan Waiting time, classification of product types and classification of mask types [39] Production quantity and work-inprocess (WIP) level 4 [82] Production for the current period that could not satisfy the aggregated demand or could meet or exceed the aggregated demand 5 [86] Amount of products to be produced and amount of raw material used Lead time, number of setups and ratio of the available WIP Sequence factor recalculated More realistic targets, i.e., the current targets predicted by the scheduling model [45] Inventory level and production target Estimated production cost [11] Which products may be produced and the number of batches of each product [53] Products to be produced and their respective quantities Assignments of the previous iteration Lowest economic amount set to zero

22 CHAPTER 2. LITERATURE REVIEW 14 Table 2.2: Literature papers that solved production lot-sizing and scheduling problems with the hierarchical and interactive approaches. The strategies proposed in this work are also presented. The table presents the classification group, the reference to the paper and the information transferred between the levels. The works are shown within the group according to their publication date. Group Reference Lot-sizing to scheduling Scheduling to lot-sizing 5 Proposed ISI Products to be produced and their respective quantities Proposed ISII Proposed ISIII Proposed ISIV Products to be produced and their respective quantities Products to be produced and their respective quantities Products to be produced and their respective quantities Makespan, jobs scheduled at the previous iteration and total setup time of the previous sequence Jobs scheduled at the previous iteration and the quantity of jobs scheduled at the previous iteration Jobs scheduled at the previous iteration, quantity of jobs scheduled at the previous iteration and total setup time of the previous sequence Production sequences, optimal total setup time, quantity of products belonging to the initial sequence and the positions of each product in the sequences Group 1, in Table 2.2, refers to the set of works using a hierarchical approach. In this case, no feedback information is returned to the lot-sizing level of the same period. As can be seen in Table 2.2, the Hierarchical Strategy I (HSI) and the Hierarchical Strategy II (HSII), which are the hierarchical strategies proposed in this work belong to Group 1. Groups 2, 3, 4 and 5 consider interactive procedures. Group 2 presents a single paper in which the strategy consists in firstly solve one decision level, fix the result and transfer the solution to the other level. Based on this fixed value, the other decision level is solved. Group 3 indicates papers using simulation to update parameters of the interactive procedure or for evaluation of the solution. Group 4 presents a single paper that uses an estimated factor in the lot-sizing problem to represent the effects of the scheduling constraints. Lastly, Group 5 summarizes papers that modify the lot-sizing model with the addition of a cut based on the scheduling solution. The Interactive Strategy I (ISI), Interactive Strategy II (ISII), Interactive Strategy III (ISIII) and Interactive Strategy IV (ISIV), which are the interactive strategies proposed in this work are classified in this group Relation between the proposed strategies and literature papers The methods discussed in this work are extensive enough to incorporate different strategies. In Table 2.3, we link the articles strategies with the algorithms here proposed. Table 2.3: Relation between the proposed strategies and the papers presented in Table 2.2. Strategy Reference associated HSI [34], [27], [46], [18], [42], [9], [37], [51], [52], [68], [30] HSII [69], [30] ISI [16], [69], [86], [82], [45], [5], [53], [39] ISII [16], [69], [86], [82], [45], [5], [11], [53], [39] ISIII [16], [69], [86], [82], [45], [5], [53], [39] ISIV [18], [16], [69], [86], [82], [45], [5], [53], [39] [34], [27], [46], [18], [42], [37], [51], [52] and [68] present procedures similar to the HSI strategy proposed in this work. In these cases, no information is transferred from the lower decision level to the upper decision level to modify the solution found by the scheduling problem of the same period. Nevertheless, most of the instances used in these generate feasible schedules. Thus, in most cases, there is not a deep analysis of the efficiency and efficacy in dealing with unfeasibility.