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This book examines the Capacitated Lot Sizing Problem (CLSP) in process industries. In almost all process industries, there are situations where products have short/long setup times, and the setup of the product and its subsequent production are carried over, across consecutive periods. The setup of a product is carried over across more than one successive period in the case of products having long setup times. A product having short setup has its setup time less than the capacity of the period in which it is setup. The setup is immediately followed by its production of the product and it may also be carried over, across successive time period(s). Many process industries require production of a product to occur immediately after its setup (without the presence of idle time between the setup and production of the product), and they also require the product to be continuously produced without any interruption. This book considers a single-machine, single-level and multiple-item CLSP problem. This book introduces the Capacitated Lot Sizing Problem with Production Carryover and Setup Crossover across periods (CLSP-PCSC). Mathematical models are proposed which are all encompassing that they can handle continuous manufacturing (as in process industries), and also situations where the setup costs and holding costs are product dependent and time independent/time dependent, with possible backorders, and with other appropriate adaptations. Comprehensive heuristics are proposed based on these mathematical models to solve the CLSP-PCSC. The performance of the proposed models and heuristics are evaluated using problem instances of various sizes. This book also covers mathematical models developed for the Capacitated Lot Sizing Problem with Production Carryover and Setup Crossover across periods, and with Sequence-Dependent Setup Times and Setup Costs (CLSP-SD-PCSC). These models allow the presence of backorders and also address real-life situations present in process industries such as production of a product starting immediately after its setup and its uninterrupted production carryover across periods, along with the presence of short/long setup times. Heuristics proposed for the CLSP-PCSC can be extended to address the CLSP problem with sequence dependent setup costs and setup times. All the models and heuristics proposed in this book address some real-life considerations present in process industries.

### Chapter 1. Introduction

In the initial phase of research in the fields of Operations Research and Management Science, researchers carried out their work on production planning under the assumption that demand was deterministic and time varying (dynamic). In due course, research on production planning evolved with the development of Harris’s EOQ (Economic Order Quantity) formula, Wilsons’s (Q,r) model, and the dynamic lot sizing model proposed by Wagner and Whitin. Harris (1913) was the first to publish the work in the lot sizing area entitled, (published as Harris (1990)). The EOQ was also known as Wilson’s lot size formula as it was used in practice by Wilson (Wilson 1934). Wagner and Whitin developed the dynamic lot sizing problem in 1958 (Wagner and Whitin 1958). Since then, researchers deal with various types of lot sizing problems for different applications (also see Manne (1958)).
Ravi Ramya, Chandrasekharan Rajendran, Hans Ziegler, Sanjay Mohapatra, K. Ganesh

### Chapter 2. CLSP: Real Life Applications and Motivation to Study Lot Sizing Problems in Process Industries

Generally, the CLSP addresses the production planning problem in discrete manufacturing industries and continuous manufacturing industries. A brief explanation about the production planning in discrete manufacturing industries and continuous manufacturing industries with relevant examples is presented in Sects. 2.1.1 and 2.1.2, respectively.
Ravi Ramya, Chandrasekharan Rajendran, Hans Ziegler, Sanjay Mohapatra, K. Ganesh

### Chapter 3. Capacitated Lot Sizing Problem with Production Carryover and Setup Crossover Across Periods (CLSP:PCSC): Mathematical Model 1 (MM1) and a Heuristic for Process Industries

The capacitated lot sizing problem (CLSP) is a lot sizing model in which the production of multiple products is allowed within a time period on a single machine, with a condition that the entire demand for a product within that period should be met from the production in that period and/or the inventory carried from the previous periods, without any backorders or lost sales. Finding a minimum cost production plan that satisfies all the demand requirements without exceeding the capacity limits of a period is the main objective of the CLSP.
Ravi Ramya, Chandrasekharan Rajendran, Hans Ziegler, Sanjay Mohapatra, K. Ganesh

### Chapter 4. Further Development: Mathematical Model 2 (MM2) and a Comprehensive Heuristic for Capacitated Lot Sizing Problem with Production Carryover and Setup Crossover Across Periods for Process Industries

In the previous chapter, a mathematical model and a heuristic are applied to the CLSP in process industries which can be applied to real-life situations in process industries such as production carryover across periods and setup crossover across periods. The heuristic proposed in Chap. 3Capacitated Lot Sizing Problem with Production Carryover and Setup Crossover Across Periods (CLSP:PCSC): Mathematical Model 1 (MM1) and a Heuristic for Process Industrieschapter.38.3 with respect to MM1:CLSP-PCSC can be easily applied when identical capacity is present across periods. However, in reality the capacity across periods may be varying. When non-identical capacity is present across periods, for allowing shift of setup/production for more periods ahead of or after the current time period, the extension of the heuristic based on MM1:CLSP-PCSC becomes tedious. In such cases the heuristic proposed in this chapter is easier to apply. Hence, in this chapter we propose a second mathematical model (MM2:CLSP-PCSC) for the CLSP-PCSC followed by a heuristic using the second mathematical model. The proposed model in this chapter is not constrained by the consideration of long setup products. The model is flexible enough to handle the process industries with small bucket setups and long bucket production runs or the scenario with large bucket setups and small production runs or a mixture of both. In other words, the proposed mathematical model and heuristic approach are flexible enough to handle or address situations in the conventional process industries such as cement and sugar industries (associated with small bucket setups and long bucket production runs), large bucket setups and small bucket production runs (associated with highly technological intensive big bucket setups and small bucket production runs such as those in highly specialized pharmaceutical processes), or a mixture of scenarios in a single process industry. Also, depending upon the industry the definition of a period may vary. It is to be noted that in all these scenarios we have real-life restrictions that once a process starts there is no interruption with the production run length, and the production has to start immediately after the completion of setup. In this book we address such a variety or mix of process-industry scenarios and the restriction in terms of continuous production and production commencement immediately after setup. This book is primarily motivated by the literature on CLSP based on the nature of continuous manufacturing industries such as chemical manufacturing, cement manufacturing, sugar industries, pharmaceuticals, hot rolling process, heat treatment, casting and injection moulding, and a real-life case study in a batch processing industry. Referring to the benchmark literature (e.g. Sung and Maravelias (2008) and Belo-Filho et al. (2013)), we find that no existing work has attempted such a mix of industrial scenarios and associated real-life constraints such as continuous production with no interruption and production commencement immediately after setup completion. Therefore, the proposed mathematical model in this chapter is also generalized in nature.
Ravi Ramya, Chandrasekharan Rajendran, Hans Ziegler, Sanjay Mohapatra, K. Ganesh

### Chapter 5. Capacitated Lot Sizing Problem with Production Carryover and Setup Crossover Across Periods Assuming Sequence-Dependent Setup Times and Setup Costs (CLSP-SD-PCSC): Mathematical Models for Process Industries

In Chaps. 3 and 4, mathematical models have been proposed for the capacitated lot sizing problem with production carryover and setup crossover across periods. Heuristics based on both the mathematical models have also been proposed. The models and heuristics address real-life situations in process industries such as production immediately after setup and uninterrupted production carryover across periods.
Ravi Ramya, Chandrasekharan Rajendran, Hans Ziegler, Sanjay Mohapatra, K. Ganesh

### Chapter 6. Summary Concerning Theoretical Developments

Lot sizing is a major decision taken during the planning of production of various products in process and manufacturing industries. The lot sizing problems can be classified into continuous lot sizing problem (economic lot scheduling problem) and dynamic lot sizing problem. The time scale considered is continuous and infinite in the continuous lot sizing problem, whereas a discrete time scale is considered in dynamic lot sizing problems. The dynamic lot sizing problems are further classified into uncapacitated and capacitated lot sizing problems based on their capacity restrictions. The capacitated lot sizing problems are further classified into small bucket and big bucket lot sizing models depending upon the number of setups that are allowed in a given time period. The discrete lot sizing and scheduling problem (DLSP), continuous setup lot sizing problem (CSLP) and the proportional lot sizing and scheduling problem (PLSP) come under the small bucket lot sizing models, and the capacitated lot sizing problem (CLSP) comes under the big bucket lot sizing model.
Ravi Ramya, Chandrasekharan Rajendran, Hans Ziegler, Sanjay Mohapatra, K. Ganesh