Capacitated lot sizing with linked lots for general product structures in job shops

doi:10.1016/j.cie.2009.10.002

Computers & Industrial Engineering

Volume 58, Issue 1, February 2010, Pages 151-164

https://doi.org/10.1016/j.cie.2009.10.002 Get rights and content

Abstract

In this paper, we propose a mixed integer programming (MIP) model for a multi-level multi resource capacitated lot sizing and scheduling problem with a set of constraints to track dependent demand balances, that is, the amount left over after allocating the available inventory to the dependent demands. A part of this leftover amount may be kept as a reservation quantity to meet dependent demands of the following period under capacity restrictions. These constraints are necessary because we assume independent demands as well as dependent demands for all items in the product structure. They also are used to tighten the domain of on hand and backorder inventory levels. Although we allow backorders for independent demands only, this is not possible for dependent demands as backorders will disturb the whole demand balance of the product structure. Determination of setup costs is a crucial task when developing lot sizing and scheduling models, especially in a capacitated manufacturing environment with backorders. In this respect, the capacitated lot sizing with linked lot sizes (CLSPL) model we formulate needs not to consider setup costs to avoid unnecessary setups thanks to the new set of constraints, and to obtain feasible lot sizes and schedules. Finally, a numerical example and computational results in a job shop environment are also given, and future research directions are provided.

Section snippets

Background and motivation

Material Requirements Planning (MRP), developed by Orlicky (1976), is the most popular production planning and scheduling system in practice. MRP provides the right part at the right time for the right customer, i.e., it aims to plan the end item requirements of the master production schedule.

First, MRP systems are characterized by their rapid adaptability to dynamic changes in a production/inventory system, and ability to determine the production and inventory requirement several periods in

Model development

Notation, indices, parameters and decision variables that are used in the rest of this paper is as follows:

Indices and sets

i, j = 1, … , V

index of end items,

i, j = V+1, … , N

index of component items,

k = 1, … , K

index of resources,

t = 1, … , T

index of time periods,

A_i

set of direct successors of component item i,

S_k

set of items that can be manufactured on resource k,

M_i

set of resources which item i can be manufactured,

Parameters

c⁽^b⁾(i)

unit backorder cost for item i,

c⁽^h⁾(i)

unit holding cost for item i,

l(i)

manufacturing lead time for item i,

e(i, j

A numerical example and computational results

In this section, an example is presented to illustrate the application of CLSPL formulation presented in previous section. Our example parameters are based on the paper by Ornek and Cengiz (2006).

Conclusions

Determining setup costs is not only a crucial work but, most of the time, somewhat arbitrary for manufacturing firms. This generally results in suboptimal solutions due to a trade-off between other cost elements such as holding and backorder costs. Hence, an approach that does not require setup cost minimization may overcome difficulties manufacturing companies face in estimating relevant costs.

On the other hand, late delivery is a fact of life in manufacturing firms as different orders compete

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There are more references available in the full text version of this article.

Cited by (21)

Improvement to an existing multi-level capacitated lot sizing problem considering setup carryover, backlogging, and emission control
2023, Manufacturing Letters
This paper presents a multi-level, multi-item, multi-period capacitated lot-sizing problem. The lot-sizing problem studies can obtain production quantities, setup decisions and inventory levels in each period fulfilling the demand requirements with limited capacity resources, considering the Bill of Material (BOM) structure while simultaneously minimizing the production, inventory, and machine setup costs. The paper proposes an exact solution to Chowdhury et al. (2018)'s[1] developed model, which considers the backlogging cost, setup carryover & greenhouse gas emission control to its model complexity. The problem contemplates the Dantzig-Wolfe (D.W.) decomposition to decompose the multi-level capacitated problem into a single-item uncapacitated lot-sizing sub-problem. To avoid the infeasibilities of the weighted problem (WP), an artificial variable is introduced, and the Big-M method is employed in the D.W. decomposition to produce an always feasible master problem. In addition, Wagner & Whitin's[2] forward recursion algorithm is also incorporated in the solution approach for both end and component items to provide the minimum cost production plan. Introducing artificial variables in the D.W. decomposition method is a novel approach to solving the MLCLSP model. A better performance was achieved regarding reduced computational time (reduced by 50%) and optimality gap (reduced by 97.3%) in comparison to Chowdhury et al. (2018)'s[1] developed model.
© 2023 Society of Manufacturing Engineers (SME). Published by Elsevier Ltd. All rights reserved.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the Scientific Committee of the NAMRI/SME.
Lot-sizing decisions for material requirements planning with hybrid uncertainties in a smart factory
2022, Advanced Engineering Informatics
Citation Excerpt :
Finally, some concluding remarks are presented in Section 7. Since MRP system was first developed by Orlicky [26] as an information management system, it has been widely used in enterprises with the purpose of providing the right components to the right customers at the right time [27]. According to the type of uncertain variables, approaches that are used to cope with uncertainty in MRP lot-sizing production problems mainly include stochastic programming, fuzzy programming, and interval programming.
Material requirements planning (MRP) is a kind of medium-term production planning, which aims to plan the end item requirements of the master production schedule over a finite planning horizon. In a smart factory, the customer requirements and the production status are varying in time, which increases the uncertainties in making lot-sizing decisions for MRP. In this study, a hybrid chance-constrained programming (HCCP) model is developed for solving an MRP problem with hybrid uncertainties, in which both randomness and fuzziness exist in a lot-sizing decision process. The objective of the HCCP model is to determine the lot sizes of all items while satisfying the stochastic demands and the fuzzy capacity constraints. The credibility and probability are incorporated into the proposed model to measure the fuzziness and randomness, respectively. In order to solve the model, relevant approaches for converting the probability-based and credibility-based constraints into the equivalent deterministic forms are proposed. Decision makers can set different confidence levels according to their own risk preferences to get different results. Finally, an example is presented to verify that the approach proposed in this paper is feasible for solving MRP problems with hybrid uncertainties.
Hybrid genetic algorithm and tabu search for finite capacity material requirement planning system in flexible flow shop with assembly operations
2016, Computers and Industrial Engineering
The finite capacity material requirement planning system (FCMRP) for industrial scale flexible flow shops is known to be strongly NP-hard. Due to very long computational time, the exact method can be inappropriate for this problem. In this paper, a new hybrid improvement algorithm for the FCMRP system in a flexible flow shop with assembly operations is proposed. The proposed algorithm is a hybrid of genetic algorithm (GA) and tabu search (TS) called HGATS. There are six primary steps in HGATS. In step 1, a production schedule is generated by variable lead-time MRP (VMRP). In step 2, dispatching and random rules are applied to generate initial sequences of orders. From step 3 to step 5, the sequences of orders are iteratively improved by characteristics of TS and GA. Finally, the start times of operations are optimally determined by linear programming. The results show that HGATS outperforms GA, TS and the existing algorithm. Furthermore, HGATS requires a practical computational time when applied to real industrial cases.
Lot sizing and job shop scheduling with compressible process times: A cut and branch approach
2015, Computers and Industrial Engineering
Citation Excerpt :
However, most part of the literature is devoted to single stage/machine cases when considering capacity issues. Oztürk and Ornek (2010) developed a MIP model for multi-period production scheduling with allowable backordering and production capacity constraint along with linked lots. Their model is an extension of capacitated lot sizing and scheduling problem (CLSP) for job shop environment.
This paper studies simultaneous lot sizing and scheduling problem in a job shop environment over a finite number of periods. The machines are flexible such that the production manager is able to change their working speeds. Also, the production schedule of the items over machines should be in consistence with technological precedence relationships. The problem is formulated as an integer linear program. Then, a number of valid inequalities are developed based on the problem structures. The proposed valid inequalities are added to the problem formulation in a “cut and branch” manner to accelerate the search process by common solvers. Finally, performance of the valid inequalities is investigated with CPLEX 12.2 on a set of randomly-generated test data.
A self-adaptive PSO for joint lot sizing and job shop scheduling with compressible process times
2015, Applied Soft Computing Journal
This paper presents mathematical modeling of joint lot sizing and scheduling problem in a job shop environment under a set of realistic working conditions. One main realistic assumption of the current study is dealing with flexible machines able to change their working speeds, known as process compressibility. The production schedules should be subject to limited available time in every planning period. Also, the model assumes that periodical sequences should be determined in a way that they obey from a fixed global sequence. Another complicating aspect of the problem is about consideration of precedence relationships for the needed processes of an item type over the corresponding machines. As the problem is proved to be strongly NP-hard, it is solved by a particle swarm optimization (PSO) algorithm. The proposed algorithm is self-controller about its working parameters, time to stop the search process, search diversification/intensification, and totally about its behavior. Performance of the algorithm is verified in terms of optimality gap using Lingo 11.0 on a set of randomly generated test data.
Operational extended model formulations for Advanced Planning and Scheduling systems
2014, Applied Mathematical Modelling
Since the basic reasoning of Manufacturing Resources Planning (MRPII) systems is flawed, a new breed of concepts called Advanced Planning and Scheduling systems (APS) have recently emerged to overcome the problems occurring on the shop floor. In this study, we develop improved and extended mixed integer programming formulations for APS systems at the factory planning level. First, we develop a basic model which explicitly considers capacity constraints, operation sequences, processing times, and due dates in a multi-machine, multi-order, multi-item environment where an item can be processed on a given set of eligible machines. The extensions to the basic model include sequence dependent setups, and transfer times between machines. We also show that our model with a little modification could be used to quote delivery times for customer orders in case due dates are not specified. We provide numerical examples and our conclusions along with future research directions.

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^☆: This manuscript was processed by Area Editor Mohamad Y. Jaber

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Capacitated lot sizing with linked lots for general product structures in job shops☆

Abstract

Section snippets

Background and motivation

Model development

Indices and sets

Parameters

A numerical example and computational results

Conclusions

European Journal of Operational Research

European Journal of Operations Research

Omega

Computers and Operations Research

Design and analysis of lean production systems

A modified framework for modeling set-up carryover in the capacitated lot sizing problem

International Journal of Production Research

A framework for modeling setup carryover in the capacitated lot sizing problem

International Journal of Production Research

Capacitated lot-sizing with linked production quantities of adjacent periods

Beyond manufacturing resource planning (MRP II)