The impact of lot-sizing in multiple product environments with congestion

doi:10.1016/j.jmsy.2014.04.003

Journal of Manufacturing Systems

Volume 33, Issue 3, July 2014, Pages 436-444

https://doi.org/10.1016/j.jmsy.2014.04.003 Get rights and content

Highlights

•
We consider a multiproduct dynamic lot sizing model in the presence of queueing behaviour.
•
Setup costs are not required as the capacity implications of setups are directly captured in the model.
•
Approximately optimal solutions are obtained and compared to a conventional lot sizing model without congestion.
•
Proposed model yields significantly better cycle time and WIP performance than conventional models.

Abstract

We present a production planning model for a multiple product single machine dynamic lot-sizing problem with congestion. Using queuing models, we develop a set of functions to capture the nonlinear relationship between the output, lot sizes and available work in process inventory levels of all products in the system. We then embed these functions in a nonlinear optimization model with continuous variables, and construct an approximate solution to the original problem by rounding the resulting fractional solution. Computational experiments show that our model with congestion provides significantly better flow time and inventory performance than a benchmark model that does not consider the effects of congestion. These advantages arise from the use of multiple smaller lots in a period instead of a single large lot as suggested by conventional fixed-charge models without congestion.

Introduction

Lot sizing problems constitute an important class of production planning models that arise when production equipment requires significant setup times to switch from the processing of one product to another [1]. The key decision variable is the lot size, the quantity of a specific product that is issued as a job to the production process. In a production setting, a large lot size yields high resource utilization due to fewer setups between products, but can result in increased finished goods inventory. Processing a large lot of one product may also delay the production of other lots, creating variability in the flow of products through the system [2]. A small lot size has advantages in processing individual orders in a timely manner and minimizing work-in-process (WIP) inventory and cycle times. However, frequent setups can consume excessive amounts of capacity, leading to long cycle times in the system [2]. Clearly, wherever possible, setup times should be reduced or eliminated, but there remain areas where this cannot be accomplished economically.

Most deterministic lot sizing models focus on the tradeoff between the fixed cost of setup that is independent of the lot size, and the holding cost of the cycle stocks due to production occurring in batches. This approach is easily defensible in a purchasing context, where the fixed cost represents the costs of placing the order and delivering the material. However, in a production context the problem is more complex. The primary difficulty centers on obtaining realistic estimates of the setup costs, particularly under time-varying resource utilization. In many industrial environments, the direct incremental cost of setup activity is often limited to the amount of scrap produced while adjusting tooling; labor and machine time costs are fixed in the time frame relevant to the lot sizing decision [3]. Hence much of the setup cost can be viewed as the opportunity cost of the capacity foregone in the setup time, which will vary over time depending on, among other things, the utilization level of the resource in question at that point in time. However, since this cost will vary over the planning horizon in the face of dynamic demands, it is difficult to estimate in practice. Hence a lot sizing model that can incorporate the effects of setups on the dynamics of the production system directly, without the need for cost parameters that are difficult to estimate, is very desirable.

The work in this paper differs from most previous lot sizing models in its explicit representation of throughput as a function of average WIP, number of lots and lot size. Setup cost is not explicitly considered since the costs of setup decisions are directly captured in the performance measures of the system. Our model can easily be modified to account for incremental, direct costs of setups, such as those arising from scrap, if necessary. We envision this model being used in tactical decision making, where a firm may want to update its lot sizes in the face of changing demands and product mixes over time. Frequent changes in manufacturing lot sizes at very short intervals are clearly difficult to implement in practice, and hence are not considered.

In this paper we present an exploratory analysis of these issues in the context of a multi-product single machine dynamic lot sizing problem. We first develop a set of nonlinear functions to capture the interactions between lot sizes, throughput, cycle time, and WIP, following the development of Karmarkar [4], [5]. We then present a nonlinear integer programming formulation for a multi-product, single machine, deterministic dynamic lot sizing problem with the objective of minimizing the total costs of WIP, finished goods inventory (FGI) with backlogging. Since this model is computationally intractable even for small instances, an approximate solution is constructed by applying a simple myopic rounding scheme to the solution obtained from a continuous relaxation. While this approach can, of course, make no claims of optimality, it permits computational experiments comparing the solutions thus obtained to those from exact integer programming models that do not consider congestion.

We find that our model provides lower cycle times and WIP levels than an alternative model that does not consider congestion. In addition, discrepancies between planned and realized performance are considerably smaller in our model, despite the extremely simple approximation approach used. These findings suggest that even with the very simple approximate solution procedure we use, the explicit consideration of congestion in lot sizing problems can be beneficial.

The next section presents a brief review of previous related work. Section 3 introduces the functions used to represent the nonlinear relation between queue size and lead times of products. Section 4 incorporates the functions developed in Section 3 into a multi-product dynamic lot sizing model. Section 5 introduces the conventional lot sizing model without congestion that is used as a benchmark in our experiments. Section 6 presents the computational experiments comparing the performance of the models. We conclude the paper with a summary of the main conclusions and highlight some possible directions for future work.

Section snippets

Previous related work

The literature on modeling production systems can be classified into two main categories. The first of these is stochastic performance analysis models, which characterize system performance measures based on a probabilistic model of the system. Within this area, queuing models have been shown to capture important aspects of system behavior [2], [6]. A fundamental insight from queuing models is that system performance measures, especially cycle times, degrade nonlinearly as utilization

Output functions for a multi-item production system with setup times

The M/G/1 queuing model, which has been shown to be effective in modeling production systems [46] is used to model congestion phenomena in the system assuming that planning periods are long enough for system to be approximately in steady state, since the function will be used in the intermediate horizon. We follow the development of Karmarkar [5], assuming deterministic processing and setup times with Markovian arrivals and a first-come first served queue discipline among lots. The lot

An integrated lot sizing model using output functions

We now present a nonlinear optimization model for dynamic lot sizing using the functions derived in the previous section. We consider a multi-product single machine production system with deterministic dynamic demand over a given planning horizon. The setup time of each product is explicitly included in the model. Completed jobs leave the machine and remain in finished goods inventory until demand is realized. Unmet demand can be backordered at a cost. We define the following notation:

Decision

A benchmark model

In our computational experiments we compare the performance of the RIM presented in the previous section with a more conventional lot sizing model by Erenguc and Mercan [42]. Using the notation given in the previous section, the model can be stated as follows:

(Erenguc and Mercan Model: EMM) $min \sum_{i} \sum_{t} h_{i t} I_{i t} + \sum_{i} \sum_{t} b_{i t} B_{i t}$ $s.t I_{i t} - B_{i t} = I_{i, t - 1} - B_{i, t - 1} + X_{i t} - D_{i t}, \forall i, t$ $X_{i t} \leq M K_{i t}, \forall i, t$ $\sum_{i} ({s^{'}}_{i} K_{i t} + {p^{'}}_{i} X_{i t}) \leq 1, \forall t$ $X_{i t} \geq 0, I_{i t} \geq 0, B_{i t} \geq 0, K_{i t} \in {0,1} \forall i, t$ In this model X_it denotes the production of product i in period t, K_it is a

Experimental design

We consider two products in our experiments, both with unit processing times. However, the setup times are 60 for product 1 and 20 for product 2, respectively. Hence one would expect the lot sizing policies for the two products to be quite different. Unit FGI and WIP holding costs are fixed at 1.5 and 1, respectively, while we use two different levels of backorder cost (10 and 100). Both formulations consider a planning horizon of 10 periods. We prefer to examine relatively small problem

Conclusions and future research

The conclusion from these experiments appears to be unequivocal: lot sizing models that do not consider congestion effects due to queuing in environments where production resources are governed by queuing behavior can produce extremely poor performance in the systems they aim to control. It is thus interesting to note the almost complete lack of such models in the literature, and the multitude of models that do not consider congestion.

The primary characteristic of the solutions produced by

Acknowledgements

The work reported in this paper is supported by NSF-TUBITAK Bilateral Cooperation Programme under Grant No: 109M018 and NSF Grant Number CMMI-0928573, and by the National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology). [NRF-2010-357-D00257]

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Technical PaperThe impact of lot-sizing in multiple product environments with congestion

Highlights

Abstract

Introduction

Section snippets

Previous related work

Output functions for a multi-item production system with setup times

An integrated lot sizing model using output functions

A benchmark model

Experimental design

Conclusions and future research

Acknowledgements

J Manuf Syst

Int J Prod Econ

Oper Res Lett

Oper Res Lett

Eur J Oper Res

Multi-level lot sizing and scheduling

Factory physics: foundations of manufacturing management

A caveat on using setup times in lot sizing

Prod Invent Manage J (First Quarter)

Capacity loading and release planning with work-in-progress (WIP) and lead-times

J Manuf Oper Manage

Manufacturing lead-times, order release and capacity loading

Stochastic models of manufacturing systems

Determining lot sizes and resource requirements: a review

Oper Res

Lot sizes lead times and in-process inventories

Manage Sci

Lotsizing in multimachine job shops

IIE Trans

Models for design and control of stochastic multi-item batch production systems

Oper Res

On production batches transfer batches and lead times

IIE Trans

Scheduling policies batch sizes and manufacturing lead times

IIE Trans

A general framework for modeling production

Manage Sci

Operations research in production planning, scheduling and inventory control

Introduction to computational optimization models for production planning in a supply chain

Operations and cost, in factory management series

Dynamic version of the economic lot size model dynamic version of the economic lot size model

Manage Sci

Deterministic production planning: algorithms and complexity

Manage Sci

Technical Paper
The impact of lot-sizing in multiple product environments with congestion