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Article Excerpt
1. Introduction
Most production-to-order companies do not have a constant flow of orders. This often leads to a varying queue of customers. Despite the uncertainty in both arrival and service of orders, customers request tight due dates, and they are also resentful of late deliveries. The companies cannot use an inventory buffer to compensate for the variations in order frequency, however, a good practical solution is to try to match the production rate to the actual workload or, when orders are statistically identical, to the length of the queue. Such a policy may bring benefits in labor cost reduction or in due date performance. This paper investigates the value of using an optimal workload-dependent policy, where "workload" means the number of orders in the system.
Our motivation for studying workload-dependent capacity planning originates from the area of Engineer-To-Order (ETO). As a definition for ETO, we quote that of Gelders (1991): "In an engineer-to-order environment a company designs and produces products to customer order." In the same paper, Gelders concludes that ETO companies need a ''fast-response capacity," which is to be considered as a general characteristic of competitive ETO production.
Similarly to ETO companies, Make-To-Order (MTO) companies also need to adjust their capacity to meet customer demand (see, for example, Vollmann et al (2005)). Apart from the motivating example for ETO in the paper of Gelders, we give three examples of MTO companies, where workload-dependent capacity management is applicable. In our first example, we describe a general manufacturing situation with an expensive bottleneck machine. In this case, the number of shifts the bottleneck machine works determines the production capacity. The capacity can be set between zero to three shifts, depending on the Workload. Our second and third examples are companies in the service industry, which can often face unforeseen variations in the number of orders. The second is a special translator service, where topic leaders share translation tasks and assign them to specialized freelance translators. The third example is a data recording company, where audio tapes are transcribed to digital text format in order to make later searches possible. While workers listen to the tapes, they type the text into a computer.
In all examples, a "fast-response capacity" is not just desirable, but also affordable. Production capacity in manufacturing does not usually require highly educated labor; labor acquisition lead times are often very short. Although, in most service industry situations labor has specific knowledge, the acquisition lead lime substantially decreased in the last decade due to the increasing use of the Internet. In the case of the translator company, the contact information of the freelance translators is carefully maintained, so distributing new tasks takes just hours. In the case of workers with a high education, capacity flexibility can be gained from using overtime or from workers contracted for variable working times (see. for example. Filho and Marcola (2001)).
In our paper, we assume that we can afford instantaneous capacity changes as an abstraction of opportunity for "fast-response capacity." For our analysis, we consider a stationary, homogeneous Poisson arrival process of orders, an exponentially distributed service time and First-Come First-Served (FCFS)-type service. The number of employees (servers) is assumed to be a decision variable that depends on the number of jobs in the system. Our objective is to minimize the total average cost per time unit, where the costs consist of labor costs per employee, hiring and firing employees, costs related to order acceptance and early or tardy order completion.
For this problem, we present our model by applying two evaluation approaches. One approach aims at exact calculation of the distribution of the throughput time (time spent in the system). The other approach, which can be used to deal with problems of a larger scale, is based on a moment-approximation of the throughput time. With respect to the effectiveness of the flexible capacity it is essential to increase and decrease capacity at the right moment. We determine the proper switching states for small-scale problem instances by searching exhaustively. The optimal strategy is compared in terms of performance with the fixed capacity rules.
A number of researchers have pointed out relations among service rate, work-in-process and due date performance. One application of due date performance measures is the evaluation of batch scheduling rules. In the scheduling rules, capacity level and work-in-process are both consistently present as parameters showing a strong relation to the due date performance (see, for example, Philipoom et at. (1993)). Lead time setting is a topic where work-in-process is also recognized as an adequate parameter, which improves the performance of the rules (see, for example, Bertrand (1983)). These two examples reveal the connection between capacity level, work-in-process and due date performance. However, there is no model in the literature that incorporates all these three closely related ideas.
We think that this paper makes a valuable contribution by being the first to study a queuing system with adaptive capacity to satisfy the objective of having light and accurate due dales. Moreover, our examples and their analysis provide insight into when a workload-dependent strategy is effective, and also the characteristics of this strategy.
The paper is organized as follows. In the next section, we describe the related literature. Then, in Section 3, we give a mathematical formulation of our problem. Results on the effectiveness and characteristics of the strategy are shown in Section 4. Finally, conclusions and plans for future extensions are presented.
2. Related literature
Decisions on capacity changes were first studied in capacity expansion problems. In the case of deterministic demand with positive trend, Chenery (1952) found that gas pipelines permanently have extra capacity. This was the basis of his "excess capacity hypothesis" that says capacity is always larger than demand; optimal overcapacity is to be investigated by looking at economies of scale. Manne (1961) revised Chenery's hypothesis when extending his model. The extension of the model with a backlog option created an environment in which the "excess capacity hypothesis" no longer held. Manne also studied stochastic, stationary demand without the backlog option. This model resulted in a smaller deviation than that suggested by Chenery s model. Luss (1982) gave a comprehensive review of the literature on capacity expansion.
Models with capacity expansion/reduction decisions and hiring/firing costs are usually studied by means of Dynamic Programming (DP). One example is the continuous-time DP model of Bentolila and Bertola (1990), where a sensitivity analysis on firing costs was presented. Rocklin et al (1984) studied a production-to-order environment with non-stationary demand, including both capacity expansion/reduction decisions and hiring/firing costs. They showed the optimality of the (S', S")-policy known from inventory theory by the means of a discrete-time DP approach. In their model, demand must always be met; if demand exceeds the available capacity, the capacity must be increased to overcome the deficit.
Pinder (1995) deduced an approximation of the optimal workload-dependent capacity control policy for stationary demand. In the model of Pinder, capacity (resources) is treated as discrete; capacity adjustments are dependent on the actual number of jobs (work), which are particular points in common with our paper. However, the workload-dependent policy class defined by Pinder seemed too broad to give an explicit formula of policy evaluation or to find an optimal solution. In addition, Pinder did not consider due date performance in any form, which is an essential part of our model. Besides, we define a less broad policy class that entails less limitations in the analysis so that we can provide additional insights.
Queuing models have made use of servers with load-dependent service times for approximate performance analysis since the fundamental work of Avi-Itzhakand Heyman (1973). These servers are apt to represent...
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