Mathematical Problems in Engineering

Volume 2015, Article ID 271902, 14 pages

http://dx.doi.org/10.1155/2015/271902

## Optimal Production Planning for Manufacturing Systems with Instantaneous Stock-Dependent Demand and Imperfect Yields

^{1}College of Management and Economics, Tianjin University, Tianjin 300072, China^{2}Department of Logistics Management, Wuhan University of Technology, Hubei 430063, China

Received 28 September 2014; Accepted 31 December 2014

Academic Editor: Pui-Sze Chow

Copyright © 2015 Longfei He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider an EPL model like manufacturing system in presence of production imperfectness and stock-demand dependence simultaneously. During the production process, the system can evolve from in-control state into out-of-control state at any random time, after which the defective items will be generated likely causing quantity loss. Meanwhile, the market demand rate is instantaneously dependent on the timely holding inventory. The manufacturer has to determine his production run length and cycle time by taking into account possible imperfect production, stock-dependent demand, and inventory holding capacity bound. We empolder a model to capture this problem and develop computational algorithm to solve it. We further conduct numerical studies to validate our model and solving method. Sensitivity analyses are reported to show the effect of parameters on the system performance.

#### 1. Introduction

Managing supply/production uncertainty and changeable demand is always one of the challenging operations since matching them is fairly hard due to instability occurring in two sides, respectively. The firm can suffer managerial issues resulting from both out-of-control production (e.g., Rosenblatt and Lee [1] and Sana [2]) and demand’s instantaneous dependence on inventory (e.g., Gupta and Vrat [3] and Urban [4]).

This study involves two aspects of imperfect production and demand-stock dependence in the OM field. Imperfect production property means that defective items can be produced as the production system evolves over time. These imperfect goods will be reworked or deposed directly. As a consequence, the inventory/production planning is spoiled. On the other hand, the demand rate is instantaneously dependent on the timely inventory holding level of quality goods. That means the random production imperfection will eventually affect demand rate through defective items. Setting longer production cycle time to obtain higher inventory can obtain more demand rate resulting in more revenue. However, the longer the production cycle time is, the larger likeliness the imperfect production happens. The annual total cost comprises annual ordering cost and annual inventory holding cost. The production cycle time has opposite effects on these two kinds of cost components. Settling the optimum trade-off between the benefits and corresponding costs for the sake of optimizing the annual total profit is accordingly a crucial problem confronting practitioners.

Our study intends to address this problem by considering the production cycle time and pure production run time length as decision variables in the modeling of the manufacturing system existing in imperfect production and demand-inventory dependence. Specifically, we research the following problem: a manufacturer schedule his/her production annually by deciding each cycle time and run length, whereas the manufacturing system may switch from the in-control state to the out-of-control state after some time on the account of continuous run of the machine, which takes place at any random time during the production process, also as presented in Shah and Shukla [5]. Moreover, this is not immediately under observation and finally causes the quantity loss. The market demand is sensitive to the on-hand effective inventory excluding the defective goods.

Although we are concentrating on a new model of production planning contextualized in uncertain yields simultaneously confronting inventory-dependent demand, our modeling processes are to some extent inspired by preceding papers. We give an elucidation on the board literature related to our study below.

The existing relevant literature mainly includes two streams: uncertain production management; inventory/production planning with inventory-dependent demand and their extensions. The production uncertainty incurred by various factors like machine failure, staff changes can always cause the entire production system instable resulting in defective products has inspired extensive studies, such as Rosenblatt and Lee [1], Lee and Rosenblatt [6], Lee and Rosenblatt [7], and a variety of their succeeding papers like Liou et al. [8], Kim and Hong [9], Chung and Hou [10], Guo et al*.* [11], Hu et al. [12], and so on. Drawing on the celebrated EOQ/EMQ models, early pioneering researches study the production planning optimization considering defection inspecting, restoration cost dependence, detection delay, and maintenance, say, Rosenblatt and Lee [1], Lee and Rosenblatt [6], and Lee and Rosenblatt [7]. Their valuable investigations firstly pull the production planning research closer to the realistic practice where there usually exist in out-of-control states for the production process incurring a lot of associated management problems. Furthermore, inspection error types are proposed to be other factors having effects on production planning in an imperfect EMQ manufacturing system; see Liou et al. [8]. From a different perspective, Kim and Hong [9] link the distinction among three types of deteriorating processes to the optimal production run length seeking. Later, Chung and Hou [10] generalize former literature to allow the shortage in the imperfect system. After those static models in this line of research, Sana [2] addresses a production-inventory model with imperfect production and time-varying demand. Their study substantially extends proceeding researches into dynamic situations. In contrast, some recent researches turn to exploring imperfect production planning in the fuzzy environment, like Guo et al. [11] and Hu et al. [12]. Although the aforementioned literature has done intensive studies on uncertain production incorporating features incurred by itself, that existing research fails to consider the change of demand pattern during the production cycle time. However, they usually assume the demand characteristics are invariable over time, which is just a very special case of our model in this study.

Observation to practice has shown that the real market demand is very complex and it is not a fixed constant, as implied by Levin et al. [13], Silver and Peterson [14], and Gupta and Vrat [3]. Incited by their ground-breaking work, a variety of studies have incorporated stock-dependent rate component into diverse classical inventory models, for instance, by considering items deterioration in Mandal and Phaujdar [15], Datta and Pal [16], and Dixit and Shah [17], involving shelf-space allocation in Baker and Urban [18], allowing stock shortage in Datta and Pal [16], and combining delayed payment schemes in Soni and Shah [19], Min et al. [20], Shah et al. [21], Teng et al. [22], and Zhou et al. [23]. A considerable amount of literature has also addressed relevant problems, like Goh [24], Wee [25], Padmanabhan and Vrat [26], Ray and Chaudhuri [27], Sarker et al. [28], Giri and Chaudhuri [29], Yan and Cheng [30], Mandal and Maiti [31], Chung and Tsai [32], and so on. To learn more about the research of this kind, one is referred to a recent literature review by Urban [4]. Almost all literature mentioned above has only focused on the dependence between demand and inventory with the implicitly obvious a priori prerequisite of completely perfect production, which usually does not fit the practice well. On the contrary, in this paper, we alter that precondition and emphasize the imperfect production frequently occurring in real world.

To the best of our knowledge, few of existing studies have considered the production imperfectness and inventory-demand dependence simultaneously. In contrast, this is just what we concentrate on in the present study. We model the production and demand consumption processes with the aforementioned characteristics as a constrained nonlinear programming system combining different scenarios. Furthermore, our model is distinguished from existing relevant models in several ways: we combine imperfect production and inventory-dependent demand together to explore the optimal production problem; we use optimizing the annual total profit as the objective function instead of annual total cost, which substantially reflects the effect of demand-inventory dependence on benefits; and we incorporate the inventory capacity upper boundary into traditional models of imperfect production and inventory holding. We select and develop a suitable computational algorithm to solve the nonlinear programming model with detailed operational steps and procedures. Subsequently, we conduct numerical examples and parameters sensitivity analysis to examine the proposed computational algorithm so as to get some managerial insights (Figures 7, 9, 10, 11, and 13).

The rest of the paper is organized as follows. In Section 2, we sketch the problem features, notations, and its assumptions. In Section 3, we discuss the various scenarios arising from our studied system and derive the associated model. In Section 4, we describe a computational algorithm for solving the constructed model. In Section 5, numerical studies and associated analysis are executed to validate the algorithm proposed. Finally, concluding remarks are given to summarize this study in Section 6.

#### 2. Problem Characteristics, Notations, and Assumptions

We study a manufacturing system described like a variant of economic production lot (EPL) model, which is distinct from classical EPL model by characterizing its specific production property and demand changing rules, respectively. On the production side, the system may randomly evolve into an out-of-control state in which a proportion of goods will be generated following some probability distribution. On the demand side, the impact of inventory level on demand is considered. The demand rate consists of the constant demand rate and the inventory level dependent demand rate addressed aswhere and .

So the dynamics where the stock is depleted at time can be presented as follows:

The production process can be described like this: for scheduled production cycle time , the manufacturer runs the production line from the beginning to time , that is, , where . The production line is closed down in interval . The manufacturing system may evolve into out-of-control situation incurring defective goods produced randomly. Denoting the production rate and the production duration yields aggregate volume during the cycle time. It may produce defects after producing continuous time and the system failure results in inferior goods in proportion of all production during , where obeys an exponential distribution with parameter . During the whole cycle time, the inventory is being consumed by the demand at a rate varying according to the associated stock level at the same time.

The following notations will be used throughout the paper.

*Parameters* : inventory level at any time (units), : sales rate at time with where and (units), : maximum system inventory capacity (units), : productivity, that is, production rate (units per year), : random variable of the time moment when production system is changing to uncontrolled state, the probability of which obeys an exponential distribution with parameter (year), : the proportion of defective goods generated after production status changing time , : fixed setup cost per order ($/order), : variable cost per unit product ($/unit), : the selling price per unit qualified product ($/unit), : the salvage value per unit defective product ($/unit), : holding cost per unit product per unit time ($/unit item/unit time).

*Decision Variables* : the whole production and supplying cycle time (year), : total production duration, (year), : total annual costs, including inventory and ordering costs ($/year), : annual inventory costs ($/year), : annual ordering costs ($/year), : total annual revenue, including sales income of qualified products and scrapped income of defective ($/year), : sales income of qualified products ($/year), : crapped income of defective ($/year), : total annual profits, ($/year).

To refine and reflect the essence of our problem, we need* assumptions* in our model as follows.(A1)The lead time is zero and the time span is unlimited.(A2)The production capacity is limited but out-of-stock is not allowed, and productivity is greater than steady section, that is .(A3)After producing continuous time, the system enters an uncontrolled state and defective rate of the product obtained is . After discovering the defective, it needs be scrap processing timely instead of entering the inventory system, where downtime obeys an exponential distribution with parameter .(A4)Production process is independent in different cycles, which is in each cycle, and the duration of controlled state is independent.(A5)During each production cycle, the beginning and ending inventory are both zero, that is, and .

#### 3. The Model

For a given production cycle time , there may be two cases in the whole production process because of the occurrence of imperfect production at time . One case is that breakdown time does not occur before the production stopping, that is, , and products obtained in during the period are all qualified products. The other is that the breakdown time occurs during the production process, that is, , and the defective rate is () from time to time . Assume the upper bound of the stocking system capacity is ; inventory levels at the beginning and the ending times of the production period are both zero, namely, and . We plot the inventory level changing trajectory of qualified products in Figure 1 with two cases, where is the stopping time of the continuous production with system under control completely, and so is when system out of control occurs.