A continuous review production–inventory system in fuzzy random environment: Minmax distribution free procedure

https://doi.org/10.1016/j.cie.2014.10.022Get rights and content

Highlights

  • Continuous review economic production quantity model with reordering strategy.

  • Uncertain demand is prescribed as a fuzzy random variable.

  • Minmax distribution free procedure is extended for fuzzy random variable.

  • Fuzzy random cost function is scalarized by fuzzy expectation and signed distance method.

Abstract

In many manufacturing systems, the production process may take some time to start the initial phase due to various reasons such as delay in installation of machines, short supply of raw materials, unavailability of workers, etc. Thus, the organization should plan accordingly so that the manufacturing process can start at the desired time. In an economic production quantity (EPQ) model, lead-time plays a significant role in ensuring that the manufacturing process starts on time. As we know, when both lead-time and demand rate are deterministic and constant, then demand during the lead-time is constant, and is referred to as zero lead-time. Moreover, when either or both of them are random variables, then lead-time demand (LTD) is a random variable. In such a case, a crucial question is: “when should the order be placed?” On the other hand, the distributional information on demand may not always be available or there may be many distribution functions in the practice, which have same mean and variance, but their frequencies are different. In this study, we develop an EPQ model in stochastic framework, wherein the distribution function of demand is unknown, but the mean and variance are known. The inventory level is continuously reviewed, and an order is placed when it reaches the reorder level. The real-life business situations are so sophisticated and floating in nature that the consideration of ‘impreciseness’ along with ‘statistical variability’ in demand parameter is more preferable. To be a part of this contingency, we further extend the model in the fuzzy random environment by considering demand rate as a fuzzy random variable (FRV). Furthermore, we mathematically analyze the cost function and propose a heuristic procedure to find the global optimum. Numerical examples with sensitivity analysis are also provided for illustration purpose.

Introduction

Since the development of the economic production quantity (EPQ) and economic order quantity (EOQ) models, both have been equally accepted and implemented by academicians as well as the business enterprises. The classical EPQ models are mainly developed in deterministic environment, wherein demand is taken as constant, time dependent, stock dependent, or price dependent; but, due to the floating nature of marketplaces, these assumptions are not very realistic. However, in recent years, authors such as Coates et al., 1996, Sarkar and Moon, 2011, Sarker and Coates, 1997, Panda et al., 2008, etc. have developed some EPQ models in stochastic framework. Coates et al., 1996, Sarker and Coates, 1997 considered manufacturing lead-time as a finite range random variable, while, demand rate and production rate both are deterministic constants. Panda et al. (2008) and Sarkar and Moon (2011) developed single period production–inventory models in stochastic environment, wherein demand during the scheduling period varies randomly, whereas production rate is taken as a deterministic constant. Ouyang, Wu, and Ho (2004) and Glock (2012) established the vendor–buyer coordination for integrated production–inventory system, by considering stochastic demand of customers and deterministic constant production rate of the vendor. Moreover, for avoiding the shortages in the vendor’s inventory system, both the models have restricted the production rate to be greater than the mean value of stochastic demand rate. This brief discussion indicates that production–inventory needs an attention of research with perturbing behavior of key parameters. Taking this in account, this study intends to incorporate stochastic and then fuzzy stochastic demand in a production–inventory model.

One major drawback of the stochastic inventory model is the description of random demand by a definite probability distribution function. Moreover, there is a tendency to use the normal distribution for the lead-time random demand, whether it is continuous review or periodic review system. Although in practice, the distributional information of demand may not be always available. However, if it is available then it may not be necessary to follow normal distribution function. Though, from the computational point of view, normal distribution is easy to use in modeling, but it does not provide the best protection against the occurrence of the other distributions with the same mean and the same variance (Gallego & Moon, 1993). According to Moon and Gallego (1994), in many cases, estimation of mean and variance of forthcoming demand is possible, while, distribution function fitting is still difficult. Moreover, there may be several probability functions that have the same mean and variance. In 1958, Scarf made the first effort to avoid the distributional information in the newsboy problem, and introduced the “minmax distribution free procedure” (MMDFP). In that model, he assumed that only mean and variance of demand are known, whereas no information about the form of the distribution function. Gallego (1992) used MMDFP in a continuous review (Q,R) system, and established a landmark result which will be discussed in a later section. In brief, MMDFP is an approach that finds the most unfavorable distribution function by fitting the two moments mean and variance, and then minimizes the total cost. Gallego and Moon (1993) extended the Scarf (1958) newsboy problem in several directions, and developed the Scarf’s ordering rule that is easier to understand and remember. Moon and Gallego (1994) implemented the MMDFP in many stochastic inventory models such as continuous review system, periodic review system, where only the mean and the variance are known. Tajbakhsh (2010) used this approach in continuous review system with fill rate service doctrine. Recently, Janakiraman, Park, Seshadri, and Wu (2013) established some robust results supported by rigorous mathematical analysis for limited demand information in the newsboy and multi-period inventory systems. Considerable research has been carried out by using the MMDFP, namely, Sarkar and Moon, 2011, Ouyang and Chuang, 2000, Moon and Choi (1994), etc. Our intention is to extend the MMDFP technique for fuzzy random demand. Hence, we turn our discussion in this direction.

Today’s highly competitive business environment and volatile nature of marketplaces perturb the key parameters of inventory system. Consequently, besides randomness, vagueness is also intrinsic in the key parameters. Hence, consideration of only randomness in the key parameters of inventory system is not a very pertinent approach to deal with real life situations. In 1978, Kwakernaak introduced the notion of fuzzy random variable (FRV), which simultaneously underlines randomness and fuzziness in the perception of an event. According to Shapiro (2009), randomness models the stochastic variability of all possible outcomes of a situation and describes the inherent variation associated with the environment under the consideration. On the other hand, fuzziness relates to the unsharp boundaries of the parameters of the model. There is abundant literature that either solely models uncertainty using probability distributions with known parameters, or solely considered fuzziness using membership function. But, unfortunately this facility is limited if we discuss about a hybrid type of uncertainty called FRV that simultaneously deals with fuzziness and randomness. However, we here discuss some cases of fuzzy random inventory modeling. Chang, Yao, and Ouyang (2006) and Lin (2008), respectively, extended the continuous review (Q,r) and periodic review (Q,T) systems in fuzzy random (FR) environment. Both the models assumed that lead-time demand (LTD) as a FRV, while annual demand as a fuzzy number. Dutta, Chakraborty, and Roy (2007) and Dey and Chakraborty (2009), respectively, extended the classical continuous review and periodic review inventory systems in FR framework by considering LTD as fuzzy variable and annual demand as discrete FRV. Recently, Dey and Chakraborty (2011) extended the continuous review inventory system via quality improvement and set-up cost reduction by incorporating demand rate as continuous FRV. Bag, Chakraborty, and Roy (2009) developed a production–inventory model by considering FR demand rate and deterministic constant production rate. Zhang, Zhao, and Tang (2009) and Hu, Zheng, Guo, and Ji (2010) extended the classical EPQ model in fuzzy random environment by considering the possibility of machine shifting from “in-control” state to “out-of-control” state, wherein machine shifting time is taken as a FRV. Recently, Kumar and Goswami (2013) developed a production–inventory model with learning effect consideration in production, and machine shifting hazard in fuzzy random environment, in which shifting time is taken as a FRV. Hu, Zheng, Xu, Ji, and Guo (2010) established the vendor–buyer coordination in fuzzy random environment, wherein the vendor internally produced the item with a constant rate, and the buyer faces the external FR demand of customers. Wang (2011) developed a continuous review inventory system for item with imperfect quality in fuzzy random environment by incorporating fuzzy renewal reward theorem to estimate the expected average cost per unit time.

From the above discussion, we have noticed that: (1) The distributional information of stochastic demand may not be always available. Thus some time we have to model a stochastic inventory problem without probability distribution function. (2) The demand parameter of a production–inventory system inherits randomness as well as fuzziness. (3) No production–inventory is developed for infinite time horizon with reordering strategy in stochastic or fuzzy stochastic framework. (4) “Minmax distribution free procedure” is developed only for stochastic demand, not for fuzzy stochastic demand. Thus our intention is to cover all these research gaps of the production–inventory theory. In this process, we first develop a stochastic EPQ model by using Gallego and Moon’s (1993) MMDFP. In an EPQ model, the organization internally produces the product and meets the customer demand. In any manufacturing system, the production process depends upon several factors such as set-up of machine, arrival of raw material, availability of worker, etc. Thus, as and when required, the production process may not always start instantly. In such situations lead-time is essential. The literature of inventory management provides several continuous review EOQ models by incorporating the concept of lead-time. Similar to an EOQ model, lead-time plays significant role in an EPQ model, also. If demand rate is a random variable, then LTD is also a random variable. As a custom in inventory problem, this production–inventory model also tries to find out the answer of the questions: “When to be ordered?” and “how much to be ordered?”

Furthermore, if the statistical data itself possesses fuzziness in terms of unreliable or inaccurate data (lack of proper corroboration), linguistic expressions (lack of precise information, ill-known parameters), variability of inventory situation (non-constant production process, prices), etc., then it is computationally much easier to assume those parameters as FRVs (Dey & Chakraborty, 2011). The real life business transactions are so sophisticated and floating that the consideration of statistical ‘uncertainty’ and linguistic ‘impreciseness’ in customer demand is desirable. Thus, we further extend the proposed stochastic model in fuzzy random framework by incorporating demand rate as a FRV. Consequently, LTD, total cost and length of cycle become FRVs. Fuzzy random total cost and fuzzy random cycle length generates a fuzzy random renewal process with rewards. In this process, we use fuzzy expectation and fuzzy random renewal reward theorem (Hwang, 2000, Hwang and Yang, 2011) in the mathematical formulation of the model. Furthermore, signed distance method (Yao & Wu, 2000) is used to find the equivalent deterministic cost function. As evident from the literature review and to the best of our knowledge, no production–inventory model is developed under these circumstances.

The rest of the paper is arranged as follows: Section 2 provides the notations and assumptions which are used throughout the paper. In Section 3, we derive mathematical formulation of the stochastic model. Section 4 extends the stochastic model in fuzzy random environment, and also provides the solution procedure to optimize the cost function toward the global optimum. Section 5 illustrates both the stochastic and fuzzy stochastic models with the help of numerical examples, graphical illustration and sensitivity analysis. Finally, Section 6 makes concluding remarks and suggests the future scope of this study.

Section snippets

Notations and assumptions

The following notations and assumptions are adopted to develop the proposed production–inventory model.

Mathematical formulation of the stochastic model

In this section we develop the proposed continuous review production–inventory in stochastic framework. In brief, continuous review (Q,r) system is based on reorder level detection doctrine, in which on hand inventory is continuously reviewed, and an order of size Q is placed whenever inventory reaches at level r. The order quantity Q is shipped in a single shipment in such a type of inventory system. But, in the practice, it may not be always possible to replenish the full order quantity Q in

Model in mixed environment of randomness and fuzziness

In contrast to the stochastic model discussed in Section 3, we now extend that model in fuzzy stochastic framework by incorporating fuzziness in random demand parameter. As we discussed in Section 1, a FRV provides an appropriate mathematical method or tool to deal with both statistical ’uncertainty’ and linguistic ‘impreciseness’ that emanate due to several disruptions in inventory system. The demand estimation process goes through a sequence of activities, namely, aggregation of information

Illustrative example

Here we consider some numerical examples in support of the models and algorithm developed in Sections 3 Mathematical formulation of the stochastic model, 4 Model in mixed environment of randomness and fuzziness. Example 1 captures the stochastic model which is discussed in Section 3, while Example 2 captures the fuzzy stochastic model of Section 4.

Example 1

D=10,000 units/year, A=70/order, c=30/unit, ch=0.6/unit/year, cb=1.5/unit (Moon & Gallego, 1994). Moreover, we take, P=50,000 units/year, L=2 weeks, σ

Conclusion

This paper quantifies the impact of a continuous review production–inventory system with finite production rate and stochastic/fuzzy stochastic demand rate in the reorder level strategy. The purpose of this study is twofold. First, to extend the classical EPQ model in stochastic environment by assuming that customer demand is a random variable with hidden distribution information. In this process, we adopt the MMDFP. In practice, the distributional information regarding the demand is not always

References (37)

Cited by (0)

View full text