Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5805404, 10 pages

https://doi.org/10.1155/2017/5805404

## Optimization and Customer Utilities under Dynamic Lead Time Quotation in an Type Base Stock System

Department of Architecture, Civil Engineering and Industrial Management Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan

Correspondence should be addressed to Koichi Nakade; pj.ca.hcetin@edakan

Received 23 February 2017; Revised 10 April 2017; Accepted 2 May 2017; Published 25 May 2017

Academic Editor: Huanqing Wang

Copyright © 2017 Koichi Nakade and Hiroki Niwa. 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

In a manufacturing and inventory system, information on production and order lead time helps consumers’ decision whether they receive finished products or not by considering their own impatience on waiting time. In Savaşaneril et al. (2010), the optimal dynamic lead time quotation policy in a one-stage production and inventory system with a base stock policy for maximizing the system’s profit and its properties are discussed. In this system, each arriving customer decides whether he/she enters the system based on the quoted lead time informed by the system. On the other hand, the customer’s utility may be small under the optimal quoted lead time policy because the actual lead time may be longer than the quoted lead time. We use a utility function with respect to benefit of receiving products and waiting time and propose several kinds of heuristic lead time quotation policies. These are compared with optimal policies with respect to both profits and customer’s utilities. Through numerical examples some kinds of heuristic policies have better expected utilities of customers than the optimal quoted lead time policy maximizing system’s profits.

#### 1. Introduction

In manufacturing systems the production and inventory control must be appropriate to reduce the production cost. Information on production such as advance demand, amounts of work-in-process and finished products, and machine’s failure is important to the control. For customers buying products, the information of order lead time is important to decide whether he/she buys a product or not. In addition, the consumer will be unsatisfied when the actual lead time is greater than the informed lead time. The quotation of lead time causes the actual number of customers to vary because the long quoted lead time leads to decrease of actual demand, and thus appropriate information makes the profit of the system increase.

Recently, the information on inventory or lead time to customers is discussed. For example, Duenyas and Hopp [1] develop a dynamic lead time quotation problem in a make-to-order system as an queue using an MDP formulation. Ata and Olsen [2] consider a make-to-order system where customers are dynamically quoted lead times. They recommend quotation policies for convex, concave, and convex-concave delay cost functions. Kapuscinski and Tayur [3] consider two classes of customers, where the high priority customers bring more profit to the system but delay for them leads to the higher penalty on the system. Wu et al. [4] consider a newsvendor problem with information of price and quoted lead time and determine optimal amounts of orders, prices, and quoted lead time. In Selçuk [5] a cost effective dynamic lead time quotation procedure in a single-stage controlled manufacturing system is considered and guidelines are discussed for setting the number of kanbans and the frequency of updating lead time. Slotnick [6] discusses an optimal lead time quotation policy when reputation of firm affects whether each consumer orders products or balks and discusses the relationships among order size, reputation, lead time decision, and so on. In addition, Hafızoğlu et al. [7] consider price and lead time decisions in a make-to-stock system with contract and spot customers under Poisson arrivals and exponential service times. The optima policy on price and lead time is characterized. Zhao et al. [8] consider the make-to-order manufacturing system which gives two lead time and price quotation strategies, one of which offers a single lead time and price, and the other gives the menu of pairs of lead time and prices. They discuss the better strategies under the price-sensitive and lead time sensitive consumers. The other lead time quotation and decision models are found in Keskinocak et al. [9], Ata [10], Charnsirisakskul et al. [11], Chaharsooghi et al. [12], and Xiao et al. [13].

In Savaşaneril et al. [14], the optimal lead time quotation in an base stock inventory queue for maximizing the system’s profit is discussed. In this system, if the lead time quoted to the arriving customer is long, the probability that the customer enters the system and receives service becomes small. In their paper, the model is formulated as a Markov decision process and the property of the optimal lead time quotation policy is discussed.

Literature on optimal quotation polices focuses on the optimal lead quotation policies for the system. Under such optimal lead time quotation policies, however, the exact lead time information is not quoted to customers. One reason is that the production process is under uncertainness because of failure or repair of the machine, and thus actual production time is stochastic. The other and important reason is that, under the optimal lead time policy, the system manager may not give the mean of actual lead time, even if the delay cost for quoted lead time is considered. Thus, when this optimal policy is applied, the customer’s satisfaction may be small, because he/she leaves the system by quotation of long lead time when the actual lead time is small and his/her actual waiting time for items may be longer than quoted lead time because of the setting of shorter quoted lead time than the actual lead time. Thus, for the system manager, it is important to maximize not only the profit of the system but also customer’s satisfaction. Most of researches in literature, however, do not discuss this kind of satisfaction of actual customers deeply.

As one of the customer’s utility on waiting time, the effect of delay information on the customer’s satisfaction is formulated in Guo and Zipkin [15, 16]. In these papers, the utility function of each customer is defined, which shows the degree of his/her satisfaction on waiting time in a queue. It includes the stochastic parameter representing his/her impatience for waiting time, and if the value of this parameter is high, then he/she is impatient of waiting. They defined several types of delay information and their effects are discussed theoretically and numerically. In Nakade and Niwa [17], an base stock manufacturing-inventory system is developed, but the evaluation of the utility function is inappropriate and thus the relationship between the utility and average cost is unclear. In fact, the utility function itself only includes the waiting time cost, but the evaluation of the utility as performance measure includes the delay cost for quoted lead time, which is confusing.

In this paper, the effect by the lead time quotation in an base stock manufacturing-inventory queue into the customer’s utility function is discussed. Poisson arrivals are popular because it is well known that the arrival process from large population with small probability that each person will arrive at a system approximately forms a Poisson process (e.g., see section 5.2 of Pinsky and Karlin [18]). The utility function is based on the definition in Guo and Zipkin [15], and if his/her utility is negative under the lead time information then he/she leaves the system without receiving an item and otherwise enters the system and receives it after possible waiting time. The model is formulated into a Markov decision process. Its optimal lead time quotation policy is derived, and several heuristic policies such as linear, convex, and concave lead time quotation policies are analyzed by birth and death processes numerically. Numerical results are modified and extended from Nakade and Niwa [17]. In comparison between profits and utilities, the performance measure of utility does not include the delay for quoted lead time. In addition, the relationship among expected reward, the inventory cost, and the delay cost for quoted lead time under optimal policy is discussed for each base stock and fixed delay cost. The numerical examples show that the optimal lead time quotation policy for maximizing system’s average profit has low customer’s utility, and the other simple heuristic quotation policy leads to the greater expected values of customers’ utility, although it has a bit smaller system’s profit than the optimal policy. In particular, some lead time quotation policy has both more system profit and much customer’s satisfaction in comparison with the optimal lead time quotation policy with the greater number of base stocks.

The organization of this paper is as follows. In Section 2, a lead time quotation model and a utility function in a manufacturing-inventory system with base stocks are defined. In Section 3, an average cost under a given quotation policy is determined and the optimization problem in this system is formulated as a Markov decision process. The expected utility is also derived. Numerical experiments for developing properties of optimal policies and other heuristic policies are given in Section 4, and the concluding remarks are given in Section 5.

#### 2. Lead Time Quotation Model

##### 2.1. Model Description

A manufacturing-inventory system with a single process is considered. Customers arrive in a Poisson process with rate , and the production time has an exponential distribution with rate The processing time is mutually independent among products and also independent of the arrival process. A base stock policy with base stock level is applied in this system. Figure 1 illustrates the model.