A numerical solution for a two-stage production and inventory system with random demand arrivals

Abstract

This paper is about the study of a production lot sizing problem consisting of customers, one retailer, and one manufacturer. Demand from customers arrives randomly at a retailer one unit at a time. The retailer replenishes inventory from the manufacturer upon receiving a customer's order after its inventory depleted to zero. The manufacturer's production rate is assumed to be a finite constant. A production cycle starts when the manufacturer's inventory falls to or below zero and stops when its on-hand inventory reaches its optimal level. During the uptime in a production cycle, inventory is being built while randomly arriving orders from retailer are being fulfilled. The order arrival times from customers are independently and identically distributed, hence the inventory processes at both the manufacturer and the retailer become a renewal process that is difficult to solve analytically for a general distribution of order arrival time. Therefore, a numerical approach is used in developing a search procedure to obtain the optimal solution to the problem. Employing such a numerical approach, we also investigate how optimal solutions in different cases will change over the spectrum of some key parameters of the problem.

Introduction

The coordination of different parties in a supply chain has become more and more important in current, competitive markets. Many firms are realizing that they could benefit more from an integrated inventory management system across a supply chain. The problem studied in this research is a two-stage production and inventory system involving three parties: customer, retailer (buyer) and manufacturer (vendor), in which a manufacturer produces a single-item product and provides it to a retailer, while the retailer sells the product to its customers. The customers' demand for a particular product arrives randomly at the retailer. Hence, the retailer's inventory depletes stochastically, resulting in a random cycle time at the retailer for its inventory replenishment from the manufacturer. The manufacturer makes its production plan to fulfill the retailer demand. The manufacturer builds up its inventory during the production uptime while at the same time satisfying the retailer's demand.

According to existing studies, this problem may fall into the category of joint lot sizing and finished product delivery problems, sometimes known as vendor–buyer coordination problems. Banerjee [1] was one of the first researchers to identify the vendor–buyer problem, as documented in the paper by Ben-Daya et al. [4]. Sarmiento and Nagi [14] conducted a comprehensive survey on production–distribution problems; Chen [8] summarized deterministic vendor and buyer inventory problems; and Ben-Daya et al. [2] provided a comprehensive review for the lot sizing problems involving a single vendor and one or multiple buyers.

Sarker and Parija [16] studied a single-stage production system under a fixed-quantity, periodic delivery policy with constant demand. Their objective was to find optimal batch size. In their study, they assumed that production continues in a production cycle until a certain batch size is reached. They also assumed that the production rate is greater than the demand rate to ensure no shortage. Ben-Day et al. [3] analyzed a production and shipment lot sizing problem in a single vendor–buyer supply chain by explicitly considering the transportation cost instead of having it included in the ordering cost.

In reality, customer demand is often random, either in the amount or the arrival time. This randomness of demand has significant impact not only on the decision of production and delivery policies of the retailer, but also on the vendor in a two-stage production system. Hence, the vendor–buyer inventory problem under a stochastic environment has become more interesting and has attracted the attention of other researchers. Ben-Daya et al. [4] developed an integrated, single-vendor and single-buyer model with stochastic demand and linear lead time in terms of lot size. Ouyang et al. [11] presented a single-vendor and single-buyer integrated production model with stochastic lead time. Sajadieh et al. [15] advanced an approach to determine the optimal production and shipment policy for vendor–buyer problems with stochastic lead time. One common assumption among these researchers is the deterministic nature of the order inter-arrival time at the vendor due to certain vendor–buyer policies.

Several researchers have studied the lot sizing problem in terms of the inventory holding cost with stochastic demand. Cetinkaya and Lee [5] analyzed lot size under vendor-managed inventory (VMI) with stochastic customer demand. Then, Cetinkaya et al. [6] examined the integrated inventory and transportation problem with stochastic inter-arrival time of the buyer. Later, Cetinkaya et al. [7] extended the problem to the one with random order size and stochastic inter-arrival time and presented a numerical and simulation solution. Kang and Kim [10] used a model for integrating inventory replenishment and delivery planning in a two-stage supply chain. While they employ an approximation method and derive a closed-form solution for the problem with a special case of compound Poisson demands, the inventory build-up process during production uptime is not considered in the inventory holding cost in their studies.

Other research has been conducted that addresses the two-stage production lot sizing problem, including some with generically distributed stochastic inter-arrival time of orders. However, to the best of the authors' knowledge, there have been few investigations which consider both the inventory build-up of the finished products and the arrival of orders during the production uptime.

In our research presented herein, we have considered the inventory build-up process in the inventory model. Orders from the retailer may arrive at any time during the production uptime. Section 2 defines the problem and assumptions. Section 3 discusses the mathematical model of the problem. Section 4 provides a solution focused approach and methodology to the problem, while Section 5 gives a numerical example and the computational result. Finally, Section 6 concludes the study and envisions future research.