A numerical solution for a two-stage production and inventory system with random demand arrivals
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.
Section snippets
Problem definition
The mathematical formulation and its solution for a two-stage production and inventory system in practice are very intractable. In order to study the basic characteristics of the system, we made some restrictive assumptions to simplify the problem.
Assume that a two-stage production and inventory system consists of a single vendor (manufacturer), a single buyer (retailer or distributor), and customers. Orders from customers arrive at the retailer randomly—one unit at a time and only one product
Model formulation
Based on the assumptions given in the previous section, the manufacturer's production process and the retailer's inventory process are both regenerative processes. The manufacturer's regeneration points are when its inventory drops to zero and the retailer's regeneration points are when its inventory is replenished. Therefore, the manufacturer's production process consists of production cycles with the same characteristics, which constitutes a renewal process, as does the retailer's inventory
A numerical approach
Although the analytical function of has been found, it is hard to derive the expression of for a general distribution. Even for a particular distribution, it is difficult to have a closed-form expression for in general, except for a few special distributions such as exponential distribution, in which case . The computation of other factors, , and , are relatively simple and only a few algebraic manipulations are needed with , , and .
Numerical experiments
Due to the complexity of the expression , thus the total cost function , it is difficult to obtain its first and second derivatives. Hence, obtaining the optimal solution in closed-form expression for decision variables and is very challenging. We will use an exhaustive search in a two-dimensional space of and to find their optimal values. The procedure utilizes the numerical approach discussed in Section 4 to compute the values of the objective function .
Conclusion
In this paper we have studied a two-stage production and inventory problem with random demand arrivals from customers. A stochastic inventory model is developed for the problem with the objective of minimizing the expected inventory cost. The inventory processes at both the manufacturer and the retailer are modeled as renewal processes based on the order arrival process from customers. Due to the complexity of the model, however, it is very difficult to derive an analytical solution if the
References (17)
- et al.
Integrated single vendor single buyer model with stochastic demand and variable lead time
Int J Prod Econ
(2004) - et al.
Integrated vendor-buyer cooperative models with stochastic demand in controllable lead time
Int J Prod Econ
(2004) - et al.
Developing a coordinated vendor–buyer model in two-stage supply chains with stochastic lead-times
Comput Oper Res
(2009) A joint economic lot size model for purchaser and vendor
Decis Sci
(1986)- et al.
The joint economic lot sizing problem: review and extensions
Eur J Oper Res
(2005) - et al.
Production and shipment lot sizing in a vendor-buyer supply chain with transportation cost
Eur J Oper Res
(2007) - et al.
Stock replenishment and shipment scheduling for vendor-managed inventory systems
Manage Sci
(2000) - et al.
A comparison of outbound dispatch policies for integrated inventory and transportation decisions
Eur J Oper Res
(2005)
Cited by (14)
Three-rates of production inventory models for deteriorating items with constant, linear and quadratic demand - comparative study
2024, International Journal of Operational ResearchInventory with Positive Service Time: A Survey
2021, Queueing Theory 2: Advanced TrendsOptimal control of a multi-supplier and multi-buyer supply chain system with JIT delivery
2021, European Journal of Industrial EngineeringStochastic supply chain model with imperfect production and controllable defective rate
2020, International Journal of Systems Science: Operations and LogisticsA vendor–buyer integrated inventory system with variable lead time and uncertain market demand
2020, Operational Research