Abstract
A reduction in the inventory replenishment lead-time allows reducing safety stock requirements and improving customer service. However, it might be accompanied by increased procurement costs because of premium charges imposed by suppliers, or higher transportation costs. This paper studies a single-stage variable lead-time inventory system with lead-time dependent procurement cost. Selection of the lead-time value represents finding the trade-off between benefits of lead-time reduction and increase in the procurement cost. A model for joint optimization of inventory and procurement costs is developed. Numerical studies are conducted to identify conditions under which lead-time reduction is favorable compared to procuring at the lowest cost.
Introduction
Demand uncertainty coupled with inventory replenishment lead-time creates the inventory carrying risk. Lead-time reduction and application of risk reducing inventory management policies have been shown to improve efficiency in managing inventory. Tersine and Hummingbird [1], Jayaram et al. [2] and Disney and Towill [3] have demonstrated the positive impact of physical lead-time reduction. The lead-time reduction allows using more accurate demand information in making inventory replenishment decisions, which in turn reduces safety stock requirements and improves the customer service level. However, the lead-time reduction may be associated with some additional risks and costs [4]. For instance, a shorter lead-time may lead to higher purchasing costs and unavailability of supplies due to the limited supplier capacity. Burnetas and Gilbert [5] provide examples from travel industry, where late orders are charged a premium price. They also mention premium transportation charges for short lead-time orders as a source of increasing purchasing costs in other industries. Cachon [6] has discussed examples from the retail industry. Retailers placing late orders are not eligible for discounts and face the possibility of insufficient supplies because suppliers give priority to early orders. Das and Abdel-Malek [7] report that suppliers accept short lead-time orders at a higher unit price because of their disruptive impact on suppliers’ operations. These observations lead to the problem of finding the trade-off between lead-time reduction and additional procurement costs. The trade-off between purchasing cost and delivery lead-time from the manufacturing point is discussed by Elhafsi [8] and Slotnick and Sobel [9] among others. These authors show that manufacturers are interested in providing flexible pricing in response to more attractive delivery time conditions.
Ouyang and Wu [10] and Lan et al. [11] consider a continuous review re-order point policy with variable lead-time. Costs associated with lead-time reduction are represented by the lead-time crashing cost, which is independent of the order size. They consider the random demand. The above-mentioned papers focus on computation of the optimal policies. Hariga and Ben-Daya [12] propose similar variable lead-time periodic review and base stock policies. Similarly, lead-time crashing has been investigated in two stage systems involving buyer and supplier [13]. The integrated inventory model with controllable lead-time is used in demonstrating that the buyer–suppliers system as a whole benefits from the lead-time reduction while gains for individual partners are not distributed uniformly.
Burnetas and Gilbert [5] have analyzed the trade-off between higher procurement costs against the benefit of making ordering decisions using better demand related information. The authors develop an optimal ordering policy for a short life-cycle product. The procurement cost increases along the product life cycle, while a parameter of the demand process (modeled as a Bernoulli process) is continuously updated as more accurate information becomes available. The authors demonstrate that ordering pattern depends upon characteristics of the procurement cost function.
There have been numerous investigations on early commitment policies, which are closely related to variable lead-time inventory policies in the supply chain environment (e.g., [6], [14]). Under these policies, a supplier offers a discount, if an internal supply chain customer places orders in advance. The customer increases its inventory carrying risk by committing early (i.e. using the longer lead-time) or loses an opportunity to procure at the lower cost. The primary objective of early commitment policies is finding a discount level optimizing the supply chain performance.
Early commitment policies under evolving forecasting accuracy are investigated by Ferguson [15]. He analyzes negotiation of an exchange price between a supplier and a manufacturer depending upon the supply chain power structure and the frame of commitment. Information about the external demand has two levels of accuracy (not necessarily no information and completely accurate information). This level of accuracy depends upon a choice between using either early commitment or delayed commitment. In the case of balance of power in the supply chain and, if accuracy of demand information improves quickly, the delayed commitment is preferable. However, if improvement of forecasting accuracy is small, the early commitment is preferable. The author also provides an excellent summary of literature related to early commitment. His main observations are that existing models usually consider short planning horizons, two levels of information accuracy or two procurement cost levels.
Buyer–supplier interactions are also analyzed using dual sourcing models (for instance, [16], [17], [18]), where the buyer allocates orders between suppliers with different characteristics including lead-time. The dual sourcing models are directed towards minimizing risk of suppliers’ non-performance while this paper focuses on setting the supply lead-time in the long-term collaboration framework.
In this paper, a single stage, long horizon, reorder point inventory system is studied; wherein the procurement cost increases with decreasing lead-time. An instance of such inventory problems is routine procurement of a product for which there are multiple modes of transportation at different cost levels. The objective of this paper is to find the lead-time value representing the trade-off between benefits of lead-time reduction and increase in the procurement cost. The lead-time reduction is expected to reduce the inventory cost because of more accurate demand information and lower safety stock requirements. Yet, it increases the procurement cost because the supplier sets a higher price for short lead-time orders, limited supplies force to seek more costly alternative products or a more expensive shipping mode is used to ensure the shorter lead-time. In order to describe this trade-off, a model for joint optimization of inventory and procurement costs with respect to the lead-time is developed. It is based on the traditional continuous review re-order point model. Demand is modeled as an autoregressive process. This model differs from the traditional model by inclusion of the lead-time dependent procurement cost. A procurement cost function is used to describe the dependence of the procurement cost upon the lead-time. It is assumed that the lead-time is deterministic and the procurement cost for different values of the lead-time is known in advance. Burnetas and Gilbert [5] have also used the assumption of the known procurement cost.
The model developed is compared with the traditional constant lead-time model by means of ANOVA analysis. The analysis is aimed at identifying values of parameters of the demand process and the inventory system under which the lead-time reduction is favorable compared to procuring at the lowest cost. Parameters considered for the inventory system are added value, backlogging penalty cost, fixed ordering cost and a procurement cost function according to the lead-time. Parameters considered for the demand process are strength of autocorrelation and the signal-to-noise ratio.
The model developed in this paper differs from models using the lead-time crashing approach by explicitly specifying the source of the lead-time reduction cost (i.e., increasing procurement cost), which allows for analyzing the impact between lead-time reduction and buyer–supplier interactions. This paper differs from other published papers on early commitment policies by concurrently considering long planning horizon, multiple levels of procurement cost and continuously evolving demand information. The distinguishing feature of this paper from other papers dealing with the impact of lead-time reduction on buyer–supplier interactions is that it addresses the problem solely from the buyer’s perspective and no information about the supply side except the procurement cost function is required. Serially correlated demand is used instead of traditionally considered independent random demand. While the paper focuses on first order autoregressive processes, results can be generalized to any autoregressive demand process. The model developed enables buyers to balance their procurement preferences regarding cost and delivery lead-time with inventory management decisions in the case of routine replenishments over a relatively long time period. Another main contribution of the paper to the existing body of literature is analysis of factors influencing the trade-off between procuring at a lower cost and using more accurate demand information. This analysis highlights situations when buyers are highly motivated to negotiate for lead-time reduction.
The rest of the paper is organized as follows. Section 2 describes the model for simultaneous optimization of inventory cost and procurement cost. An example and numerical evaluation of the model are presented in Section 3. Section 4 offers general conclusion on the analyzed problem.
Experimental design
Numerical studies are conducted to test the accuracy of approximations used, to evaluate the proposed model and to identify parameters of the inventory system and properties of the demand process having the most profound impact on the trade-off between the inventory cost and the procurement cost. The demand process is described using expression (1). Factors characterizing the inventory system included in the experimental design are fixed ordering cost, added value and backlogging penalty.
Conclusion
In this paper, the traditional continuous review re-order point inventory management policy has been modified by considering the variable lead-time and including the lead-time dependent procurement cost. Optimization of the total inventory and procurement cost yields values of the lead-time, the order quantity and the safety factor characterizing the trade-off between the inventory cost and the procurement cost.
The trade-off depends upon the particular structure of the procurement cost
Acknowledgment
Valuable comments and suggestions made by referees and Dr. math. Vineta Minkevicha is gratefully acknowledged. This research has been partially supported by the Henry W. Patton, Center for Engineering Education and Practice, University of Michigan-Dearborn, and by the European Social Fund within the Latvian National Program “Support for the carrying out doctoral study program and post-doctoral-research”.
References
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An empirical study of time-based competition in the North American automotive supplier industry
