Lost Sales Costs Research Articles

Managing Perishable Inventory Systems with Multiple Demand Classes

In this paper, we study a multi-period stochastic perishable inventory system with multiple demand classes that require products of different ages. The firm orders the product with a positive leadtime and sells it to multiple demand classes, each only accepting products with remaining lifetime longer than a threshold. In each period, after demand realization, the firm decides how to allocate the on-hand inventory to different demand classes with different backorder or lost-sale cost. At the end of each period, the firm can dispose inventory of any age. We formulate this problem as a Markov decision process and characterize the optimal ordering, allocation, and disposal policies. When unfulfilled demand is backlogged, we show that the optimal order quantity is decreasing in the inventory levels and is more sensitive to the inventory level of fresher products, the optimal allocation policy is a sequential rationing policy, and the optimal disposal policy is characterized by a set of thresholds. For the lost-sale case, we show that the optimal allocation and disposal policies have the same structure but the optimal ordering policy may be different. Based on the structure of the optimal policy, we develop an efficient heuristic that is at most 4% away from the optimal cost in our numerical examples. Using numerical studies, we show that the ordering and allocation policies are close to optimal even if the firm cannot intentionally dispose products. Moreover, ignoring the difference between demand classes and using a simple allocation policy (e.g., FIFO) can significantly increase the total cost. We examine how the firm can improve the control of perishable items and show that the benefit of decreasing the leadtime is more significant than that of increasing the lifetime of the products or that of decreasing the acceptance threshold of the demand. The analysis is extended to systems with age dependent disposal cost and stochastic supply.

CAN-ORDER POLICY WITH ITEM GROUP IN JOINT REPLENISHMENT PROBLEM

In this paper, multi-item inventory management is considered. The associated problem is called the joint replenishment problem (JRP). In this model, warehouse sells multiple items for customer and replenishes items from a supplier. For the replenishing, ordering cost is charged independent on the amount of orders. For this situation, can-order policy is often used. Under a can-order policy, items are ordered when their inventory level drops to or below their re-order level, and other item with an inventory level at or below its can-order level can be included in this order. In this situation, by considering can-order level and grouping the items can reduce ordering cost. In this research, setting of a can-order level and grouping are considered. The main objective in this model is minimizing the number of items lost sales cost, holding cost and ordering cost. In a numerical experiment, inventory movement are simulated.

An EPQ model for deteriorating items with variable demand rate and allowable shortages

The fundamental assumption of an economic order quantity (EOQ) model is that 100% of items in an ordered lot are perfect. This assumption is not always pertinent for production processes because of process deterioration or other factors. This is an economic production quantity model for deteriorating items. To fulfil the market demands and expectations, the production rate is taken as a function of demand rate. The demand and deterioration of the products are time dependent function. Shortages are allowed and partially backlogged. The backlogging phenomenon in the literature is often modelled using backordering and lost sale costs. The backlogging option gets used only when it is economic to do so. The inventory policy proposed here considers the optimal production run time, production quantity and shortage period such that the total average cost can be minimised. Numerical examples are provided to illustrate and sensitivity analyses of optimal solutions are given for the proposed inventory model.

An integrated inventory model with variable holding cost under two levels of trade-credit policy

This paper presents an integrated vendor-buyer model for deteriorating items. We assume that the deterioration follows a constant rate with respect to time. The vendor allows a certain credit period to buyer in order to promote the market competition. Keeping in mind the competition of modern age, the stock-dependent demand rate is included in the formulated model which is a new policy to attract more customers. Shortages are allowed for the model to give the model more realistic sense. Partial backordering is offered for the interested customers, and there is a lost-sale cost during the shortage interval. The traditional parameter of holding cost is considered here as time-dependent. Henceforth, an easy solution procedure to find the optimal order quantity is presented so that the total relevant cost per unit time will be minimized. The mathematical formation is explored by numerical examples to validate the proposed model. A sensitivity analysis of the optimal solution for important parameters is also carried out to modify the result of the model.

Optimal control of a continuous-time W-configuration assemble-to-order system

We analyze a W-configuration assemble-to-order system with random lead times, random arrival of demand, and lost sales, in continuous time. Specifically, we assume exponentially distributed production and demand inter-arrival times. We formulate the problem as an infinite-horizon Markov decision process. We deviate from the standard approach by first characterizing a region (the recurrent region) of the state space where all properties of the cost function hold. We then characterize the optimal policy within this region. In particular, we show that within the recurrent region components are always produced. We also characterize the optimal component allocation policy which specifies whether an arriving product demand should be fulfilled. Our analysis reveals that the optimal allocation policy is counter-intuitive. For instance, even when one product dominates the other, in terms of lost sale cost and lost sale cost rate (i.e., demand rate times the lost sale cost), its demand may not have absolute priority over the other product's demand. We also show that the structure of the optimal policy remains the same for systems with batch production, Erlang distributed production times, and non-unitary product demand. Finally, we propose efficient heuristics that can be either used as an approximation to the optimal policy or can be used as a starting policy for the common algorithms that are used to obtain the optimal policy in an effort to reduce their computational time.

A quantitative and simulation model for managing sudden supply delay with fuzzy demand and safety stock

In this paper, a recovery model is developed for managing sudden supply delays that affect retailers’ economic order quantity model. For this, a mathematical model is developed that considers fuzzy demand and safety stock, and generates a recovery plan for a finite future period immediately after a sudden supply delay. An efficient heuristic solution is developed that generates the recovery plan after a sudden supply delay. An experiment with scenario-based analysis is conducted to test our heuristic and to analyse the results. To assess the quality and consistency of solutions, the performance of the proposed heuristic is compared with the performance of the generalised reduced gradient method, which is widely applied in constrained mathematical programming. A simulation model is also designed to bring the recovery model closer to real-world processes. Several numerical examples are presented and a sensitivity analysis is performed to demonstrate the effects of various parameters on the performance of the heuristic method. The results show that safety stock plays an important role in recovery from sudden supply delays, and there is a trade-off between backorder and lost sales costs in the recovery plan. With the help of the proposed model, supply chain decision-makers can make accurate and prompt decision regarding recovery plans in case of sudden supply delay.

A non-myopic dynamic inventory routing and pricing problem

A new framework for the design of a dynamic non-myopic inventory and delivery network between suppliers and retailers under the assumption of elastic demand—one that simultaneously incorporates inventory, routing, and pricing—is proposed. The developed queuing approximation method is based on optimal tolling of queues. We propose a dynamic approach for a supplier who has to deliver products to a number of retailers while maximizing social welfare through dynamic pricing that accounts for customer waiting times, inventory holding, lost-sales costs, and delivery costs. The proposed non-myopic model increases the social welfare by up to 17% compared to the marginal pricing case.

Affinely Adjustable Robust Model for Multiperiod Production Planning Under Uncertainty

Demand forecasting is an important factor in production planning, but future demand is not easy to forecast in practice. We consider a multiperiod, multiproduct production planning problem under demand uncertainty with constrains for raw materials, manufacturing capacity, and inventory. Under the assumption that probability distribution of demand is not available, two types of robust optimization models are proposed. First, a robust counterpart is developed to determine the here-and-now decision. Next, an affinely adjustable robust counterpart is developed to determine the wait-and-see decisions by approximating a robust solution with a linear decision rule. The robust models find an optimal solution that is always feasible and less sensitive against all realized demand within a given uncertainty set, in order to minimize production, procurement, inventory, and lost sales costs even in the worst case. Numerical studies demonstrated that, without knowing probability distribution of future demand, the affinely adjustable robust counterpart approach could outperform the robust counterpart and deterministic model in terms of the average cost, the standard deviation of the realized cost, and the worst-case scenario cost. The proposed method is much better than the others, especially when penalty cost due to lost sales is high and unknown demand is left skewed.

Relation between lead time dependent demand and capacity flexibility in a two-stage supply chain with lost sales

Supply chain literature shows that speed of delivery, or equivalently provided delivery lead time, is a crucial performance factor and influences the expected customer demand. However, short delivery lead times imply stochastic demand peaks, for which production economics literature indicates that flexible capacity can be applied to mitigate their negative effects. In this paper a two stage supply chain with flexible capacity at the supplier and lead time dependent demand at the customer is investigated. Specifically, the random customer demand depends on the delivery lead time provided by the supplier and demand which cannot be fulfilled is lost. An analytical model is developed which includes predefined steady capacity and a demand dependent decision for flexible capacity. The trade-off between lost sales due to the provided delivery lead time, lost sales due to a lack of capacity, inventory cost and the cost for flexible capacity is discussed. Numerical results indicate that high capacity flexibility is needed if lost sales cost, demand variance, or customer lead time sensitivity is high. Furthermore, the positive effect of information sharing within the supply chain is evaluated.

Integrated location-inventory modelling under forward and reverse product flows in the used merchandise retail sector: A multi-echelon formulation

This study presents a joint three-echelon location inventory model for a donation-demand driven industry in which the main warehouse (MW), distribution centers (DC), retail stores (RS) and donation-only centers (ADCs) exist. This unique inventory-location problem involves demand and supply uncertainties, coverage radius limitations, service level requirements, and multiple products consideration. Each retailer has two classes of products flowing from the assigned DC due to demands minus donations occurring in that retailer. The proposed model simultaneously determines the number of DCs to open, DC locations, and assignments of retailers to the open DCs for particular product types. The objective is to minimize the total annual cost including: facility location costs, transportation costs, inventory costs, and the lost sale costs. Due to the complexity of the problem, the proposed model structure allows for relaxing complicating constraints through recourse to Lagrangian relaxation. The use of robust branch-cut and price heuristics solves the mixed integer nonlinear problem to obtain a lower bound and a distance-based heuristic to get an upper bound. We formulate essential features of this novel problem, solve several numerical example problems and evaluate solution performance. We believe this is a novel problem environment, and that this initial study extends integrated location-inventory modeling to a new context.

Probabilistic multi-item inventory model with varying mixture shortage cost under restrictions.

This paper proposed a new general probabilistic multi-item, single-source inventory model with varying mixture shortage cost under two restrictions. One of them is on the expected varying backorder cost and the other is on the expected varying lost sales cost. This model is formulated to analyze how the firm can deduce the optimal order quantity and the optimal reorder point for each item to reach the main goal of minimizing the expected total cost. The demand is a random variable and the lead time is a constant. The demand during the lead time is a random variable that follows any continuous distribution, for example; the normal distribution, the exponential distribution and the Chi square distribution. An application with real data is analyzed and the goal of minimization the expected total cost is achieved. Two special cases are deduced.

Planning for ramp-ups and new product introductions in the automotive industry: Extending sales and operations planning

Sales and operations planning for ramp-ups and new product introductions in the automotive industry differs significantly from steady-state situations. We identify the ramp-up challenges in collaboration with a large European OEM. To support sales and operations planning for ramp-ups and new product introductions, we develop a mixed-integer linear programming model. In our numerical study, we show that it is necessary to consider complexity in sales and operations planning for ramp-up and new product introduction. Besides lost sales costs, complexity and demand volume are factors that need to be considered in the trade-off for the optimal sequence and timing of the start of production of variants. Furthermore, our analysis shows that rework quantities must be considered during ramp-up planning and that an extreme emphasis on fairness among markets results in less efficient operations and negatively affects sales.

AN INVENTORY RECOVERY MODEL FOR AN ECONOMIC LOT SIZING PROBLEM WITH DISRUPTION

Supply chains face risks from various unexpected events that make disruptions almost inevitable. This paper presents a disruption recovery model for a single stage production and inventory system, where finished product supply is randomly disrupted for periods of random duration. A production facility that manufactures a single product following the Economic Production Quantity policy is considered. The model is solved using a search algorithm combined with a penalty function method to find the best recovery plan. It is shown that the optimal recovery schedule is dependent on the extent of the disruption, as well as the back order cost and lost sales cost parameters. The proposed model is seen to be a very useful tool for manufacturers to make quick decisions on the optimal recovery plan after the occurrence of a disruption.

Lot‐sizing model with advance payment pricing and disruption in supply under planned partial backordering

AbstractInventory management is a group of policies used to enhance company efficiencies in using the inventory and decrease its related costs. Inventory management policies operate efficiently only in the normal market environment. When the manager faces disruption, the common decisions do not lead to optimality. In this paper, a lot‐sizing model for when the retailer may face disruption is developed. A portion of purchasing cost should be spent before delivery by several predetermined equal‐sized advanced payments; shortage is permitted and will be partially back‐ordered. The total cost function includes purchasing, holding, capital, backordering, and lost sale costs. The optimal decisions are obtained after proving the convexity of the expected total cost function. Finally, a solution algorithm is presented and several numerical examples and analysis are provided to show that the developed model and derived results can be applied to real‐world problems.

Planning for shortages? Net Present Value analysis for a deteriorating item with partial backlogging

This paper develops inventory models to help answer strategic questions concerning whether planning for shortages offers financial benefits. A production-inventory system producing a deteriorating product in batches at a finite production rate with partial backordering is considered. Customers pay a deposit when placing a backorder. Backordered items receive a discount on the sales price. As lost sales may lead to customers not returning, the demand rate may depend on the fraction of lost sales. We develop a cash-flow based profit maximising Net Present Value (NPV) model without the inventory cost parameters commonly used in this context: unit holding cost, unit backorder cost, unit deterioration cost, and unit lost sales cost. The model finds the optimal inventory policy just like NPV models that discount the traditional parameters but has the advantage of not needing to estimate the value of the traditional parameters. It is shown that in models based on discounting the traditional parameters, the parameters are not exogenously determinable but are non-trivial functions of non-financial endogenous system parameters such as the production rate, annual demand rate, and backorder rate. Extensive numerical experiments illustrate how cash-flow NPV models provide insights into the value of planning for shortages and strategic choices about the design of the production-inventory system. It also provides insight into the classical problem of how to interpret unit backorder cost and unit lost sales cost. The study indicates that these insights cannot be reliably obtained from NPV models based on discounting unit backorder costs and unit lost sales costs.

Supply Chain Management with Online Customer Selection

We consider new online variants of supply chain management models, where in addition to production decisions, one also has to actively decide on which customers to serve. Specifically, customers arrive sequentially during a selection phase, and one has to decide whether to accept or reject each customer upon arrival. If a customer is rejected, then a lost-sales cost is incurred. Once the selection decisions are all made, one has to satisfy all the accepted customers with minimum possible production cost. The goal is to minimize the total cost of lost sales and production. A key feature of the model is that customers arrive in an online manner, and the decision maker does not require any information about future arrivals. We provide two novel algorithms for online customer selection problems, which are based on repeatedly solving offline subproblems that ignore previously made decisions. For many important settings, our algorithms achieve small constant competitive ratio guarantees. That is, for any sequence of arriving customers, the cost incurred by the online algorithm is within a fixed constant factor of the cost incurred by the respective optimal solution that has full knowledge upfront on the sequence of arriving customers. Finally, we provide a computational study on the performance of these algorithms when applied to the economic lot sizing and joint replenishment problems with online customer selection.

Joint optimization of condition-based maintenance and production lot-sizing

Due to the development of sensor technologies nowadays, condition-based maintenance (CBM) programs can be established and optimized based on the data collected through condition monitoring. The CBM activities can significantly increase the uptime of a machine. However, they should be conducted in a coordinated way with the production plan to reduce the interruptions. On the other hand, the production lot size should also be optimized by taking the CBM activities into account. Relatively fewer works have been done to investigate the impact of the CBM policy on production lot-sizing and to propose joint optimization models of both the economic manufacturing quantity (EMQ) and CBM policy. In this paper, we evaluate the average long-run cost rate of a degrading manufacturing system using renewal theory. The optimal EMQ and CBM policy can be obtained by minimizing the average long-run cost rate that includes setup cost, inventory holding cost, lost sales cost, predictive maintenance cost and corrective maintenance cost. Unlike previous works on this topic, we allow the use of continuous time and continuous state degradation processes, which broadens the application area of this model. Numerical examples are provided to illustrate the utilization of our model.

Optimal production control of a manufacturing-remanufacturing system with delivery activity

This work considers a manufacturing-remanufacturing system composed of two parallel machines, a manufacturing inventory, a purchase warehouse, a recovery inventory and customers who demand a constant quantity of product. Stochastic fluid model is adopted to describe the system and to take into account machine failure, returned products, remanufacturing products and delivery activity. The objective of this paper is to establish an optimal production control which allows minimizing the sum of inventory, transportation and lost sales costs. Perturbation analysis is applied to the stochastic fluid model to determine the optimal manufacturing inventory level that describes the optimal production control. The trajectories of the manufacturing inventory level are studied and the perturbation analysis estimates are evaluated. The unbiasedness of these estimates is proved and then they are implemented in an optimization algorithm which determines the optimal manufacturing inventory in the presence of returned products and delivery time.

A heterogeneous reliable location model with risk pooling under supply disruptions

This paper investigates a facility location model that considers the disruptions of facilities and the cost savings from the inventory risk-pooling effect and economies of scale. Facilities may have heterogeneous disruption probabilities. When a facility fails, its customers may be reassigned to other surviving ones to hedge against lost-sales costs. We first develop both an exact and an approximate expression for the nonlinear inventory cost, and then formulate the problem as a nonlinear integer programming model. The objective is to minimize the expected total cost across all possible facility failure scenarios. To solve this problem, we design two methods, an exact approach using special ordered sets of type two (SOS2) and a heuristic based on Lagrangian relaxation. We test the model and algorithms on data sets with up to 150 nodes. Computational results show that the proposed algorithms can solve the problem efficiently in reasonable time. Managerial insights on the optimal facility deployment, customer assignments and inventory control strategies are also drawn.

Two inventory systems with trapezoidal-type demand rate and time-dependent deterioration and backlogging

The demand rate for seasonable/fashionable products, in general, fluctuates through their life cycle, initially increases during the growth phase, then remains practically constant in the maturity phase, and finally decreases in the decline phase. Consequently, their demand rate can be reasonably depicted by a trapezoidal-type pattern, which includes constant, increasing, decreasing and ramp-type demand patterns as special cases. Recently, inventory models for deteriorating items with trapezoidal-type demand have been developed to minimize the total cost over the planning horizon (see Cheng & Wang, 2009; Skouri & Konstantaras, 2009; Cheng et al., 2011; Lin et al., 2014). However, in these models the purchasing cost is excluded from the total cost. As a result, this implies an underestimation of the lost sales cost. In the present paper, previous studies regarding inventory models with trapezoidal-type demand rate are extended and completed by (1) discussing two inventory systems starting with and without shortages, (2) including the purchasing cost into the total cost, (3) extending the deterioration rate to any time-varying rate, (4) taking into consideration the time value of money (5) maximizing the net present value of total profit. Therefore, many previous inventory models become special cases of the present model.