An inventory sharing and allocation method for a multi-location service parts logistics network with time-based service levels

We consider a model to allocate stock levels at warehouses in a service parts logistics network. The network is a two-echelon distribution system with one central warehouse with infinite capacity and a number of local warehouses, each facing Poisson demands from geographically dispersed customers. Each local warehouse uses a potentially different base stock policy. The warehouses are collectively required to satisfy time-based service targets: Certain percentages of overall demand need to be satisfied from facilities within specified time windows. These service levels not only depend on the distance between customers and the warehouses, but also depend on the part availabilities at the warehouses. Moreover, the warehouses share their inventory as a way to increase achieved service levels, i.e., when a local warehouse is out of stock, demand is satisfied with an emergency shipment from another close-by warehouse. Observing that the problem of finding minimum-cost stock levels is an integer non-linear program, we develop an implicit enumeration-based method which adapts an existing inventory sharing model from the literature, prioritizes the warehouses for emergency shipments, and makes use of a lower bound. The results show that the proposed inventory sharing strategy results in considerable cost reduction when compared to the no-sharing case and the method is quite efficient for the considered test problems.     

Designing multi-echelon service parts networks with finite repair capacity

We propose an approach to model and solve the joint problem of facility location, inventory allocation and capacity investment in a two echelon, single-item, service parts supply chain with stochastic demand. The objective of the decision problem is to minimize the total expected costs associated with (1) opening repair facilities, (2) assigning each field service location to an opened facility, (3) determining capacity levels of the opened repair facilities, and (4) optimizing inventory allocation among the locations. Due to the size of the problem, computational efficiency is essential. The accuracy of the approximations and effectiveness of the approach are analyzed with two numerical studies. The approach provides optimal results in 90% of scenarios tested and was within 2% of optimal when it did not.We explore the impact of capacity utilization, inventory availability, and lead times on the performance of the approach. We show that including tactical considerations jointly with strategic network design resulted in additional cost savings from 3% to 12%. Our contribution is the development of a practical model and approach to support the decision making process of joint facility location and multi-echelon inventory optimization.     

An approximation algorithm for a facility location problem with stochastic demands and inventories

We propose a 2-approximation algorithm for a facility location problem with stochastic demands. At open facilities, inventory is kept such that arriving requests find a zero inventory with (at most) some pre-specified probability. Costs incurred are expected transportation costs, facility operating costs and inventory costs.     

A simulation optimization approach for a two-echelon inventory system with service level constraints

In this paper, we present a simulation optimization algorithm for solving the two-echelon constrained inventory problem. The goal is to determine the optimal setting of stocking levels to minimize the total inventory investment costs while satisfying the expected response time targets for each field depot. The proposed algorithm is more adaptive than ordinary optimization algorithms, and can be applied to any multi-item multi-echelon inventory system, where the cost structure and service level function resemble what we assume. Empirical studies are performed to compare the efficiency of the proposed algorithms with other existing simulation algorithms.     

Improved approximation for the capacitated inventory access point problem

The Inventory Access Point (IAP) is the single-item lot-sizing problem where a single customer faces demands in a discrete planning horizon, and the goal is to find a replenishment policy that minimizes the total inventory and ordering costs. While the uncapacitated version is polynomial, only a 3-approximation is known for the capacitated case. We improve this factor to 2.619 and, as a byproduct, we also improve the best factor for SIRPFL, which is a variant with multiple depots and customers.     

Coordinated control of distribution supply chains in the presence of fuzzy customer demand

This paper considers a single product inventory control in a Distribution Supply Chain (DSC). The DSC operates in the presence of uncertainty in customer demands. The demands are described by imprecise linguistic expressions that are modelled by discrete fuzzy sets. Inventories at each facility within the DSC are replenished by applying periodic review policies with optimal order up-to-quantities. Fuzzy customer demands imply fuzziness in inventory positions at the end of review intervals and in incurred relevant costs per unit time interval. The determination of the minimum of defuzzified total cost of the DSC is a complex problem which is solved by applying decomposition; the original problem is decomposed into a number of simpler independent optimisation subproblems, where each retailer and the warehouse determine their optimum periodic reviews and order up-to-quantities. An iterative coordination mechanism is proposed for changing the review periods and order up-to-quantities for each retailer and the warehouse in such a way that all parties within the DSC are satisfied with respect to total incurred costs per unit time interval. Coordination is performed by introducing fuzzy constraints on review periods and fuzzy tolerances on retailers and warehouse costs in local optimisation subproblems.     

On the stochastic uncapacitated dynamic single-item lotsizing problem with service level constraints

In this paper, we consider the uncapacitated single-item dynamic lotsizing problem with stochastic period demands and backordering. We present a model formulation that minimizes the setup and holding costs with respect to a constraint on the probability that the inventory at the end of any period does not become negative (α service level) and, alternatively, to a fill rate constraint (β service level). In contrast to earlier model formulations which consider the cycle α service level (α_c) and which approximate the on hand inventory by the net inventory, we include the exact on hand inventory into the model formulation. Therefore, the models are also applicable in situations with very low service levels.     

Sequential location-allocation problems on chains and trees with probabilistic link demands

In this paper, we consider multiperiod minisum location problems on networks in which demands can occur continuously on links according to a uniform probability density function. In addition, demands may change dynamically over time periods and at most one facility can be located per time period. Two types of networks are considered in conjunction with three behavioral strategies. The first type of network discussed is a chain graph. A myopic strategy and long-range strategy for locatingp-facilities are considered, as is a discounted present worth strategy for locating two facilities. Although these problems are generally nonconvex, effective methods are developed to readily identify all local and global minima. This analysis forms the basis for similar problems on tree graphs. In particular, we construct algorithms for the 3-facility myopic problem and the 2-facility long-range and discounted cost problems on a tree graph. Extensions and suggestions for further research on problems involving more general networks are provided.     

Pooling through lateral transshipments in service parts systems

We study the inventory management problem of a service center operating in a decentralized service parts network. The service centers collaborate through inventory and service pooling, and through sharing information on the inventory status. Upon demand arrival, a service center may request a part from the other center, in which case a payment is made. Under this competitive and collaborative environment, we first characterize the optimal operating policy of an individual service center. Through computational analysis we identify the conditions under which pooling is most beneficial to the service center, and make an assessment of different pooling strategies which are commonly adopted in practice and in the literature. Finally, we analyze the effect of interaction between the centers on the benefit of pooling.     

A single-period inventory placement problem for a supply system with the satisficing objective

Consider the inventory placement problem in an N-stage supply system facing a stochastic demand for a single planning period. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy demand. Stocking decisions are made before demand occurs. Unsatisfied demands are lost. The revenue, salvage value, ordering, transformation, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. The objective is to maximize the probability of achieving a given target profit level.     

Joint optimization of level of repair analysis and spare parts stocks

In the field of after sales service logistics for capital goods, generally, METRIC type methods are used to decide where to stock spare parts in a multi-echelon repair network such that a target availability of the capital goods is achieved. These methods generate a trade-off curve of spares investment costs versus backorders. Backorders of spare parts lead to unavailability of the capital goods. Inputs in the spare parts stocking problem are decisions on (1) which components to repair upon failure and which to discard, and (2) at which locations in the repair network to perform the repairs and discards. The level of repair analysis (LORA) can be used to make such decisions in conjunction with the decisions (3) at which locations to deploy resources, such as test equipment that are required to repair, discard, or move components. Since these decisions significantly impact the spare parts investment costs, we propose to solve the LORA and spare parts stocking problems jointly. We design an algorithm that finds efficient solutions. In order for the algorithm to be exact and because of its computational complexity, we restrict ourselves to two-echelon, single-indenture problems. In a computational experiment, we show that solving the joint problem is worthwhile, since we achieve a cost reduction of over 43% at maximum (5.1% on average) compared with using a sequential approach of first solving a LORA and then the spare parts stocking problem.     

Approximation algorithms for deterministic continuous-review inventory lot-sizing problems with time-varying demand

This work deals with the continuous time lot-sizing inventory problem when demand and costs are time-dependent. We adapt a cost balancing technique developed for the periodic-review version of our problem to the continuous-review framework. We prove that the solution obtained costs at most twice the cost of an optimal solution. We study the numerical complexity of the algorithm and generalize the policy to several important extensions while preserving its performance guarantee of two. Finally, we propose a modified version of our algorithm for the lot-sizing model with some restricted settings that improves the worst-case bound.     

Capacitated,balanced, sequential location-allocation problems on chains and trees

This paper considers finite horizon, multiperiod, sequential, minisum location-allocation problems on chain graphs and tree networks. The demand has both deterministic and probabilistic components, and increases dynamically from period to period. The problem is to locate one additionalcapacitated facility in each of thep specified periods, and to determine the service allocations of the facilities, in order to optimally satisfy the demand on the network. In this context, two types of objective criteria or location strategies are addressed. The first is a myopic strategy in which the present period cost is minimized sequentially for each period, and the second is a discounted present worth strategy. For the chain graph, we analyze ap-facility problem under both these criteria, while for the tree graph, we analyze a 3-facility myopic problem, and a 2-facility discounted present worth problem. All these problems are nonconvex, and we specify a finite set of candidate solutions which may be compared in order to determine a global optimal solution.     

Optimising a general repair kit problem with a service constraint

Field services are a particular type of after-sales service performed at the customer’s location where technicians repair malfunctioning machines. The inventory decisions about which spare part types to take to the repair site and in what quantities is called the repair kit problem. This problem is characterized by an order-based performance measure since a customer is only satisfied when all required spare parts are available to fix the machine. As a result, the service level in the decision making process is defined as a job fill rate. In this paper we derive a closed-form expression for the expected service level and total costs for the repair kit problem in a general setting, where multiple units of each part type can be used in a multi-period problem. Such an all-or-nothing strategy is a new characteristic to investigate, but commonly used in practice. Namely, items are only taken from the inventory when all items to perform the repair are available in the right quantity. We develop a new algorithm to determine the contents of the repair kit both for a service and cost model while incorporating this new expression for the job fill rate. We show that the algorithm finds solutions which differ on average 0.2% from optimal costs. We perform a case study to test the performance of the algorithm in practice. Our approach results in service level improvements of more than 30% against similar holding costs.     

Consolidating Facilities for the Connecticut Department of Transportation: a Case-Study

The Connecticut Department of Transportation (ConnDOT) was charged with investigating the possibility of reducing costs by operating with fewer than 13 equipment repair facilities. Using a mixed-integer programming model, we helped ConnDOT identify options for achieving savings without sacrificing quality of service. A key feature of the model is that it establishes repair capacities based on the past performance of benchmark facilities within the system. As we developed recommendation, we gave serious consideration to factors outside the context of the model and to feedback received from ConnDOT personnel. Ultimately, we proposed two options-one containing ten facilities and the other 11. The net present values of savings offered by these options are estimated at over $5.0 million and $3.1 million, respectively. While the 11-facility option is lower in savings,it has advantages in that it: (1) satisfies concerns highlighted by ConnDOT Maintenance; (2) is easier to implement than the ten-facility option; and (3) represents less of a current commitment, which is beneficial if future consideration of additional changes is likely. These advantages contributed to the 11-facility option being the one most widely preferred within ConnDOT.     

Modelling inventories in a network

If each inventory in a supply chain, comprising a network of production and transportation facilities, is optimized separately by some EOQ formula, the overall result may be far from optimal. If the objective of inventories is to reduce the variability of production and delivery rates, then this can be modelled as a network system with feedback links. The propagation of demands through the network is described by certain “propagation vectors”, and the optimization is with respect to certain feedback factors.     

Incorporating inventory and routing costs in strategic location models

We consider a supply chain design problem where the decision maker needs to decide the number and locations of the distribution centers (DCs). Customers face random demand, and each DC maintains a certain amount of safety stock in order to achieve a certain service level for the customers it serves. The objective is to minimize the total cost that includes location costs and inventory costs at the DCs, and distribution costs in the supply chain. We show that this problem can be formulated as a nonlinear integer programming model, for which we propose a Lagrangian relaxation based solution algorithm. By exploring the structure of the problem, we find a low-order polynomial algorithm for the nonlinear integer programming problem that must be solved in solving the Lagrangian relaxation sub-problems. We present computational results for several instances of the problem with sizes ranging from 40 to 320 customers. Our results show the benefits of having an integrated supply chain design framework that includes location, inventory, and routing decisions in the same optimization model.     

A Bayesian Approach to Estimating Decision Parameters in a Replacement Inventory System

This paper deals with the estimation of decision parameters in a special class of inventory systems in which the rate of stock depletion depends on the internal operations of the parent organization instead of market demands. Stocking decisions for this type of inventory systems are unique in that demands for the stocked items are internal demands, sporadic in some instances, infrequent in others, and almost never the direct results of conscious economic decisions. To cope with the peculiar situation, Bayesian point and interval estimation procedures are used. The estimated probability measures provide a basis on which stocking decisions pertinent to this type of inventory systems can be made or evaluated. Specific statistical postulates are made, and a numerical example is given for illustration. The possibility of applying the estimated results to one of the existing inventory models is briefly discussed at the end.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.