Dmitry Krass

Facility Reliability Issues in Network p-Median Problems: Strategic Centralization and Co-Location Effects

In this paper we analyze a facility location model where facilities may be subject to disruptions, causing customers to seek service from the operating facilities. We generalize the classical p-Median problem on a network to explicitly include the failure probabilities, and analyze structural and algorithmic aspects of the resulting model. The optimal location patterns are seen to be strongly dependent on the probability of facility failure - with facilities becoming more centralized, or even co-located, as the failure probability grows. Several exact and heuristic solution approaches are developed. Extensive numerical computations are performed

The generalized maximal covering location problem

We consider a generalization of the maximal cover location problem which allows for partial coverage of customers, with the degree of coverage being a non-increasing step function of the distance to the nearest facility. Potential application areas for this generalized model to locating retail facilities are discussed.We show that, in general, our problem is equivalent to the uncapacitated facility location problem. We develop several integer programming formulations that capitalize on the special structure of our problem. Extensive computational analysis of the solvability of our model under a variety of conditions is presented.

Competitive facility location and design problem

Abstract We develop a spatial interaction model that seeks to simultaneously optimize location and design decisions for a set of new facilities. The facilities compete for customer demand with pre-existing competitive facilities and with each other. The customer demand is assumed to be elastic, expanding as the utility of the service offered by the facilities increases. Increases in the utility can be achieved by increasing the number of facilities, design improvements, or locating facilities closer to the customer. We show that our model is able to capture some of the principal trade-offs involved in facility location and design decisions, including demand cannibalization, market expansion, and design/location trade-offs. Managerial insights are obtained through sensitivity analysis of the model and through several illustrative examples. An efficient near-optimal solution approach, with adjustable error bound, is developed for the special case where only a finite number of design alternatives are available. Several heuristic approaches capable of handling large instances are also presented.

The gradual covering decay location problem on a network

Abstract In covering problems it is assumed that there is a critical distance within which the demand point is fully covered, while beyond this distance it is not covered at all. In this paper we define two distances. Within the lower distance a demand point is fully covered and beyond the larger distance it is not covered at all. For a distance between these two values we assume a gradual coverage decreasing from full coverage at the lower distance to no coverage at the larger distance.

FLOW INTERCEPTING SPATIAL INTERACTION MODEL: A NEW APPROACH TO OPTIMAL LOCATION OF COMPETITIVE FACILITIES.

Abstract In this paper we propose a flexible new model for the location of competitive facilities that may derive their demand for service from both special-purpose purchase trips by their customers and from “intercepting” customers passing by a facility while en route to another destination on the network. Our model combines the features of the spatial interaction and flow interception models. An efficient heuristic procedure is developed, with worst case analysis provided. We also develop a tight upper bound and a branch-and-bound scheme for our model. Results of a set of computational experiments are presented.

Dynamic lot sizing with returning items and disposals

We analyze a version of the Dynamic Lot Size (DLS) model where demands can be positive and negative and disposals of excess inventory are allowed. Such problems arise naturally in several applications areas, including retailing where previously sold items are returned to the point of sale and re-enter the inventory stream (such returns can be viewed as negative demands), and in managing kits of spare parts for scheduled maintenance of aircraft (where excess spares are returned to the depot), among other applications. Both the procurement of new items and the disposal of excess inventory decisions are considered within the framework of deterministic time-varying demands, concave holding, procurement and disposal costs and a finite time horizon (disposal of excess inventory at a profit is also allowed). By analyzing the structure of optimal policies, several useful properties are derived, leading to an efficient dynamic programming algorithm. The new model is shown to be a proper generalization of the classical Dynamic Lot Sizing Model, and the computational complexity of our algorithm is compared with that of the standard algorithms for the DLS model. Both the theoretical worst-case complexity analysis and a set of computational experiments are undertaken. The proposed methodology appears to be quite adequate for dealing with realistic-sized problems.

Locating service facilities to reduce lost demand

We analyze the problem of locating a set of service facilities on a network when the demand for service is stochastic and congestion may arise at the facilities. We consider two potential sources of lost demand: (i) demand lost due to insufficient coverage; and (ii) demand lost due to congestion. Demand loss due to insufficient coverage arises when a facility is located too far away from customer locations. The amount of demand lost is modeled as an increasing function of the travel distance. The second source of lost demand arises when the queue at a facility becomes too long. It is modeled as the proportion of balking customers in a Markovian queue with a fixed buffer length. The objective is to find the minimum number of facilities, and their locations, so that the amount of demand lost from either source does not exceed certain pre-set levels. After formulating the model, we derive and investigate several different integer programming formulations, focusing in particular on alternative representations of closest assignment constraints. We also investigate a wide variety of heuristic approaches, ranging from simple greedy-type heuristics, to heuristics based on time-limited branch and bound, tabu search, and random adaptive search heuristics. The results of an extensive set of computational experiments are presented and discussed.

Locating Multiple Competitive Facilities: Spatial Interaction Models with Variable Expenditures

We develop a new framework for location of competitive facilities by introducing non-constant expenditure functions into spatial interaction location models. This framework allows us to capture two key effects – market expansion and cannibalization – within the same model.We develop algorithmic approaches for finding optimal or near-optimal solutions for several models that arise from choosing a specific form of the expenditure functions.

Competitive facility location model with concave demand

Abstract We consider a spatial interaction model for locating a set of new facilities that compete for customer demand with each other, as well as with some pre-existing facilities to capture the “market expansion” and the “market cannibalization” effects. Customer demand is assumed to be a concave non-decreasing function of the total utility derived by each customer from the service offered by the facilities. The problem is formulated as a non-linear Knapsack problem, for which we develop a novel solution approach based on constructing an efficient piecewise linear approximation scheme for the objective function. This allows us to develop exact and α-optimal solution approaches capable of dealing with relatively large-scale instances of the model. We also develop a fast Heuristic Algorithm for which a tight worst-case error bound is established.

Inventory models with minimal service level constraints

Abstract This paper investigates inventory models in which the stockout cost is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met in every period. The minimal service level approach has the virtue of simplifying the computation of an optimal ordering policy, because the optimal reorder level is solely determined by the minimal SLC and demand distributions. It is found that above a certain “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to level policy, which is independent on the cost parameters. This shows the minimal SLC models to be qualitatively different from their shortage cost counterparts. We also demonstrate that the “imputed shortage cost” transforming a minimal SLC model to a shortage cost model does not generally exist. The minimal SLC approach is extended to models with negligible set-up costs. The optimality of myopic base-stock policies is established under mild conditions.