Demand Allocation in Systems with Multiple Inventory Locations and Multiple Demand Sources
TL;DR: It is shown that the optimal demand allocations are always discrete, with demand from each source always fulfilled entirely from a single inventory location, and this discreteness property extends to systems with other forms of supply processes.
Abstract: We consider the problem of allocating demand that originates from multiple sources among multiple inventory locations. Demand from each source arrives dynamically according to an independent Poisson process. The cost of fulfilling each order depends on both the source of the order and its fulfillment location. Inventory at all locations is replenished from a shared production facility with a finite production capacity and stochastic production times. Consequently, supply lead times are load dependent and affected by congestion at the production facility. Our objective is to determine an optimal demand allocation and optimal inventory levels at each location so that the sum of transportation, inventory, and backorder costs is minimized. We formulate the problem as a nonlinear optimization problem and characterize the structure of the optimal allocation policy. We show that the optimal demand allocations are always discrete, with demand from each source always fulfilled entirely from a single inventory location. We use this discreteness property to reformulate the problems as a mixed-integer linear program and provide an exact solution procedure. We show that this discreteness property extends to systems with other forms of supply processes. However, we also show that supply systems exist for which the property does not hold. Using numerical results, we examine the impact of different parameters and provide some managerial insights.
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TL;DR: In this article, the authors propose a method for solving the p-center problem on trees and demonstrate the duality of covering and constraining p-Center problems on trees.
Abstract: Ingredients of Locational Analysis (J. Krarup & P. Pruzan). The p-Median Problem and Generalizations (P. Mirchandani). The Uncapacitated Facility Location Problem (G. Cornuejols, et al.). Multiperiod Capacitated Location Models (S. Jacobsen). Decomposition Methods for Facility Location Problems (T. Magnanti & R. Wong). Covering Problems (A. Kolen & A. Tamir). p-Center Problems (G. Handler). Duality: Covering and Constraining p-Center Problems on Trees (B. Tansel, et al.). Locations with Spatial Interactions: The Quadratic Assignment Problem (R. Burkard). Locations with Spatial Interactions: Competitive Locations and Games (S. Hakimi). Equilibrium Analysis for Voting and Competitive Location Problems (P. Hansen, et al.). Location of Mobile Units in a Stochastic Environment (O. Berman, et al.). Index.
451 citations
01 Jan 2016
TL;DR: In this article, a multilocation newsboy problem with normal demand at each location and identical linear holding and penalty cost functions is considered, and an expression is derived for the resulting expected holding and penalties as a function of demand parameters for each location (means, variances, and correlation coefficients).
Abstract: This paper concerns a multilocation newsboy problem with normal demand at each location and identical linear holding and penalty cost functions at each location. Consolidation of demand from several facilities is considered, and an expression is derived for the resulting expected holding and penalty costs as a function of the demand parameters for each location (means, variances, and correlation coefficients). The expression is used to demonstrate that (i) the expected holding and penalty costs in a decentralized system exceed those in a centralized system; (ii) the magnitude of the saving depends on the correlation of demands; and (iii) if demands are identical and uncorrelated, the costs increase as the square root of the number of consolidated demands. (FACILITIES/EQUIPMENT PLANNING; INVENTORY/PRODUCTIONOPERATING CHARACTERISTICS; INVENTORY/PRODUCTION-STOCHASTIC MODELS)
53 citations
TL;DR: A mixed integer nonlinear programming model (MINLP) is formulated to minimize the total expected cost of the system and a Lagrangian heuristic is proposed to optimize the base-stock levels at both echelons.
Abstract: We study the problem of designing a two-echelon spare parts inventory system consisting of a central plant and a number of service centers each serving a set of customers with stochastic demand. Processing and storage capacities at both levels of facilities are limited. The manufacturing process is modeled as a queuing system at the plant. The goal is to optimize the base-stock levels at both echelons, the location of service centers, and the allocation of customers to centers simultaneously, subject to service constraints. A mixed integer nonlinear programming model (MINLP) is formulated to minimize the total expected cost of the system. The problem is NP-hard and a Lagrangian heuristic is proposed. We present computational results and discuss the trade-off between cost and service.
46 citations
TL;DR: In this paper, the authors derived necessary and sufficient conditions to relate the fill rate requirement of each customer to the resources needed in the system and provided a new approach to study the value of resource pooling in a system with differentiated service requirements.
Abstract: Resource pooling strategies have been widely used in industry to match supply with demand. However, effective implementation of these strategies can be challenging. Firms need to integrate the heterogeneous service level requirements of different customers into the pooling model and allocate the resources (inventory or capacity) appropriately in the most effective manner. The traditional analysis of inventory pooling, for instance, considers the performance metric in a centralized system and does not address the associated issue of inventory allocation. Using Blackwell’s Approachability Theorem, we derive a set of necessary and sufficient conditions to relate the fill rate requirement of each customer to the resources needed in the system. This provides a new approach to studying the value of resource pooling in a system with differentiated service requirements. Furthermore, we show that with “allocation flexibility,” the amount of safety stock needed in a system with independent and identically distribut...
30 citations
References
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Book•
03 Mar 1993
TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.
Abstract: COMPREHENSIVE COVERAGE OF NONLINEAR PROGRAMMING THEORY AND ALGORITHMS, THOROUGHLY REVISED AND EXPANDED"Nonlinear Programming: Theory and Algorithms"--now in an extensively updated Third Edition--addresses the problem of optimizing an objective function in the presence of equality and inequality constraints. Many realistic problems cannot be adequately represented as a linear program owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any constraints. The "Third Edition" begins with a general introduction to nonlinear programming with illustrative examples and guidelines for model construction.Concentration on the three major parts of nonlinear programming is provided: Convex analysis with discussion of topological properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programmingOptimality conditions and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditionsAlgorithms and their convergence, with a presentation of algorithms for solving both unconstrained and constrained nonlinear programming problemsImportant features of the "Third Edition" include: New topics such as second interior point methods, nonconvex optimization, nondifferentiable optimization, and moreUpdated discussion and new applications in each chapterDetailed numerical examples and graphical illustrationsEssential coverage of modeling and formulating nonlinear programsSimple numerical problemsAdvanced theoretical exercisesThe book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques. The logical and self-contained format uniquely covers nonlinear programming techniques with a great depth of information and an abundance of valuable examples and illustrations that showcase the most current advances in nonlinear problems.
6,259 citations
Book•
24 Jan 2000
TL;DR: In this article, one item with a constant demand rate and time-varying demands is described. But, the model is based on a single item with constant lead times.
Abstract: 1 General Introduction2 Systems and Models3 One Item with a Constant Demand Rate4 Time-Varying Demands5 Several Products and Locations6 Stochastic Demand: One Item with Constant Leadtimes7 Stochastic Leadtimes: The Structure of the Supply System8 Several Items with Stochastic Demands9 Time-Varying, Stochastic Demand: Policy Optimization Bibliography Appendix A: Optimization and Convexity Appendix B: Dynamical Systems Appendix C: Probability and Stochastic Processes Appendix D: Notational Conventions
1,709 citations
"Demand Allocation in Systems with M..." refers background or methods in this paper
...If we approximate the distribution of Qj by a normal distribution with matching mean E Qj and variance Var Qj (see Chapters 6 and 7 of Zipkin 2000 for an extensive discussion of the appropriateness of the normal approximation for this and for other inventory contexts), then expected total cost for…...
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...…of the following relationship between the z-transform of Qj , gQj , and the Laplace transform of the supply lead time Lj , fLj (see Chapter 7 of Zipkin 2000): gQj z = fLj ̂j 1− z (27) from which we can derive the mean and variance of Qj as E Qj =E Lj ̂j = ̂j/ j and Var Qj =Var Lj ̂2j + E Lj ̂j…...
[...]
...Note that relaxing the integrality of the basestock level is in line with standard treatments in the inventory literature (Zipkin 2000) and in the analysis of make-to-stock queues (Buzacott and Shantikumar 1993, Wein 1992, Zipkin 1995)....
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Book•
10 Feb 1989
TL;DR: An integrated treatment of applied stochastic processes and queueing theory, with an emphasis on time-averages and long-run behavior.
Abstract: An integrated treatment of applied stochastic processes and queueing theory, with an emphasis on time-averages and long-run behavior. Theory demonstrates practical effects, such as priorities, pooling of queues, and bottlenecks. Appropriate for Sr/Grad courses in queueing theory in Operations Research, Computer Science, Statistics, or IE departments.
1,689 citations
"Demand Allocation in Systems with M..." refers background in this paper
...The second difficulty can be addressed by using one of the many reasonably good approximations of expected queue size in a GI/G/1 queue; see, for example, Wolff (1989), Whitt (1983, 1993), and Buzacott and Shanthikumar (1993)....
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Book•
01 Jan 1993
TL;DR: In this article, the evolution of manufacturing system models: an example of a single stage "produce-to-order" system and a single-stage "buy-and-buy" system is presented.
Abstract: 1 Discrete Part Manufacturing Systems 2 Evolution of Manufacturing System Models: An Example 3 Single Stage 'Produce-to-Order' Systems 4 Single Stage 'Produce-to-Stock' Systems 5 Flow Lines 6 Transfer Lines 7 Dynamic Job Shops 8 Flexible Machining Systems 9 Flexible Assembly Systems 10 Multiple Cell Manufacturing Systems 11 Unresolved Issues: Directions for Future Research Appendix A: Standard Probability Distributions Appendix B: Some Notions of Stochastic Ordering Appendix C: Nonparametric Families of Distributions
1,565 citations
1,050 citations