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Showing papers in "Operations Research in 2004"


Journal ArticleDOI
TL;DR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

3,364 citations


Journal ArticleDOI
TL;DR: This work shows that when the demand model is additive, the profit-to-go functions arek-concave and hence an ( s, S, p) policy is optimal and introduces a new concept, the symmetrick-con cave functions, and applies it to provide a characterization of the optimal policy.
Abstract: We analyze a finite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite horizon. We show that when the demand model is additive, the profit-to-go functions arek-concave and hence an ( s, S, p) policy is optimal. In such a policy, the period inventory is managed based on the classical ( s, S) policy and price is determined based on the inventory position at the beginning of each period. For more general demand functions, i.e., multiplicative plus additive functions, we demonstrate that the profit-to-go function is not necessarilyk-concave and an ( s, S, p) policy is not necessarily optimal. We introduce a new concept, the symmetrick-concave functions, and apply it to provide a characterization of the optimal policy.

468 citations


Journal ArticleDOI
TL;DR: This paper considers a dynamic VRPTW with stochastic customers, where the goal is to maximize the number of serviced customers and presents a multiple scenario approach (MSA) that continuously generates routing plans for scenarios including known and future requests.
Abstract: The multiple vehicle routing problem with time windows (VRPTW) is a hard and extensively studied combinatorial optimization problem. This paper considers a dynamic VRPTW with stochastic customers, where the goal is to maximize the number of serviced customers. It presents a multiple scenario approach (MSA) that continuously generates routing plans for scenarios including known and future requests. Decisions during execution use a distinguished plan chosen, at each decision, by a consensus function. The approach was evaluated on vehicle routing problems adapted from the Solomon benchmarks with a degree of dynamism varying between 30% and 80%. They indicate that MSA exhibits dramatic improvements over approaches not exploiting stochastic information, that the use of consensus function improves the quality of the solutions significantly, and that the benefits of MSA increase with the (effective) degree of dynamism.

420 citations


Journal ArticleDOI
TL;DR: This paper considers the scheduling problems arising when two agents, each with a set of nonpreemptive jobs, compete to perform their respective jobs on a common processing resource, and addresses the complexity of various problems.
Abstract: We consider the scheduling problems arising when two agents, each with a set of nonpreemptive jobs, compete to perform their respective jobs on a common processing resource. Each agent wants to minimize a certain objective function, which depends on the completion times of its jobs only. The objective functions we consider in this paper are maximum of regular functions (associated with each job), number of late jobs, and total weighted completion times. We obtain different scenarios, depending on the objective function of each agent, and on the structure of the processing system (single machine or shop). For each scenario, we address the complexity of various problems, namely, finding the optimal solution for one agent with a constraint on the other agent's cost function, finding single nondominated schedules (i.e., such that a better schedule for one of the two agents necessarily results in a worse schedule for the other agent), and generating all nondominated schedules.

401 citations


Journal ArticleDOI
TL;DR: A queueing system with multitype customers and flexible (multiskilled) servers that work in parallel is considered and a very simple generalizedcµ-rule is shown to minimizes both instantaneous and cumulative queueing costs, asymptotically, over essentially all scheduling disciplines, preemptive or non-preemptive.
Abstract: We consider a queueing system with multitype customers and flexible (multiskilled) servers that work in parallel. IfQ iis the queue length of typei customers, this queue incurs cost at the rate ofC i ( Q i ), whereC i (.) is increasing and convex. We analyze the system in heavy traffic (Harrison and Lopez 1999) and show that a very simple generalizedcµ-rule (Van Mieghem 1995) minimizes both instantaneous and cumulative queueing costs, asymptotically, over essentially all scheduling disciplines, preemptive or non-preemptive. This rule aims at myopically maximizing the rate of decrease of the instantaneous cost at all times, which translates into the following: when becoming free, serverj chooses for service a typei customer such thati ? arg max i C' i ( Q i )µ ij , where µ ijis the average service rate of typei customers by serverj.An analogous version of the generalizedcµ-rule asymptotically minimizes delay costs. To this end, let the cost incurred by a typei customer be an increasing convex functionC i ( D) of its sojourn timeD. Then, serverj always chooses for service a customer for which the value ofC' i ( D) µ ijis maximal, whereD andi are the customer's sojourn time and type, respectively.

345 citations


Journal ArticleDOI
TL;DR: This paper presents a new best-fit heuristic for the two-dimensional rectangular stock-cutting problem and demonstrates its effectiveness by comparing it against other published approaches and suggesting an efficient implementation of this heuristic.
Abstract: This paper presents a new best-fit heuristic for the two-dimensional rectangular stock-cutting problem and demonstrates its effectiveness by comparing it against other published approaches. A placement algorithm usually takes a list of shapes, sorted by some property such as increasing height or decreasing area, and then applies a placement rule to each of these shapes in turn. The proposed method is not restricted to the first shape encountered but may dynamically search the list for better candidate shapes for placement. We suggest an efficient implementation of our heuristic and show that it compares favourably to other heuristic and metaheuristic approaches from the literature in terms of both solution quality and execution time. We also present data for new problem instances to encourage further research and greater comparison between this and future methods.

319 citations


Journal ArticleDOI
TL;DR: A stochastic general equilibrium inventory model for an oligopoly, in which all inventory constraint parameters are endogenously determined, is developed and it is shown that in all of the above settings a Nash equilibrium of infinite-horizon stationary strategies exists and that it is of a simple structure.
Abstract: This paper develops a stochastic general equilibrium inventory model for an oligopoly, in which all inventory constraint parameters are endogenously determined. We propose several systems of demand processes whose distributions are functions of all retailers' prices and all retailers' service levels. We proceed with the investigation of the equilibrium behavior of infinite-horizon models for industries facing this type of generalized competition, under demand uncertainty.We systematically consider the following three competition scenarios. (1) Price competition only: Here, we assume that the firms' service levels are exogenously chosen, but characterize how the price and inventory strategy equilibrium vary with the chosen service levels. (2) Simultaneous price and service-level competition: Here, each of the firms simultaneously chooses a service level and a combined price and inventory strategy. (3) Two-stage competition: The firms make their competitive choices sequentially. In a first stage, all firms simultaneously choose a service level; in a second stage, the firms simultaneously choose a combined pricing and inventory strategy with full knowledge of the service levels selected by all competitors. We show that in all of the above settings a Nash equilibrium of infinite-horizon stationary strategies exists and that it is of a simple structure, provided a Nash equilibrium exists in a so-called reduced game.We pay particular attention to the question of whether a firm can choose its service level on the basis of its own (input) characteristics (i.e., its cost parameters and demand function) only. We also investigate under which of the demand models a firm, under simultaneous competition, responds to a change in the exogenously specified characteristics of the various competitors by either: (i) adjusting its service level and price in the same direction, thereby compensating for price increases (decreases) by offering improved (inferior) service, or (ii) adjusting them in opposite directions, thereby simultaneously offering better or worse prices and service.

302 citations


Journal ArticleDOI
TL;DR: A new integer programming formulation for the CVRP based on a two-commodity network flow approach is described and a lower bound derived from the linear programming (LP) relaxation of the new formulation which is improved by adding valid inequalities in a cutting-plane fashion is presented.
Abstract: The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. In this paper, we describe a new integer programming formulation for the CVRP based on a two-commodity network flow approach. We present a lower bound derived from the linear programming (LP) relaxation of the new formulation which is improved by adding valid inequalities in a cutting-plane fashion. Moreover, we present a comparison between the new lower bound and lower bounds derived from the LP relaxations of different CVRP formulations proposed in the literature. A new branch-and-cut algorithm for the optimal solution of the CVRP is described. Computational results are reported for a set of test problems derived from the literature and for new randomly generated problems.

268 citations


Journal ArticleDOI
TL;DR: An identity in law is used between the integral of geometric Brownian motion over a finite time interval [0,t] and the state at timet of a one-dimensional diffusion process with affine drift and linear diffusion to express Asian option values in terms of spectral expansions associated with the diffusion infinitesimal generator.
Abstract: Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian option when the underlying asset price follows geometric Brownian motion. We use an identity in law between the integral of geometric Brownian motion over a finite time interval [0,t] and the state at timet of a one-dimensional diffusion process with affine drift and linear diffusion and express Asian option values in terms of spectral expansions associated with the diffusion infinitesimal generator. The first formula is an infinite series of terms involving Whittaker functionsM andW. The second formula is a single real integral of an expression involving Whittaker functionW plus (for some parameter values) a finite number of additional terms involving incomplete gamma functions and Laguerre polynomials. The two formulas allow accurate computation of continuously sampled arithmetic Asian option prices.

258 citations


Journal ArticleDOI
TL;DR: This paper proposes both an exact and a heuristic method for the car pooling problem, based on two integer programming formulations of the problem, which transforms the solution of a Lagrangean lower bound into a feasible solution.
Abstract: Car pooling is a transportation service organized by a large company which encourages its employees to pick up colleagues while driving to/from work to minimize the number of private cars travelling to/from the company site. The car pooling problem consists of defining the subsets of employees that will share each car and the paths the drivers should follow, so that sharing is maximized and the sum of the path costs is minimized. The special case of the car pooling problem where all cars are identical can be modeled as a Dial-a-Ride Problem. In this paper, we propose both an exact and a heuristic method for the car pooling problem, based on two integer programming formulations of the problem. The exact method is based on a bounding procedure that combines three lower bounds derived from different relaxations of the problem. A valid upper bound is obtained by the heuristic method, which transforms the solution of a Lagrangean lower bound into a feasible solution. The computational results show the effectiveness of the proposed methods.

246 citations


Journal ArticleDOI
TL;DR: A new approach to stochastic inventory/routing that approximates the future costs of current actions using optimal dual prices of a linear program and an efficient algorithm to both generate and eliminate itineraries during solution of the linear programs and control policy are considered.
Abstract: We consider a new approach to stochastic inventory/routing that approximates the future costs of current actions using optimal dual prices of a linear program. We obtain two such linear programs by formulating the control problem as a Markov decision process and then replacing the optimal value function with the sum of single-customer inventory value functions. The resulting approximation yields statewise lower bounds on optimal infinite-horizon discounted costs. We present a linear program that takes into account inventory dynamics and economics in allocating transportation costs for stochastic inventory routing. On test instances we find that these allocations do not introduce any error in the value function approximations relative to the best approximations that can be achieved without them. Also, unlike other approaches, we do not restrict the set of allowable vehicle itineraries in any way. Instead, we develop an efficient algorithm to both generate and eliminate itineraries during solution of the linear programs and control policy. In simulation experiments, the price-directed policy outperforms other policies from the literature.

Journal ArticleDOI
TL;DR: This paper describes a contact center with two channels, one for real-time telephone service, and another for a postponed call-back service offered with a guarantee on the maximum delay until a reply is received and proposes a staffing rule that satisfies a set of operational constraints on the performance of the system.
Abstract: Organizations worldwide use contact centers as an important channel of communication and transaction with their customers. This paper describes a contact center with two channels, one for real-time telephone service, and another for a postponed call-back service offered with a guarantee on the maximum delay until a reply is received. Customers are sensitive to both real-time and call-back delay and their behavior is captured through a probabilistic choice model. The dynamics of the system are modeled as anM/M/N multiclass system. We rigorously justify that as the number of agents increases, the system's load approaches its maximum processing capacity. Based on this observation, we perform an asymptotic analysis in the many-server, heavy traffic regime to find an asymptotically optimal routing rule, characterize the unique equilibrium regime of the system, approximate the system performance, and finally, propose a staffing rule that picks the minimum number of agents that satisfies a set of operational constraints on the performance of the system.

Journal ArticleDOI
TL;DR: The development and implementation of a system to solve a vehicle routing and delivery scheduling problem at Albert Heijn resulted in savings of 4% of distribution costs in its first year of implementation and is expected to yield 12%-20% savings as the firm expands its usage.
Abstract: Albert Heijn, BV, a supermarket chain in the Netherlands, faces a vehicle routing and delivery scheduling problem once every three to six months. Given hourly demand forecasts for each store, travel times and distances, cost parameters, and various transportation constraints, the firm seeks to determine a weekly delivery schedule specifying the times when each store should be replenished from a central distribution center, and to determine the vehicle routes that service these requirements at minimum cost. We describe the development and implementation of a system to solve this problem at Albert Heijn. The system resulted in savings of 4% of distribution costs in its first year of implementation and is expected to yield 12%-20% savings as the firm expands its usage. It also has tactical and strategic advantages for the firm, such as in assessing the cost impact of various logistics and marketing decisions, in performance measurement, and in competing effectively through reduced lead time and increased frequency of replenishment.

Journal ArticleDOI
TL;DR: The resulting system is a multiclass, multiserver queueing system with state-dependent arrival rates that minimizes real-time delay subject to the deadline of the postponed service mode and performs better than a system in which customers make decisions based on steady-state waiting-time information.
Abstract: Motivated by practices in customer contact centers, we consider a system that offers two modes of service: real-time and postponed with a delay guarantee. Customers are informed of anticipated delays and select their preferred option of service. The resulting system is a multiclass, multiserver queueing system with state-dependent arrival rates. We propose an estimation scheme for the anticipated real-time delay that is asymptotically correct, and a routing policy that is asymptotically optimal in the sense that it minimizes real-time delay subject to the deadline of the postponed service mode. We also show that our proposed state-dependent scheme performs better than a system in which customers make decisions based on steady-state waiting-time information. Our results are derived using an asymptotic analysis based on "many-server" limits for systems with state-dependent parameters.

Journal ArticleDOI
TL;DR: A combined interior-point and active-set method for solving the minimum-volume n-dimensional ellipsoid that must contain m given points a1, a2, a3 to solve the convex constrained problem in data mining and robust statistics.
Abstract: We present a practical algorithm for computing the minimum-volume n-dimensional ellipsoid that must contain m given points a1,',am ∈ ℝn. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30,000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.

Journal ArticleDOI
TL;DR: This paper considers an overbooking problem with multiple reservation and inventory classes, in which the multiple inventory classes may be used as substitutes to satisfy the demand of a given reservation class (perhaps at a cost).
Abstract: This paper considers an overbooking problem with multiple reservation and inventory classes, in which the multiple inventory classes may be used as substitutes to satisfy the demand of a given reservation class (perhaps at a cost). The problem is to jointly determine overbooking levels for the reservation classes, taking into account the substitution options. Such problems arise in a variety of revenue management contexts, including multicabin aircraft, back-to-back scheduled flights on the same leg, hotels with multiple room types, and mixed-vehicle car rental fleets. We model this problem as a two-period optimization problem. In the first period, reservations are accepted given only probabilistic knowledge of cancellations. In the second period, cancellations are realized and surviving customers are assigned to the various inventory classes to maximize the net benefit of assignments (e.g., minimize penalties). For this formulation, we show that the expected revenue function is submodular in the overbooking levels, which implies the natural property that the optimal overbooking level in one reservation class decreases with the number of reservations held in the other reservation classes. We then propose a stochastic gradient algorithm to find the joint optimal overbooking levels. We compare the decisions of the model to those produced by more naive heuristics on some examples motivated by airline applications. The results show that accounting for substitution when setting overbooking levels has a small, but still significant, impact on revenues and costs.

Journal ArticleDOI
TL;DR: This paper considers the trade-off between inventory cost, direct shipment cost, and facility location cost in such a system and shows that the moderate size distribution network design problem can be solved efficiently via this approach.
Abstract: In this paper, we study the distribution network design problem integrating transportation and infinite horizon multiechelon inventory cost function. We consider the trade-off between inventory cost, direct shipment cost, and facility location cost in such a system. The problem is to determine how many warehouses to set up, where to locate them, how to serve the retailers using these warehouses, and to determine the optimal inventory policies for the warehouses and retailers. The objective is to minimize the total multiechelon inventory, transportation, and facility location costs. To the best of our knowledge, none of the papers in the area of distribution network design has explicitly addressed the issues of the 2-echelon inventory cost function arising from coordination of replenishment activities between the warehouses and the retailers. We structure this problem as a set-partitioning integer-programming model and solve it using column generation. The pricing subproblem that arises from the column generation algorithm gives rise to a new class of the submodular function minimization problem. We show that this pricing subproblem can be solved inO( n?log? n) time, wheren is the number of retailers. Computational results show that the moderate size distribution network design problem can be solved efficiently via this approach.

Journal ArticleDOI
TL;DR: This work addresses a periodic-review, stochastic, capacitated, finite and infinite horizon production system faced by a manufacturer who has the ability to obtain advance demand information, and illustrates howvance demand information can be a substitute for capacity and inventory.
Abstract: Manufacturers make production decisions and carry inventory to satisfy uncertain demand. When holding and shortage costs are high, carrying inventory could be even more expensive for acapacitated production system. Recent developments in information technology and sales strategies enabled firms to acquire, collect, or induce advance demand information. We address a periodic-review, stochastic, capacitated, finite and infinite horizon production system faced by a manufacturer who has the ability to obtain advance demand information. We establish optimal policies and characterize their behavior with respect to capacity, fixed costs, advance demand information, and the planning horizon. With a numerical study, we quantify the value of advance demand information and additional capacity for specific problem instances. We illustrate how advance demand information can be a substitute for capacity and inventory.

Journal ArticleDOI
TL;DR: This paper considers scheduling problems where a set of original jobs has already been scheduled to minimize some cost objective, when a new set of jobs arrives and creates a disruption, and provides either an efficient algorithm or a proof that such an algorithm is unlikely to exist.
Abstract: This paper considers scheduling problems where a set of original jobs has already been scheduled to minimize some cost objective, when a new set of jobs arrives and creates a disruption. The decision maker needs to insert the new jobs into the existing schedule without excessively disrupting it. Two classes of models are considered. First, we minimize the scheduling cost of all the jobs, subject to a limit on the disruption caused to the original schedule, where this disruption is measured in various ways. In the second class, a total cost objective, which includes both the original cost measure and the cost of disruption, is minimized. For both classes and various costs based on classical scheduling objectives, and for almost all problems, we provide either an efficient algorithm or a proof that such an algorithm is unlikely to exist. We also show how to extend both classes of models to deal with multiple disruptions in the form of repeated arrivals of new jobs. Our work refocuses the extensive literature on scheduling problems towards issues of rescheduling, which are important because of the frequency with which disruptions occur in manufacturing practice.

Journal ArticleDOI
TL;DR: In this article, the authors compare the Kalman filter algorithm to LSQR, an iterative algorithm proposed by Paige and Saunders (1982) for the solution of large-scale least-squares problems.
Abstract: The problem of estimating and predicting Origin-Destination (OD) tables is known to be important and difficult. In the specific context of Intelligent Transportation Systems (ITS), the dynamic nature of the problem and the real-time requirements make it even more intricate. We consider here a least-square modeling approach for solving the OD estimation and prediction problem, which seems to offer convenient and flexible algorithms. The dynamic nature of the problem is represented by an autoregressive process, capturing the serial correlations of the state variables. Our formulation is inspired from Cascetta et al. (1993) and Ashok and Ben-Akiva (1993). We compare the Kalman filter algorithm to LSQR, an iterative algorithm proposed by Paige and Saunders (1982) for the solution of large-scale least-squares problems. LSQR explicitly exploits matrix sparsity, allowing to consider larger problems likely to occur in real applications. We show that the LSQR algorithm significantly decreases the computation effort needed by the Kalman filter approach for large-scale problems. We also provide a theoretical number of flops for both algorithms to predict which algorithm will perform better on a specific instance of the problem.

Journal ArticleDOI
TL;DR: This analysis provides the structure of the firm's optimal resource investment strategy as a function of demand parameters and investment costs, and shows that the flexible resource investment decision follows athreshold policy.
Abstract: We study the optimal resource investment decision faced by a two-product, price-setting firm that operates in a monopolistic setting and employs a postponed pricing scheme. The firm has the option to invest in dedicated resources as well as a more expensive, flexible resource that can satisfy both products. While the resource investment decision is made under demand uncertainty, pricing and resource allocation decisions are postponed to the time when demand curves are realized. Our analysis provides the structure of the firm's optimal resource investment strategy as a function of demand parameters and investment costs, and shows that the flexible resource investment decision follows athreshold policy. We also show that it can be optimal for the firm to invest in the flexible resource even when demand patterns are perfectly positively correlated. The reason for flexible capacity investment in this case is financial rather than risk pooling. On the other hand, we show that it can be optimal for the firm not to invest in the flexible resource even when demand patterns are perfectly negatively correlated. The flexible resource investment decision in this case depends on the profitability of the two products. Based on our analysis, we provide principles on the firm's optimal resource investment decision.

Journal ArticleDOI
TL;DR: This paper develops an approach based on performance targets to assess a preference function for a multiobjective decision under uncertainty that yields preference functions that are strategically equivalent to conventional multiattribute utility functions, but the target-oriented approach is more natural for some classes of decisions.
Abstract: This paper develops an approach based on performance targets to assess a preference function for a multiobjective decision under uncertainty. This approach yields preference functions that are strategically equivalent to conventional multiattribute utility functions, but the target-oriented approach is more natural for some classes of decisions. In some situations, the target-oriented preference conditions are analogous to reliability theory conditions for series or parallel failure modes in a system. In such cases, reinterpreting the conditions using reliability concepts can be useful in assessing the preference function. The target-oriented approach is also a generalization of common forms of goal programming. The approach has particular applicability for resource allocation decisions where the outcome of the decision is significantly determined by the actions of other stakeholders to the decision, such as new product development or decision making in a controversial regulated environment.

Journal ArticleDOI
TL;DR: A Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes is considered, where the total arrival rate and the number of servers both become large in such a way that the system's traffic intensity parameter approaches one.
Abstract: We consider a Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes. Customers waiting to be served may abandon the queue, and there is a cost penalty associated with such abandonments. Service rates, abandonment rates, and abandonment penalties are generally different for the different classes. The problem studied is that of dynamically scheduling the various classes. We consider the Halfin-Whitt heavy traffic regime, where the total arrival rate and the number of servers both become large in such a way that the system's traffic intensity parameter approaches one. An approximating diffusion control problem is described and justified as a purely formal (that is, nonrigorous) heavy traffic limit. The Hamilton-Jacobi-Bellman equation associated with the limiting diffusion control problem is shown to have a smooth (classical) solution, and optimal controls are shown to have an extremal or "bang-bang" character. Several useful qualitative insights are derived from the mathematical analysis, including a "square-root rule" for sizing large systems and a sharp contrast between system behavior in the Halfin-Whitt regime versus that observed in the "conventional" heavy traffic regime. The latter phenomenon is illustrated by means of a numerical example having two customer classes.

Journal ArticleDOI
TL;DR: This note discusses the relationships among three assumptions that appear frequently in the pricing/revenue management literature, and provides proofs and examples to show that none of these conditions implies any other.
Abstract: This note discusses the relationships among three assumptions that appear frequently in the pricing/revenue management literature. These assumptions are mostly needed for analytical tractability, and they have the common property of ensuring a well-behaved "revenue function." The three assumptions are decreasing marginal revenue with respect to demand, decreasing marginal revenue with respect to price, and increasing price elasticity of demand. We provide proofs and examples to show that none of these conditions implies any other. However, they can be ordered from strongest to weakest over restricted regions, and the ordering depends upon the region.

Journal ArticleDOI
TL;DR: The big square small square global optimization search in the plane is modified with a big triangle small triangle approach and the triangulation of the feasible region is obtained by using Voronoi diagrams.
Abstract: In this paper we propose to modify the big square small square global optimization search in the plane with a big triangle small triangle approach. The triangulation of the feasible region is obtained by using Voronoi diagrams. The resulting algorithm was tested on the obnoxious facility location and the attraction-repulsion Weber problems with excellent results.

Journal ArticleDOI
TL;DR: This research addresses two common tools for reducing product development lead times: overlapping of development stages and crashing of development times for the first time in the product development literature, thus facilitating analysis of the interdependencies between overlapping and crashing.
Abstract: This research addresses two common tools for reducing product development lead times: overlapping of development stages and crashing of development times. For the first time in the product development literature, a formal model addresses both tools concurrently, thus facilitating analysis of the interdependencies between overlapping and crashing. The results exhibit the necessity of addressing overlapping and crashing concurrently, and exhibit general characteristics of optimal overlapping/crashing policies. The impact of different evolution/sensitivity constellations on optimal policies is investigated, and comprehensive guidelines for structuring development processes are provided. For the special case of linear costs, an efficient procedure is presented that generates the efficient time-cost trade-off curves and determines the corresponding optimal overlapping/crashing policies. The impact of key parameters and the robustness regarding their estimates is illustrated with a simple two-stage example.

Journal ArticleDOI
Ward Whitt1
TL;DR: A diffusion approximation for the queue-length stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution,n servers, andm extra waiting spaces) is developed, focusing especially upon the steady-state delay probability.
Abstract: We develop a diffusion approximation for the queue-length stochastic process in theG/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution,n servers, andm extra waiting spaces). We use the steady-state distribution of that diffusion process to obtain approximations for steady-state performance measures of the queueing model, focusing especially upon the steady-state delay probability. The approximations are based on heavy-traffic limits in whichn tends to infinity as the traffic intensity increases. Thus, the approximations are intended for largen.For theGI/M/n/8 special case, Halfin and Whitt (1981) showed that scaled versions of the queue-length process converge to a diffusion process when the traffic intensity? napproaches 1 with (1 -? n )v n ? I for 0

Journal ArticleDOI
TL;DR: An algorithm for the valuation and optimal operation of hydroelectric and thermal power generators in deregulated electricity markets is presented, designed to incorporate a wide class of spot price models that can exhibit the same time-dependent, mean-reverting dynamics and price spikes as those observed in most electricity markets.
Abstract: We present an algorithm for the valuation and optimal operation of hydroelectric and thermal power generators in deregulated electricity markets. Real options theory is used to derive nonlinear partial-integro-differential equations (PIDEs) for the valuation and optimal operating strategies of both types of facilities. The equations are designed to incorporate a wide class of spot price models that can exhibit the same time-dependent, mean-reverting dynamics and price spikes as those observed in most electricity markets. Particular attention is paid to the operational characteristics of real power generators. For thermal power plants, these characteristics include variable start-up times and costs, control response time lags, minimum generating levels, nonlinear output functions, and structural limitations on ramp rates. For hydroelectric units, head effects and environmental constraints are addressed. We illustrate the models with numerical examples of a pump storage facility and a thermal power plant. This PIDE framework can achieve high levels of computational speed and accuracy while incorporating a wide range of spot price dynamics and operational characteristics.

Journal ArticleDOI
TL;DR: This paper demonstrates optimal policies for capacitated serial multiechelon production/inventory systems by demonstrating that a modified echelon base-stock policy is optimal in a two-stage system when there is a smaller capacity at the downstream facility.
Abstract: This paper demonstrates optimal policies for capacitated serial multiechelon production/inventory systems. Extending the Clark and Scarf (1960) model to include installations with production capacity limits, we demonstrate that a modified echelon base-stock policy is optimal in a two-stage system when there is a smaller capacity at the downstream facility. This is shown by decomposing the dynamic programming value function into value functions dependent upon individual echelon stock variables. We show that the optimal structure holds for both stationary and nonstationary stochastic customer demand. Finite-horizon and infinite-horizon results are included under discounted-cost and average-cost criteria.

Journal ArticleDOI
TL;DR: The current paper revisits a published Korean telecommunication analysis and presents a new and simple approach to execute the IDEA through the standard linear DEA models, finding that imprecise data can be easily converted into exact data.
Abstract: Data Envelopment Analysis (DEA) requires that the data for all inputs and outputs are known exactly. When some outputs and inputs are unknown decision variables, such as bounded and ordinal data, the DEA model becomes a nonlinear programming problem and is called imprecise DEA (IDEA). The nonlinear IDEA program can be converted into a linear program by an algorithm based upon scale transformations and variable alterations. Such an algorithm requires a set of special computational codes for each evaluation, because a different objective function and a different constraint with a set of new variables are present for each unit under evaluation. The current paper revisits a published Korean telecommunication analysis, and, by so doing, presents a new and simple approach to execute the IDEA through the standard linear DEA models. This greatly enhances the applicability of IDEA in applications, and the IDEA analysis is no longer limited to obtaining the efficiency scores. The key to the new approach lies in the finding that imprecise data can be easily converted into exact data. Based upon the exact data, models can be developed to determine all possible multiple optimal solutions in imprecise data, and to perform efficiency sensitivity analysis in IDEA.