A note on “resource flexibility with responsive pricing”

We revisit the capacity investment decision problem studied in the article “Resource Flexibility with Responsive Pricing” by Chod and Rudi [Operations Research 53, (2005) 532–548]. A monopolist firm producing two dependent (substitutable or complementary) products needs to determine the capacity of one flexible resource under demand risk so as to maximize its expected profit. Product demands are linear functions of the prices of both products, and the market potentials are random and correlated. We perform a comparative statics analysis on how demand variability and correlation impact the optimal capacity and the resulting expected profit. In particular, C&R study this problem under the following assumptions/approximations: (i) demand intercepts follow a bivariate Normal distribution; (ii) demand uncertainty is of an additive form; (iii) and under approximate expressions for the optimal capacity and optimal expected profit. We revisit Propositions 2, 3, 4, 5, and 10 of C&R without these assumptions and approximations, and show that these results continue to hold (i) for the exact expressions for the optimal expected profit and optimal capacity, and (ii) under any arbitrary continuous distribution of demand intercepts. However, we also show that the additive demand uncertainty is a critical assumption for the C&R results to hold. In particular, we provide a case of multiplicative uncertainty under which the C&R results (Propositions 2 and 3) fail. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Requirements of an “optimizer” for computer simulations

Computer simulation has many advantages. However, one major disadvantage is that, in all too many cases, the attempt to use computer simulation to find an optimum solution to a problem rapidly degenerates into a trial-and-error process. Techniques for overcoming this disadvantage, i. e., for making optimization and computer simulation more compatible, are applicable at two points in the development of the overall computer simulation. Techniques which are used within actual construction of the mathematical models comprising the simulation will be labeled as internal methods, while those which are used after the simulation has been completely developed will be termed external methods Because external methods appear to offer the largest potential payoff, discussion is restricted to these methods, which are essentially search techniques. In addition, the development of an “Optimizer” computer program based on these techniques is suggested Although drawbacks to the use of search techniques in the computer simulation framework exist, these techniques do offer potential for “optimization.” The modification of these techniques to satisfy the requirements of an “Optimizer” is discussed.     

Optimal burn‐in procedure for periodically inspected systems

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we study burn‐in procedure for a system that is maintained under periodic inspection and perfect repair policy. Assuming that the underlying lifetime distribution of a system has an initially decreasing and/or eventually increasing failure rate function, we derive upper and lower bounds for the optimal burn‐in time, which maximizes the system availability. Furthermore, adopting an age replacement policy, we derive upper and lower bounds for the optimal age parameter of the replacement policy for each fixed burn‐in time and a uniform upper bound for the optimal burn‐in time given the age replacement policy. These results can be used to reduce the numerical work for determining both optimal burn‐in time and optimal replacement policy. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

“simple-approximate” battle-outcome-prediction conditions for variable-coefficient lanchester-type equations of modern warfare

This article considers combat between two homogeneous forces modeled by variable- coefficient Lanchester-type equations of modern warfare and develops new “simple-approximate” battle-outcome-prediction conditions for military engagements terminated by two different types of prescribed conditions being met (fixed-force-level-breakpoint battles and fixed-force-ratio-breakpoint battles). These battle-outcome-prediction conditions are sufficient (but not necessary) to determine the outcome of battle without having to explicitly compute the force-level trajectories, and they are characterized by their simplicity, requiring no advanced mathematical knowledge or tabulations of “special functions” for their application. Integrability properties of the Lanchester attrition-rate coefficients figure prominently in their results, and involved in their development is a generalization of Lanchester's famous square law to variable-coefficient Lanchester-type combat and several other novel mathematical developments for the analysis of ordinary differential equations. Examples are given, with the attack of a mobile force against a static defensive position (both sides armed with weapons whose firepower is range dependent) being examined in detail.     

Managing waiting times of backordered demands in single‐stage (Q,r) inventory systems

We present a service constrained (Q, r) model that minimizes expected holding and ordering costs subject to an upper bound on the expected waiting time of demands that are actually backordered. We show that, after optimizing over r, the average cost is quasiconvex in Q for logconcave continuous lead time demand distributions. For logconcave discrete lead time demand distributions we find a single‐pass efficient algorithm based on a novel search stopping criterion. The algorithm also allows for bounds on the variability of the service measure. A brief numerical study indicates how the bounds on service impact the optimal average cost and the optimal (Q, r) choice. The discrete case algorithm can be readily adapted to provide a single pass algorithm for the traditional model that bounds the expected waiting time of all demands (backordered or not). © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 557–573, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10028     

Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r,s)‐out‐of‐(m,n):F system

A 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system consists of m × n components, and fails if and only if k or more components fail in an r × s submatrix. This system can be treated as a reliability model for TFT liquid crystal displays, wireless communication networks, etc. Although an effective method has been developed for evaluating the exact system reliability of small or medium‐sized systems, that method needs extremely high computing time and memory capacity when applied to larger systems. Therefore, developing upper and lower bounds and accurate approximations for system reliability is useful for large systems. In this paper, first, we propose new upper and lower bounds for the reliability of a 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system. Secondly, we propose two limit theorems for that system. With these theorems we can obtain accurate approximations for system reliabilities when the system is large and component reliabilities are close to one. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Reliability bounds for the NBUE distributions

In this paper, we study upper and lower bounds on the reliability in new better than used in expectation (NBUE) life distribution class with fixed first two moments. By a constructive proof, we determine the upper bounds on the reliability analytically in different regions and show that these bounds are sharp. For the lower bounds, similar results are obtained except in one region. For that region, a conjecture is given for further study. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 781–797, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10035     

Two‐machine flow shop scheduling with common due window to minimize weighted number of early and tardy jobs

This article studies two due window scheduling problems to minimize the weighted number of early and tardy jobs in a two‐machine flow shop, where the window size is externally determined. These new scheduling models have many practical applications in real life. However, results on these problems have rarely appeared in the literature because of a lack of structural and optimality properties for solving them. In this article, we derive several dominance properties and theorems, including elimination rules and sequencing rules based on Johnsos order, lower bounds on the penalty, and upper bounds on the window location, which help to significantly trim the search space for the problems. We further show that the problems are NP‐hard in the ordinary sense only. We finally develop efficient pseudopolynomial dynamic programming algorithms for solving the problems. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Computing bounds for the optimal value in linear programming

Consider a standard linear programming problem and suppose that there are bounds available for the decision variables such that those bounds are not violated at an optimal solution of the problem (but they may be violated at some other feasible solutions of the problem). Thus, these bounds may not appear explicitly in the problem, but rather they may have been derived from some prior knowledge about an optimal solution or from the explicit constraints of the problem. In this paper, the bounds on variables are used to compute bounds on the optimal value when the problem is being solved by the simplex method. The latter bounds may then be used as a termination criteria for the simples iterations for the purpose of finding a “sufficiently good” near optimal solution. The bounds proposed are such that the computational effort in evaluating them is insignificant compared to that involved in the simplex iterations. A numerical example is given to demonstrate their performance.     

Bounds on the availability function

This paper gives bounds on the availability function for an alternating renewal process with exponential failure and general repair times. A bound on the error is also given. Several of the bounds with greatest practical consequence are worked out and illustrated. Repair distributions for which a lower bound on availability is easily computed are gamma (integer shape parameter), log normal, and Weibull. Finally, some simulation results for log normal repair versus gamma repair are given.     

非匹配摄动系统协方差矩阵的定界估计

基于Ｌｙａｐｕｎｏｖ 稳定性理论,在系统矩阵及噪声输入矩阵均存在非匹配摄动的一般情形下,研究随机系统稳态状态协方差的定界估计问题,通过求解两个代数Ｒｉｃｃａｔｉ方程,给出了该协方差矩阵上下界的估计式,并提供了一个说明性算例     

A duality‐based relaxation and decomposition approach for inventory distribution systems

We propose a new method for making the inventory replenishment decisions in distribution systems. In particular, we consider distribution systems consisting of multiple retailers that face random demand and a warehouse that supplies the retailers. The method that we propose is based on formulating the distribution problem as a dynamic program, and relaxing the constraints that ensure the nonnegativity of the shipments to the retailers, by associating Lagrange multipliers with them. We show that our method provides lower bounds on the value functions, and a good set of values for the Lagrange multipliers can be obtained by maximizing a concave function in a relatively straightforward manner. Computational experiments indicate that our method can provide significant improvements over the traditional approaches for making the inventory replenishment decisions, in terms of both the tightness of the lower bounds on the value functions and the performance of the policies. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

A multi‐stage stochastic programming approach for network capacity expansion with multiple sources of capacity

In networks, there are often more than one sources of capacity. The capacities can be permanently or temporarily owned by the decision maker. Depending on the nature of sources, we identify the permanent capacity, spot market capacity, and contract capacity. We use a scenario tree to model the uncertainty, and build a multi‐stage stochastic integer program that can incorporate multiple sources and multiple types of capacities in a general network. We propose two solution methodologies for the problem. Firstly, we design an asymptotically convergent approximation algorithm. Secondly, we design a cutting plane algorithm based on Benders decomposition to find tight bounds for the problem. The numerical experiments show superb performance of the proposed algorithms compared with commercial software. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 600–614, 2017     

Approximation methods for the analysis of a multicomponent,multiproduct assemble‐to‐order system

In this article, we consider the performance evaluation of a multicomponent, multiproduct assemble‐to‐order (ATO) system. Each component is managed independently using a base‐stock policy at a supply facility with limited production capacity and an infinite buffer. The arrivals of demands follow a multivariate Poisson process and unfilled demands are backlogged. Because exact analysis of the proposed system is not feasible, we propose two approximation methods which provide upper and lower bounds for various performance measures such as fill rate, average waiting time, and average number of backorders of the proposed system. Our computational experiments demonstrate the effectiveness of the two approximation methods under various system settings. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Technical note: Find a hidden “treasure”

This paper deals with a two searchers game and it investigates the problem of how the possibility of finding a hidden object simultaneously by players influences their behavior. Namely, we consider the following two‐sided allocation non‐zero‐sum game on an integer interval [1,n]. Two teams (Player 1 and 2) want to find an immobile object (say, a treasure) hidden at one of n points. Each point i ∈ [1,n] is characterized by a detection parameter λ_i (μ_i) for Player 1 (Player 2) such that p_i(1 ? exp(?λ_ix_i)) (p_i(1 ? exp(?μ_iy_i))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort x_i (y_i) applied at point i where p_i ∈ (0,1) is the probability that the object is hidden at point i. Player 1 (Player 2) undertakes the search by allocating the total amount of effort X(Y). The payoff for Player 1 (Player 2) is 1 if he detects the object but his opponent does not. If both players detect the object they can share it proportionally and even can pay some share to an umpire who takes care that the players do not cheat each other, namely Player 1 gets q₁ and Player 2 gets q₂ where q₁ + q₂ ≤ 1. The Nash equilibrium of this game is found and numerical examples are given. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Benders decomposition with alternative multiple cuts for a multi‐product closed‐loop supply chain network design model

In this article, we consider a multi‐product closed‐loop supply chain network design problem where we locate collection centers and remanufacturing facilities while coordinating the forward and reverse flows in the network so as to minimize the processing, transportation, and fixed location costs. The problem of interest is motivated by the practice of an original equipment manufacturer in the automotive industry that provides service parts for vehicle maintenance and repair. We provide an effective problem formulation that is amenable to efficient Benders reformulation and an exact solution approach. More specifically, we develop an efficient dual solution approach to generate strong Benders cuts, and, in addition to the classical single Benders cut approach, we propose three different approaches for adding multiple Benders cuts. These cuts are obtained via dual problem disaggregation based either on the forward and reverse flows, or the products, or both. We present computational results which illustrate the superior performance of the proposed solution methodology with multiple Benders cuts in comparison to the branch‐and‐cut approach as well as the traditional Benders decomposition approach with a single cut. In particular, we observe that the use of multiple Benders cuts generates stronger lower bounds and promotes faster convergence to optimality. We also observe that if the model parameters are such that the different costs are not balanced, but, rather, are biased towards one of the major cost categories (processing, transportation or fixed location costs), the time required to obtain the optimal solution decreases considerably when using the proposed solution methodology as well as the branch‐and‐cut approach. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

The sequencing of “related” jobs

Given n jobs and a single facility, and the fact that a subset of jobs are “related” to each other in such a manner that regardless of which job is completed first, its utility is hampered until all other jobs in the same subset are also completed, it is desired to determine the sequence which minimizes the cost of tardiness. The special case of pairwise relationship among all jobs is easily solved. An algorithm for the general case is given through a dynamic programming formulation.     

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

In this article, we analyze a discrete‐time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady‐state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state‐dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions—from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.     

A bounded dual (all-integer) integer programming algorithm with an objective cut

In this article, we describe a new algorithm for solving all-integer, integer programming problems. We generate upper bounds on the decision variables, and use these bounds to create an advanced starting point for a dual all-integer cutting plane algorithm. In addition, we use a constraint derived from the objective function to speed progress toward the optimal solution. Our basic vehicle is the dual all-integer algorithm of Gomory, but we incorporate certain row- and column-selection criteria which partially avoid the problem of dual-degenerate iterations. We present the results of computational testing.