Duality for quasi-concave programs with application to economics

A duality theory is developed for mathematical programs with strictly quasi-concave objective functions to be maximized over a convex set. This work broadens the duality theory of Rockafellar and Peterson from concave (convex) functions to quasi-concave (quasi-convex) functions. The theory is closely related to the utility theory in economics. An example from economic planning is examined and the solution to the dual program is shown to have the properties normally associated with market prices.     

Note: “Simultaneous optimization of efficiency and performance balance measures in single-machine scheduling problems”

n independent jobs are to be scheduled nonpreemptively on a single machine so as to minimize some performance measure. Federgruen and Mosheiov [2] show that a large class of such scheduling problems can be optimized by solving either a single instance or a finite sequence of instances of the so-called SQC problem, in which all the jobs have a fixed or controllable common due date and the sum of general quasiconvex functions of the job completion times is to be minimized. In this note we point out that this is not always true. In particular, we show that the algorithm proposed in [2] does not always find a global optimal schedule to the problem of minimizing the weighted sum of the mean and variance of job completion times. © 1996 John Wiley & Sons, Inc.     

The maximum collection problem with time-dependent rewards

We consider a routing problem where the objective is to maximize the sum of the rewards collected at the nodes visited. Node rewards are decreasing linear functions of time. Time is spent when traveling between pairs of nodes, and while visiting the nodes. We propose a penalty-based greedy (heuristic) algorithm and a branch-and-bound (optimal) algorithm for this problem. The heuristic is very effective in obtaining good solutions. We can solve problems with up to 20 nodes optimally on a microcomputer using the branch-and-bound algorithm. We report our computational experience with this problem. © 1996 John Wiley & Sons, Inc.     

Analysis of a bicriteria location model

In this article a bicriteria model, formed by the weighted sum of the minisum and minimax functions for a single-location problem, is investigated. It is shown that all efficient solutions generated by either constrained model are also properly efficient. The bicriteria model and the constrained models are theoretically equivalent, but it is more efficient and simpler to generate nondominated solutions using the constrained criterion approach. When solving the bicriteria model, a critical range is found for which all properly efficient solutions are generated.     

均匀分布集群目标射击火力分配优化数值仿真方法

基于所提出的火力分配方案,使用数值积分方法将对均匀分布集群目标射击的火力分配优化问题转化为由多变量初等函数表示的积分和函数在单位多面正方体内的最小值求解问题,火力分配优化的瞄准点坐标可以尝试利用成熟的Matlab优化计算软件来直接求解,算例说明所给出方法对于火力分配优化问题的深入研究是有裨益的.     

Minimizing sums and products of linear fractional functions over a polytope

In this paper, we develop efficient deterministic algorithms for globally minimizing the sum and the product of several linear fractional functions over a polytope. We will show that an elaborate implementation of an outer approximation algorithm applied to the master problem generated by a parametric transformation of the objective function serves as an efficient method for calculating global minima of these nonconvex minimization problems if the number of linear fractional terms in the objective function is less than four or five. It will be shown that the Charnes–Cooper transformation plays an essential role in solving these problems. Also a simple bounding technique using linear multiplicative programming techniques has remarkable effects on structured problems. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 583–596, 1999     

Optimal flowshop schedules with no intermediate storage space

In this paper the problem of finding an optimal schedule for the n-job, M-machine flowshop scheduling problem is considered when there is no intermediate space to hold partially completed jobs and the objective function is to minimize the weighted sum of idle times on all machines. By assuming that jobs are processed as early as possible, the problem is modeled as a traveling salesman problem and solved by known solution techniques for the traveling salesman problem. A sample problem is solved and a special case, one involving only two machines, is discussed.     

Determining optimal sizes of bounded batches with rejection via quadratic min‐cost flow

In this article, we consider a single machine scheduling problem, in which identical jobs are split into batches of bounded sizes. For each batch, it is allowed to produce less jobs than a given upper bound, that is, some jobs in a batch can be rejected, in which case a penalty is paid for each rejected job. The objective function is the sum of several components, including the sum of the completion times, total delivery cost, and total rejection cost. We reduce this problem to a min‐cost flow problem with a convex quadratic function and adapt Tamir's algorithm for its solution. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 217–224, 2017     

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

A simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017     

A note on the sum of a linear and linear-fractional function

The sum of a linear and linear-fractional function is investigated in terms of quasi-convexity and quasi-concavity. From this we obtain some insight into the nature of local optima of these functions useful in algorithms.     

Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date

We consider a single-machine scheduling problem in which all jobs have the same due date and penalties are assessed for both early and late completion of jobs. However, earliness and tardiness are penalized at different rates. The scheduling objective is to minimize either the weighted sum of absolute deviations (WSAD) or the weighted sum of squared deviations (WSSD). For each objective we consider two versions of the problem. In the unconstrained version an increase in the due date does not yield any further decrease in the objective function. We present a constructive algorithm for the unconstrained WSAD problem and show that this problem is equivalent to the two-parallel, nonidentical machine, mean flow-time problem. For the unconstrained WSSD and the constrained WSAD and WSSD problems we propose implicit enumeration procedures based on several dominance conditions. We also report on our computational experience with the enumeration procedures.     

Minimizing the sum of absolute lateness in single-machine and multimachine scheduling

Kanet addressed the problem of scheduling n jobs on one machine so as to minimize the sum of absolute lateness under a restrictive assumption on their common due date. This article extends the results to the problem of scheduling n jobs on m parallel identical processors in order to minimize the sum of absolute lateness. Also, a heuristic algorithm for a more general version with no restriction on the common due date, for the problem of n-job single-machine scheduling is presented and its performance is reported.     

面向多最优解组合优化问题的决策求解算法

针对具有固定物品总和、多最优解特征的组合优化问题,以固定总和实数子集问题和购买鸡翅问题为例,给出了这类多最优解组合优化问题的形式化表示。在分析枚举等经典算法基础上,提出了基于整数状态表示和实数状态表示的0-1决策递归搜索多最优解动态规划算法。针对该算法在最优解数量较大时,时间复杂度趋向O(mⁿ)的问题,提出了基于相同决策路径合并和基于0-x决策的两种改进算法。实验中两种改进算法的计算时间基本符合与O(nb+nm)的正比关系,表明对于这类多最优解组合优化问题具有良好的求解性能。     

The permutation flow shop problem with sum-of-completion times performance criterion

In this article we address the non-preemptive flow shop scheduling problem for minimization of the sum of the completion times. We present a new modeling framework and give a novel game-theoretic interpretation of the scheduling problem. A lower-bound generation scheme is developed by solving appropriately defined linear assignment problems. This scheme can also be used as a heuristic approach for the solution of the problem with satisfactory results. Its main use, however, is as a bounding scheme within a branch-and-bound procedure. Our branch-and-bound procedure improves significantly upon the best available enu-merative procedures in the current literature. Extensive computational results are used to qualify the above statements. © 1993 John Wiley & Sons, Inc.     

Coordinated scheduling of customer orders for quick response

The scheduling problem addressed in this paper concerns a manufacturer who produces a variety of product types and operates in a make‐to‐order environment. Each customer order consists of known quantities of the different product types, and must be delivered as a single shipment. Periodically the manufacturer schedules the accumulated and unscheduled customer orders. Instances of this problem occur across industries in manufacturing as well as in service environments. In this paper we show that the problem of minimizing the weighted sum of customer order delivery times is unary NP‐hard. We characterize the optimal schedule, solve several special cases of the problem, derive tight lower bounds, and propose several heuristic solutions. We report the results of a set of computational experiments to evaluate the lower bounding procedures and the heuristics, and to determine optimal solutions. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Multiple subset sum with inclusive assignment set restrictions

In a traditional multiple subset sum problem (MSSP), there is a given set of items and a given set of bins (or knapsacks) with identical capacities. The objective is to select a subset of the items and pack them into the bins such that the total weight of the selected items is maximized. However, in many applications of the MSSP, the bins have assignment restrictions. In this article, we study the subset sum problem with inclusive assignment set restrictions, in which the assignment set of one item (i.e., the set of bins that the item may be assigned to) must be either a subset or a superset of the assignment set of another item. We develop an efficient 0.6492‐approximation algorithm and test its effectiveness via computational experiments. We also develop a polynomial time approximation scheme for this problem. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Acceleration of the HAP approach for the multifacility location problem

This article considers the problem of locating multiple new facilities to minimize the cost function consisting of the sum of weighted distances among new facilities and between new and existing facilities. The hyperboloid approximate procedure (HAP) is probably the most widely used approach for solving this problem. In this article, an optimality condition for this problem is derived and a method to accelerate the convergence rate of the HAP for the case of Euclidean distances is presented. From the numerical results presented in this article, it can be concluded that the performance of the new algorithm is superior to the performance of the original HAP.     

Least-absolute-value estimators for one-way and two-way tables

This paper concerns itself with the problem of estimating the parameters of one-way and two-way classification models by minimization of the sum of the absolute deviations of the regression function from the observed points. The one-way model reduces to obtaining a set of medians from which optimal parameters can be obtained by simple arithmetic manipulations. The two-way model is transformed into a specially structured linear programming problem, and two algorithms are presented to solve this problem. The occurrence of alternative optimal solutions in both models is discussed, and numerical examples are presented.     

Composite unimodality

This article defines a class of univariate functions termed composite unimodal, and shows how their minimization admits an effective search procedure, albeit one not as efficient as is Fibonacci search for unimodal functions. An approximate Lagrangian approach to an important real-world logistics problem is seen to yield a surrogate problem whose objective function is composite unimodal. The mathematical form of this objective function is likely to be encountered in solving future real-world problems. © 1993 John Wiley & Sons, Inc.     

Single-machine scheduling with dynamic arrivals: Decomposition results and an improved algorithm

This article considers the single-machine dynamic scheduling problem where the jobs have different arrival times and the objective is to minimize the sum of completion times. This problem is known to be strongly NP-hard. We develop decomposition results for this problem such that a large problem can be solved by combining optimal solutions for several smaller problems. The decomposition results can be used with any implicit enumeration method to develop an optimal algorithm. Our computational experiment indicates that the computational efficiency of the currently best available branch-and-bound algorithm can be improved with the use of our decomposition results. © 1996 John Wiley & Sons, Inc.