A labeling algorithm for the sensitivity ranges of the assignment problem

In view of the simplex-type algorithm, the assignment problem is inherently highly degenerate. It may be the optimal basis has changed, but the optimal assignment is unchanged when parameter variation occurs. Degeneracy then makes sensitivity analysis difficult, as well as makes the classical Type I range, which identifies the range the optimal basis unchanged, impractical. In this paper, a labeling algorithm is proposed to identify two other sensitivity ranges – Type II range and Type III range. The algorithm uses the reduced cost matrix, provided in the final results of most solution algorithms for AP, to determine the Type II range which reflects the stability of the current optimal assignment. Thus, the algorithm generates a streamlined situation from searching the optimal solution until performing the sensitivity analysis of the assignment problem. The Type III range, reflecting the flexibility of optimal decision making, can be obtained immediately after the Type II range is determined. Numerical examples are presented to demonstrate the algorithm.     

Determining Type II sensitivity ranges of the fractional assignment problem

This paper proposes iterative labeling algorithms to determine the Type II sensitivity ranges of the fractional assignment problem. Unlike the traditional sensitivity range which keeps the current optimal basis remaining optimal, the Type II sensitivity range is the range that keeps the current optimal assignment remaining optimal. Focusing only on the non-degenerate basic variables makes the Type II sensitivity range more practical. Three cases of perturbation, each with two kinds, are discussed. An example is presented to demonstrate the proposed algorithms.     

Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant.For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: “what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?”. The differences of the results between the linear and convex quadratic optimization problems are highlighted too.     

Single machine SLK/DIF due window assignment problem with learning effect and deteriorating jobs

We study a single-machine scheduling and due window assignment problem, in which job processing times are defined by functions of their starting times (deteriorating effect) and positions in the sequence (learning effect). The problem is to determine the optimal due windows and the processing sequence simultaneously to minimize costs for earliness, tardiness, the window location, window size and makespan. We show that the problem remains polynomially solvable under the proposed model for two popular due window assignment methods: The slack due date assignment method (usually referred to as SLK) and the unrestricted due date assignment method (usually referred to as DIF).     

Type II sensitivity analysis of cost coefficients in the degenerate transportation problem

This paper focuses on sensitivity analysis of the degenerate transportation problem (DTP) when perturbation occurs on one cost coefficient. The conventional Type I sensitivity analysis of the transportation problem (TP) determines the perturbation ranges for the invariant optimal basis. Due to different degenerate optimal basic solutions yielding different Type I ranges, the Type I range is misleading for the DTP. Type II sensitivity analysis, which determines the perturbation ranges for the invariant shipping pattern, is more practical for the DTP. However, it is too tedious to obtain Type II ranges by enumerating all optimal basic solutions and all primal optimal basic solutions while getting the union of each corresponding Type I ranges. Here, we propose two labeling algorithms to determine the Type II ranges of the cost coefficient. Besides, three lemmas are provided for obtaining the upper bound or lower bound of the Type II ranges of the cost coefficient directly under specific conditions of the DTP. A numerical example is given to demonstrate the procedure of the proposed labeling algorithms and computational results have been provided.     

Using bounds on the data in linear programming: The tolerance approach to sensitivity analysis

In contrast to traditional sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent variations in a number of parameters. A primary focus of this approach is to determine a maximum tolerance percentage for selected right-hand-side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value. Similarly, the approach yields a maximum tolerance percentage for selected objective function coefficients. This paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages.     

Robustness of the efficient DMUs in data envelopment analysis

By means of modified versions of CCR model based on evaluation of a decision making unit (DMU) relative to a reference set grouped by all other DMUs, sensitivity analysis of the CCR model in data envelopment analysis (DEA) is studied in this paper. The methods for sensitivity analysis are linear programming problems whose optimal values yield particular regions of stability. Sufficient and necessary conditions for upward variations of inputs and for downward variations of outputs of an (extremely) efficient DMU which remains efficient are provided. The approach does not require calculation of the basic solutions and of the inverse of the corresponding optimal basis matrix. The approach is illustrated by two numerical examples.     

Unifying Optimal Partition Approach to Sensitivity Analysis in Conic Optimization

We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as functions of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the optimal partition approach to sensitivity analysis in LP and SDP, the range of perturbations for which the optimal partition remains constant can be computed by solving two conic optimization problems. Under a weaker notion of nondegeneracy, this range is simply given by a minimum ratio test. We discuss briefly the properties of the optimal value function under such perturbations.     

Optimal assignment of resources to strengthen the weakest link in an uncertain environment

The weakest link principle applies to many real-life situations where a system is as productive as its bottleneck. The purpose of this paper is to show how the efficiency of a manufacturing or service process (i.e., the strength of a chain) may be maximized by optimal allocation of resources to improve the performance of the bottleneck (i.e., strengthen the weakest link) in an uncertain environment. Specifically, we consider two different versions of the stochastic bottleneck assignment problem (SBAP), which is a variation of the classic assignment problem (AP), with respective goals of minimizing the expected longest processing time and maximizing the expected lowest production rate. It is proven that each of the two intrinsically difficult SBAPs is reducible to an efficiently solvable AP provided that the processing times or the production rates are independent random variables following some families of probability distributions.     

Profit maximization in flexible serial queueing networks

We analyze the tradeoff between efficiency and service quality in tandem systems with flexible servers and finite buffers. We reward efficiency by assuming that a revenue is earned each time a job is completed, and penalize poor service quality by incorporating positive holding costs. We study the dynamic assignment of servers to tasks with the objective of maximizing the long-run average profit. For systems of arbitrary size, structured service rates, and linear or nonlinear holding costs, we determine the server assignment policy that maximizes the profit. For systems with two stations, two servers with arbitrary service rates, and linear holding costs, we show that the optimal server assignment policy is of threshold type and determine the value of this threshold as a function of the revenue and holding cost. The threshold can be interpreted as the best possible buffer size, and hence our results prove the equivalence of addressing service quality via a holding cost and via limiting the buffer size. Furthermore, we identify the optimal buffer size when each buffer space comes at a cost. We provide numerical results that suggest that the optimal policy also has a threshold structure for nonlinear holding costs. Finally, for larger systems with arbitrary service rates, we propose effective server assignment heuristics.     

分派问题保持最优解不变的充分必要条件

进一步讨论了在保持分派问题最优解不变的情况下,效率矩阵元素的变化范围.这些变化范围是保持分派问题最优解不变的充要条件.     

Sensitivity analysis in convex quadratic optimization: invariant support set interval

In sensitivity analysis one wants to know how the problem and the optimal solutions change under the variation of the input data. We consider the case when variation happens in the right-hand side of the constraints and/or in the linear term of the objective function. We are interested to find the range of the parameter variation in Convex Quadratic Optimization (CQO) problems where the support set of a given primal optimal solution remains invariant. This question has been first raised in Linear Optimization (LO) and known as Type II (so-called Support Set Invariancy) sensitivity analysis. We present computable auxiliary problems to identify the range of parameter variation in support set invariancy sensitivity analysis for CQO. It should be mentioned that all the given auxiliary problems are LO problems and can be solved by an interior point method in polynomial time. We also highlight the differences between the characteristics of support set invariancy sensitivity analysis for LO and CQO.     

Sensitivity analysis in linear optimization: Invariant support set intervals

Sensitivity analysis is one of the most interesting and preoccupying areas in optimization. Many attempts are made to investigate the problem’s behavior when the input data changes. Usually variation occurs in the right hand side of the constraints and/or the objective function coefficients. Degeneracy of optimal solutions causes considerable difficulties in sensitivity analysis. In this paper we briefly review three types of sensitivity analysis and consider the question: what is the range of the parameter, where for each parameter value, an optimal solution exists with exactly the same set of positive variables that the current optimal solution has. This problem is coming from managerial requirements. Managers want to know in what range of variation of sources or prices in the market can they keep the installed production lines active, and only production’s levels would change.     

Sensitivity analysis in fuzzy number linear programming problems

In this paper, we generalize the concept of sensitivity analysis in fuzzy number linear programming (FLNP) problems by applying fuzzy simplex algorithms and using the general linear ranking functions on fuzzy numbers. The purpose of sensitivity analysis is to determine changes in the optimal solution of FNLP problem resulting from changes in the data. If the change affects the optimality of the basis, we perform primal pivots to achieve optimality by use of the fuzzy primal simplex method. Whenever the change destroys the feasibility of the optimal basis, we perform dual pivots to achieve feasibility by use of the fuzzy dual simplex method.     

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.     

Optimal maintenance service contract negotiation with aging equipment

In recent years, there has been a growing trend to out-source service operations in which the equipment maintenance is carried out by an external agent rather than in-house. Often, the agent (service provider) offers more than one option and the owners of equipment (customers) are faced to the problem of selecting the optimal option, under the terms of a contract. In the current work, we develop a model and report results to determine the agent’s optimal strategy for a given type of contract. The model derives in a non-cooperative game formulation in which the decisions are taken by maximizing expected profits. This work extends previous models by considering the realistic case of equipments having an increasing failure intensity due to imperfect maintenance, instead of the standard assumption that considers failure times are exponentially distributed (constant failure intensity). We develop a model using a linear function of time to characterize the failure intensity. The main goal, for the agent, is to determine the pricing structure in the contract and the number of customers to service. On the other hand, for the clients, the main goal is to define the period between planned actions for preventive maintenance and the time to replace equipments. In order to give a complete characterization of the results, we also carry out a sensitivity analysis over some of the factors that would influence over the failure intensity.     

Categorized Assignment Scheduling: a Tabu Search Approach

The assignment problem (AP) and bottleneck assignment problem (BAP) are well studied in operational research literature. In this paper we consider two related problems which simultaneously generalize both AP and BAP. Unlike AP and BAP, these generalizations are strongly NP-complete. We propose two heuristics to solve these generalized problems: one based on a greedy principle and the other based on tabu search. Computational results are presented which show that these heuristics, when used together, produce good quality solutions. Our adaptation of tabu search also gives some new insight into the application of tabu search on permutation problems.     

Scheduling policies using marked/phantom slot algorithms

We address the problem of schedulingM customer classes in a single-server system, with customers arriving in one ofN arrival streams, as it arises in scheduling transmissions in packet radio networks. In general,NM and a customer from some stream may join one of several classes. We consider a slotted time model where at each scheduling epoch the server (channel) is assigned to a particular class (transmission set) and can serve multiple customers (packets) simultaneously, one from every arrival stream (network node) that can belong to this class. The assignment is based on arandom polling policy: the current time slot is allocated to theith class with probability _i. Our objective is to determine the optimal probabilities by adjusting them on line so as to optimize some overall performance measure. We present an approach based on perturbation analysis techniques, where all customer arrival processes can be arbitrary, and no information about them is required. The basis of this approach is the development of two sensitivity estimators leading to amarked slot and aphantom slot algorithm. The algorithms determine the effect of removing/ adding service slots to an existing schedule on the mean customer waiting times by directly observing the system. The optimal slot assignment probabilities are then used to design adeterministic scheduling policy based on the Golden Ratio policy. Finally, several numerical results based on a simple optimization algorithm are included.This work was supported by the Naval Research Laboratory under contracts N000014-91-J-2025 and N000014-92-J-2017, by the National Science Foundation under grant EID-9212122, and by the Rome Laboratory under contract F30602-94-C-0109.     

System optimal and user equilibrium time-dependent traffic assignment in congested networks

This paper formulates two dynamic network traffic assignment models in which O-D desires for the planning horizon are assumed known a priori: the system optimal (SO) and the user equilibrium (UE) time-dependent traffic assignment formulations. Solution algorithms developed and implemented for these models incorporate a traffic simulation model within an overall iterative search framework. Experiments conducted on a test network provide the basis for a comparative analysis of system performance under the SO and UE models.