Linear programming with multiple fuzzy goals

This paper describes the use of fuzzy set theory in goal programming (GP) problems. In particular, it is demonstrated how fuzzy or imprecise aspirations of the decision maker (DM) can be quantified through the use of piecewise linear and continuous functions. Models are then presented for the use of fuzzy goal programming with preemptive priorities, with Archimedean weights, and with the maximization of the membership function corresponding to the minimum goal. Examples are also provided.     

Symbolic integration of logic in MILP branch and bound methods for the synthesis of process networks

This paper deals with the branch and bound solution of process synthesis problems that are modelled as mixed-integer linear programming (MILP) problems. The symbolic integration of logic relations between potential units in a process network is proposed in the LP based branch and bound method to expedite the search for the optimal solution. The objective of this integration is to reduce the number of nodes that must be enumerated by using the logic to decide on the branching of variables and to determine by symbolic inference whether additional variables can be fixed at each node. The important feature of this approach is that it does not require additional constraints in the MILP and the logic can be systematically generated for process networks. Strategies for performing the integration are proposed that use the disjunctive and conjunctive normal form representations of the logic, respectively. Computational results will be presented to illustrate that substantial savings can be achieved.     

Goal programming models and their duality relations for use in evaluating security portfolio and regression relations

The literature on multiple objective programming contains numerous examples in which goal programming is used to plan a selection of inputs to secure desired outputs that will conform ‘as closely as possible’ to a collection of (possibly conflicting) objectives. In this paper the orientation is changed from selection to evaluation and the dual variables associated with goal programming are brought into play for this purpose. The body of the paper is devoted to an example in portfolio planning modelled along lines like those used by Konno and Yamazaki where closeness to risk and return objective is measured in sums of absolute deviations. An appendix then shows how such a use of dual variables may be applied to evaluate least absolute value (LAV) regressions relative to their sensitivity to data variations. Simple numerical examples are used to illustrate the potential uses of these dual variable values for evaluation in more complex situations that include determining whether an efficiency frontier has been attained.     

Goal Programming and Agricultural Planning

Goal Programming is similar in structure to linear programming, but offers a more flexible approach to planning problems by allowing a number of goals which are not necessarily compatible to be taken into account, simultaneously. The use of linear programming in farm planning is reviewed briefly. Consideration is given to published evidence of the goals of farmers, and ways in which these goals can be represented. A goal programming model of a 600 acre mixed farm is described and evaluated. Advantages and shortcomings of goal programming in relation to linear programming are considered. It is found that goal programming can be used as a means of generating a range of possible solutions to the planning problem.     

A fuzzy goal programming method with imprecise goal hierarchy

Two most widely used approaches to treating goals of different importance in goal programming (GP) are: (1) weighted GP, where importance of goals is modelled using weights, and (2) preemptive priority GP, where a goal hierarchy is specified implying infinite trade-offs among goals placed in different levels of importance. These approaches may be too restrictive in modelling of real life decision making problems. In this paper, a novel fuzzy goal programming method is proposed, where the hierarchical levels of the goals are imprecisely defined. The imprecise importance relations among the goals are modelled using fuzzy relations. An additive achievement function is defined, which takes into consideration both achievement degrees of the goals and degrees of satisfaction of the fuzzy importance relations. Examples are given to illustrate the proposed method.     

Binary fuzzy goal programming

Goal programming is an important technique for solving many decision/management problems. Fuzzy goal programming involves applying the fuzzy set theory to goal programming, thus allowing the model to take into account the vague aspirations of a decision-maker. Using preference-based membership functions, we can define the fuzzy problem through natural language terms or vague phenomena. In fact, decision-making involves the achievement of fuzzy goals, some of them are met and some not because these goals are subject to the function of environment/resource constraints. Thus, binary fuzzy goal programming is employed where the problem cannot be solved by conventional goal programming approaches. This paper proposes a new idea of how to program the binary fuzzy goal programming model. The binary fuzzy goal programming model can then be solved using the integer programming method. Finally, an illustrative example is included to demonstrate the correctness and usefulness of the proposed model.     

Dynamic Programming in Digital Communications: Viterbi Decoding to Turbo Multiuser Detection

Many important signal processing tasks in digital communications are based on integer programming problems whose raw complexity is extremely high. Such problems include the decoding of convolutional codes, channel equalization, multiuser detection, and the joint performance of these tasks. In each of these problems, the high complexity arises from the need to perform simultaneous processing on long sequences of finite-valued symbols in order to optimally detect or decode them. Fortunately, the complexity of these optimization problems can often be greatly reduced through the use of dynamic programming, which efficiently finds optimal [e.g., maximum likelihood (ML) or maximum a posteriori probability (MAP)] decisions in long sequences of symbols. This paper reviews four decades of progress in this area: the Viterbi algorithm for ML decoding of convolutional codes of the 1960s; the ML sequence detectors for channel equalization and the BCJR algorithm for MAP decoding of convolutional codes of the 1970s; the ML and MAP multiuser detectors of the 1980s; and combinations of these through the turbo processing of the 1990s.     

Goal programming using multiple objective tabu search

Goal programming (GP) is one of the most commonly used mathematical programming tools to model multiple objective optimisation (MOO) problems. There are numerous MOO problems of various complexity modelled using GP in the literature. One of the main difficulties in the GP is to solve their mathematical formulations optimally. Due to difficulties imposed by the classical solution techniques there is a trend in the literature to solve mathematical programming formulations including goal programmes, using the modern heuristics optimisation techniques, namely genetic algorithms (GA), tabu search (TS) and simulated annealing (SA). This paper uses the multiple objective tabu search (MOTS) algorithm, which was proposed previously by the author to solve GP models. In the proposed approach, GP models are first converted to their classical MOO equivalent by using some simple conversion procedures. Then the problem is solved using the MOTS algorithm. The results obtained from the computational experiment show that MOTS can be considered as a promising candidate tool for solving GP models.     

Time Schedule Optimization of Satellites

For an effective monitoring of environmental activities by a satellite, the main goal is to achieve maximum coverage throughout a short period of time. Due to technical restrictions of the satellite hardware, it is only possible to use the camera for a limited time period per orbit, while rotating around the Earth. The main goal is to determine the time intervals for which the coverage with respect to given target areas is maximized and the restrictions such as contact to ground stations are satisfied. To obtain a continuous objective function for the sequential quadratic programming method, the coverage is modelled by polygons integrated on a sphere. For acceptable computing times, special investigations had to be made in order to achieve fast gradient computations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computationally Efficient Solution of a Multiproduct,Two-Stage Distribution—Location Problem

A two-stage distribution planning problem, in which customers are to be served with different commodities from a number of plants, through a number of intermediate warehouses is addressed. The possible locations for the warehouses are given. For each location, there is an associated fixed cost for opening the warehouse concerned, as well as an operating cost and a maximum capacity. The demand of each customer for each commodity is known, as are the shipping costs from a plant to a possible warehouse and thereafter to a customer. It is required to choose the locations for opening warehouses and to find the shipping schedule such that the total cost is minimized. The problem is modelled as a mixed-integer programming problem and solved by branch and bound. The lower bounds are calculated through solving a minimum-cost, multicommodity network flow problem with capacity constraints. Results of extensive computational experiments are given.     

Lower bound for the mean project completion time in dynamic PERT networks

We apply the stochastic dynamic programming to obtain a lower bound for the mean project completion time in a PERT network, where the activity durations are exponentially distributed random variables. Moreover, these random variables are non-static in that the distributions themselves vary according to some randomness in society like strike or inflation. This social randomness is modelled as a function of a separate continuous-time Markov process over the time horizon. The results are verified by simulation.     

A capacitated hub location problem in freight logistics multimodal networks

In this paper we deal with a capacitated hub location problem arising in a freight logistics context; in particular, we have the need of locating logistics platforms for containers travelling via road and rail. The problem is modelled on a weighed multimodal network. We give a mixed integer linear programming model for the problem, having the goal of minimizing the location and shipping costs. The proposed formulation presents some novel features for modelling capacity bounds that are given both for the candidate hub nodes and the arcs incident to them; further, the containerised origin-destination (\(o-d)\) demand can be split among several platforms and different travelling modes. Note that here the network is not fully connected and only one hub for each \(o-d\) pair is used, serving both to consolidate consignments on less transport connections and as reloading point for a modal change. Results of an extensive computational experimentation performed with randomly generated instances of different size and capacity values are reported. In the test bed designed to validate the proposed model all the instances up to 135 nodes and 20 candidate hubs are optimally solved in few seconds by the commercial solver CPLEX 12.5.     

Bounds for codes by semidefinite programming

Delsarte’s method and its extensions allow one to consider the upper bound problem for codes in two-point homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that using as variables power sums of distances, this problem can be considered as a finite semidefinite programming problem. This method allows one to improve some linear programming upper bounds. In particular, we obtain new bounds of one-sided kissing numbers.     

Goal Programming: the RPMS Network Approach

This paper presents an application of the Resource Planning and Management Systems (RPMS) network approach to the goal programming (GP) problem as an alternative to several other approaches used in the GP process. The RPMS approach to solving the GP problem is illustrated through an example. Sensitivity analysis is also discussed.     

A103 (1997) 18–27 in heavily congested cities

A minimum cost network flow model is proposed to deal with the problem of bus trip scheduling in heavily congested cities. Real instances of scheduling problems in Bangkok are analysed and solved by means of a particular network formulation. The produced solutions are at least as good as those generated by known procedures. Characterisations which make the network approach superior, are very fast solution times, the ease with which the approach can be explained to management, flexibility, and the possibility of doing sensitivity analysis. The modelling technique can be easily extended to situations more complex than a single bus route, and can be useful in less congested urban environments than Bangkok. The approach also illustrates how goal programming principles can be applied in a network model.     

Goal network programs: A specialized algorithm and an application

This paper presents a specialized network procedure for the solution of pure goal network programs with preemptive priorities. The specialization solves such goal network programs efficiently since it requires a modification only in the pricing rule of the network simplex algorithm. The algorithm is used to solve multiple goal network programs that arise in the assignment of naval personnel to jobs. Computational experience indicates that the specialization dominates the sequential linear goal programming procedure.     

用AHP方法确定多目标的优先等级

本文给出了一种确定目标规划问题中多目标的优先等级的方法—— AHP方法 .     

A penalty linear programming method using reduced-gradient basis-exchange techniques

We offer a variant of the piecewise-linear penalty-function approach to linear programming which was proposed by Conn [5]. Our variant makes use of computational techniques which are more closely related to those in existing computer codes for linear programming and which can be more readily adapted for large sparse problems that were the techniques described by Conn. An experimental code for small dense problems has been prepared and some experience with it is reported.     

Revised multi-choice goal programming for multi-period,multi-stage inventory controlled supply chain model with popup stores in Guerrilla marketing

In this paper, we consider a supply chain network design problem with popup stores which can be opened for a few weeks or months before closing seasonally in a marketplace. The proposed model is multi-period and multi-stage with multi-choice goals under inventory management constraints and formulated by 0–1 mixed integer linear programming. The design tasks of the problem involve the choice of the popup stores to be opened and the distribution network design to satisfy the demand with three multi-choice goals. The first goal is minimization of the sum of transportation costs in all stages; the second is to minimization of set up costs of popup stores; and the third goal is minimization of inventory holding and backordering costs. Revised multi-choice goal programming approach is applied to solve this mixed integer linear programming model. Also, we provide a real-world industrial case to demonstrate how the proposed model works.     

A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities

Airline seat inventory control is the allocation of seats in the same cabin to different fare classes such that the total revenue is maximized. Seat allocation can be modelled as dynamic stochastic programs, which are computationally intractable in network settings. Deterministic and probabilistic mathematical programming models are therefore used to approximate dynamic stochastic programs. The probabilistic model, which is the focus of this paper, has a nonlinear objective function, which makes the solution of large-scale practical instances with off-the-shelf solvers prohibitively time consuming. In this paper, we propose a Lagrangian relaxation (LR) method for solving the probabilistic model by exploring the fact that LR problems are decomposable. We show that the solutions of the LR problems admit a simple analytical expression which can be resolved directly. Both the booking limit policy and the bid-price policy can be implemented using this method. Numerical simulations demonstrate the effectiveness of the proposed method.