A multi-objective optimal allocation model for irrigation water resources under multiple uncertainties

This paper proposed a multi-objective optimal water resources allocation model under multiple uncertainties. The proposed model integrated the chance-constrained programming, semi-infinite programming and integer programming into an interval linear programming. Then, the developed model is applied to irrigation water resources optimal allocation system in Minqin’s irrigation areas, Gansu Province, China. In this study, the irrigation areas’ economic benefits, social benefits and ecological benefits are regarded as the optimal objective functions. As a result, the optimal irrigation water resources allocation plans of different water types (surface water and groundwater) under different hydrological years (wet year, normal year and dry year) and probabilities are obtained. The proposed multi-objective model is unique by considering water-saving measures, irrigation water quality impact factors and the dynamic changes of groundwater exploitable quantity in the irrigation water resources optimal allocation system under uncertain environment. The obtained results are valuable for supporting the adjustment of the existing irrigation patterns and identify a desired water-allocation plan for irrigation under multiple uncertainties.     

Project portfolio designing using data envelopment analysis and De Novo optimisation

In a rapidly evolving economic world, projects become tools to support organization goals. Project portfolio is set of all projects that are implemented in the organisation at a time. Possible projects are characterized by sets of inputs and outputs, where inputs are resources for project realisation and outputs measure multiple goals of the organisation. The data envelopment analysis (DEA) is an appropriate approach to select efficient projects. The organisation has its total resources in limited quantities. Designing a portfolio of efficient projects not exceeding the limited resources does not always lead to the most efficient portfolio. De Novo optimisation is an approach for designing optimal systems by reshaping the feasible set. The paper proposes a new approach for project portfolio designing based on a systemic combination of DEA model and De Novo optimisation approach. A total available budget is a restriction on project portfolio. The proposed concept provides designing of optimal project portfolio with the minimal budget. Performance measures of the designed project portfolio are the efficiency of the portfolio and the effectiveness of outputs. Possible extensions of the concept are formulated and discussed.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

A multi-level Taguchi-factorial two-stage stochastic programming approach for characterization of parameter uncertainties and their interactions: An application to water resources management

This paper presents a multi-level Taguchi-factorial two-stage stochastic programming (MTTSP) approach for supporting water resources management under parameter uncertainties and their interactions. MTTSP is capable of performing uncertainty analysis, policy analysis, factor screening, and interaction detection in a comprehensive and systematic way. A water resources management problem is used to demonstrate the applicability of the proposed approach. The results indicate that interval solutions can be generated for the objective function and decision variables, and a variety of decision alternatives can be obtained under different policy scenarios. The experimental data obtained from the Taguchi’s orthogonal array design are helpful in identifying the significant factors affecting the total net benefit. Then the findings from the multi-level factorial experiment reveal the latent interactions among those important factors and their curvature effects on the model response. Such a sequential strategy of experimental designs is useful in analyzing the interactions for a large number of factors in a computationally efficient manner.     

Optimal budget for system design series network DEA model

Wei and Chang (2011a) developed optimal system design (OSD) data envelopment analysis (DEA) models to design a decision-making unit (DMU)’s optimal system, in which the DMU could encounter the well-known economic phenomenon of budget congestion. To show how to verify the optimal budget and budget congestion, they develop a solution method. In this paper, we note that their method is incorrect for the OSD network DEA model in general. A new approach is developed to derive the DMU’s corresponding optimal budgets and to check for the existence of budget congestion not only for the OSD DEA models but also for the OSD network DEA models. In addition, the proposed approach is computationally economical. Finally, two numerical examples are used to illustrate the proposed approach.     

Interval goal programming for S-shaped penalty function

The approach of Jones and Tamiz (1995) [Jones, D.F., Tamiz, M., 1995. Expanding the flexibility of goal programming via preference modeling techniques. Omega 23, 41–48] has been accepted as the most efficient approach in the field of interval goal programming (IGP). Although several modifications to the original approach have been proposed recently [Vitoriano, B., Romero, C., 1999. Extended interval goal programming. Journal of the Operational Research Society 50, 1280–1283; Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396], all of them cannot formulate IGP with an S-shaped penalty function. In order to improve the utility of IGP, we extend the model of Chang (2006) [Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396] to be able to model an S-shaped penalty function. The newly formulated model is more concise and compact than the method of Li and Yu (2000) and it can easily be applied to a decision problem with the S-shaped penalty function. Finally, an illustrative example (i.e. how to build an appropriate E-learning system) is included for demonstrating the usefulness of the proposed model.     

Marginal Indemnification Function formulation for optimal reinsurance

In this paper, we propose to combine the Marginal Indemnification Function (MIF) formulation and the Lagrangian dual method to solve optimal reinsurance model with distortion risk measure and distortion reinsurance premium principle. The MIF method exploits the absolute continuity of admissible indemnification functions and formulates optimal reinsurance model into a functional linear programming of determining an optimal measurable function valued over a bounded interval. The MIF method was recently introduced to analyze the reinsurance model but without premium budget constraint. In this paper, a Lagrangian dual method is applied to combine with MIF to solve for optimal reinsurance solutions under premium budget constraint. Compared with the existing literature, the proposed integrated MIF-based Lagrangian dual method provides a more technically convenient and transparent solution to the optimal reinsurance design. To demonstrate the practicality of the proposed method, analytical solution is derived on a particular reinsurance model that involves minimizing Conditional Value at Risk (a special case of distortion function) and with the reinsurance premium being determined by the inverse-S shaped distortion principle.     

An integrated model for screening cargo containers

This paper focuses on detecting nuclear weapons on cargo containers using port security screening methods, where the nuclear weapons would presumably be used to attack a target within the United States. This paper provides a linear programming model that simultaneously identifies optimal primary and secondary screening policies in a prescreening-based paradigm, where incoming cargo containers are classified according to their perceived risk. The proposed linear programming model determines how to utilize primary and secondary screening resources in a cargo container screening system given a screening budget, prescreening classifications, and different device costs. Structural properties of the model are examined to shed light on the optimal screening policies. The model is illustrated with a computational example. Sensitivity analysis is performed on the ability of the prescreening in correctly identifying prescreening classifications and secondary screening costs. Results reveal that there are fewer practical differences between the screening policies of the prescreening groups when prescreening is inaccurate. Moreover, devices that can better detect shielded nuclear material have the potential to substantially improve the system’s detection capabilities.     

Optimal value range in interval linear programming

We deal with the linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. We present a general approach to the situation the feasible set is described by an arbitrary linear interval system. Moreover, certain dependencies between the constraint matrix coefficients can be involved. As long as we are able to characterize the primal and dual solution set (the set of all possible primal and dual feasible solutions, respectively), the bounds of the objective function result from two nonlinear programming problems. We demonstrate our approach on various cases of the interval linear programming problem (with and without dependencies).     

An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters

Real engineering design problems are generally characterized by the presence of many often conflicting and incommensurable objectives. Naturally, these objectives involve many parameters whose possible values may be assigned by the experts. The aim of this paper is to introduce a hybrid approach combining three optimization techniques, dynamic programming (DP), genetic algorithms and particle swarm optimization (PSO). Our approach integrates the merits of both DP and artificial optimization techniques and it has two characteristic features. Firstly, the proposed algorithm converts fuzzy multiobjective optimization problem to a sequence of a crisp nonlinear programming problems. Secondly, the proposed algorithm uses H-SOA for solving nonlinear programming problem. In which, any complex problem under certain structure can be solved and there is no need for the existence of some properties rather than traditional methods that need some features of the problem such as differentiability and continuity. Finally, with different degree of α we get different α-Pareto optimal solution of the problem. A numerical example is given to illustrate the results developed in this paper.     

Uncertainty and value of information when allocating resources within and between healthcare programmes

A two-stage stochastic mathematical programming formulation has been developed to optimally allocate resources within and between healthcare programmes when there is an exogenous budget and the parameters of the healthcare models are variable and uncertain. This formulation solves the optimal resource allocation problem and calculates the expected value of acquiring additional information to resolve the uncertainties within the allocation. It is shown that the proposed formulation has several advantages over the chance constrained and robust mathematical programming methods.     

Decentralization of responsibility for site decontamination projects: A budget allocation approach

A major problem currently confronting central governments is how to optimally allocate resources for decontamination of polluted sites. ‘Optimally’ here refers to obtaining maximum environmental benefits with the limited resources available. An important issue in budget allocation is that of decentralization, given the magnitude of the information flows between regional and central level necessary in a fully centralized approach. This paper investigates the use of mathematical programming models to support allocation procedures to obtain maximum environmental effectiveness and economic efficiency. We consider the situation where regional authorities provide limited, summary information to the central government, which then allocates budgets. The central government aims to maximize total environmental benefits, subject to a central budget constraint (and constraints on other resources). The problem can be formulated as a mixed integer programming problem, but the size of the problem precludes the search for optimal solutions. We present an effective heuristic and include computational results on its performance.     

A facility location and installation of resources model for level of repair analysis

The Level of Repair Analysis – LORA – is an analytic methodology aimed at determining: (i) the optimal location of facilities that compose a maintenance structure; (ii) the quantity of required resources in each facility; and (iii) the best repair policies, i.e., rules that determine if a given component should be discarded or repaired, and where those actions should take place. This work presents a mixed-integer programming model for LORA that is more comprehensive than others in the literature, being suitable to many practical situations. The model was applied to 15 substantial real world problems, and considering distinct maintenance policies to some of them, resulted in 22 different solutions, all of which could be achieved by a commercial Mixed-Integer Programming (MIP) solver in reasonable times.     

Simulation and optimization for crop water allocation based on crop water production functions and climate factor under uncertainty

In this paper, the uncertainty methods of interval and functional interval are introduced in the research of the uncertainty of crop water production function itself and optimal allocation of water resources in the irrigation area. The crop water production functions in the whole growth period under uncertainty and the optimal allocation of water resources model in the irrigation area under uncertainty are established, and the meteorological factor is considered in the model. It can promote the practical application of the uncertain methods, reflect the complexity and uncertainty of the actual situation, and provide more reliable scientific basis for using water resources economically, fully improving irrigation efficiency, and keeping the sustainable development of the irrigated area. This approach has important value on theoretical and practical for the optimal irrigation schedule, and has very broad prospects for research and development to other related agriculture water management.     

An interval-parameter fuzzy two-stage stochastic program for water resources management under uncertainty

This study presents an interval-parameter fuzzy two-stage stochastic programming (IFTSP) method for the planning of water-resources-management systems under uncertainty. The model is derived by incorporating the concepts of interval-parameter and fuzzy programming techniques within a two-stage stochastic optimization framework. The approach has two major advantages in comparison to other optimization techniques. Firstly, the IFTSP method can incorporate pre-defined water policies directly into its optimization process and, secondly, it can readily integrate inherent system uncertainties expressed not only as possibility and probability distributions but also as discrete intervals directly into its solution procedure. The IFTSP process is applied to an earlier case study of regional water resources management and it is demonstrated how the method efficiently produces stable solutions together with different risk levels of violating pre-established allocation criteria. In addition, a variety of decision alternatives are generated under different combinations of water shortage.     

An Iterated Local Search for the Budget Constrained Generalized Maximal Covering Location Problem

Capacitated covering models aim at covering the maximum amount of customers’ demand using a set of capacitated facilities. Based on the assumptions made in such models, there is a unique scenario to open a facility in which each facility has a pre-specified capacity and an operating budget. In this paper, we propose a generalization of the maximal covering location problem, in which facilities have different scenarios for being constructed. Essentially, based on the budget invested to construct a given facility, it can provide different service levels to the surrounded customers. Having a limited budget to open the facilities, the goal is locating a subset of facilities with the optimal opening scenario, in order to maximize the total covered demand and subject to the service level constraint. Integer linear programming formulations are proposed and tested using ILOG CPLEX. An iterated local search algorithm is also developed to solve the introduced problem.     

Checking weak optimality of the solution to linear programming with interval right-hand side

The interval linear programming (IvLP) is a method for decision making under uncertainty. A weak feasible solution to IvLP is called weakly optimal if it is optimal for some scenario of the IvLP. One of the basic and difficult tasks in IvLP is to check whether a given point is weak optimal. In this paper, we investigate linear programming problems with interval right-hand side. Some necessary and sufficient conditions for checking weak optimality of given feasible solutions are established, based on the KKT conditions of linear programming. The proposed methods are simple, easy to implement yet very effective, since they run in polynomial time.     

Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm

This paper deals with chance constraints based reliability stochastic optimization problem in the series system. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources. The assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval (0, 1). However, in real life situations, the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number. In this work, we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components. Then, the chance constraints of the problem are converted into the equivalent deterministic form. The transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by Big-M penalty technique. Then to solve this problem, we have developed a real coded genetic algorithm (GA) for integer variables with tournament selection, uniform crossover and one-neighborhood mutation. To illustrate the model two numerical examples have been solved by our developed GA. Finally to study the stability of our developed GA with respect to the different GA parameters, sensitivity analyses have been done graphically.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.