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This chapter presents the survey of selected linear and mixed integer programming multi-objective portfolio optimization. The definitions of selected percentile risk measures are presented. Some contrasts and similarities of the different types of portfolio formulations are drawn out. The survey of multi-criteria methods devoted to portfolio optimization such as weighting approach, lexicographic approach, and reference point method is also presented. This survey presents the nature of the multi-objective portfolio problems focuses on a compromise between the construction of objectives, constraints, and decision variables in a portfolio and the problem complexity of the implemented mathematical models. There is always a trade-off between computational time and the size of an input data, as well as the type of mathematical programming formulation with linear and/or mixed integer variables.

This paper seeks to construct a model for inventory management for multiple periods. The model considers not only the usual parameters, but also price quantity discount, storage and batch size constraints.

Mixed 0‐1 integer programming is applied to solve the multi‐period inventory problem and to determine an appropriate inventory level for each period. The total cost of materials in the system is minimized and the optimal purchase amount in each period is determined.

The proposed model is applied in colour filter inventory management in thin film transistor‐liquid crystal display (TFT‐LCD) manufacturing because colour filter replenishment has the characteristics of price quantity discount, large product size, batch‐sized purchase and forbidden shortage in the plant. Sensitivity analysis of major parameters of the model is also performed to depict the effects of these parameters on the solutions.

The proposed model can be tailored and applied to other inventory management problems.

Although many mathematical models are available for inventory management, this study considers some special characteristics that might be present in real practice. TFT‐LCD manufacturing is one of the most prosperous industries in Taiwan, and colour‐filter inventory management is essential for TFT‐LCD manufacturers for achieving competitive edge. The proposed model in this study can be applied to fulfil the goal.

Negative effects of habitat isolation that arise from landscape fragmentation can be mitigated, by connecting natural areas through a network of habitat corridors. To increase the permeability of a given network, i.e. to decrease the resistance to animal movements through this network, often many developments can be made. The available financial resources being limited, the most effective developments must be chosen. This optimization problem, suggested in Finke and Sonnenschein, can be treated by heuristics and simulation approaches, but the method is heavy and the obtained solutions are sub‐optimal. The aim of the paper is to show that the problem can be efficiently solved to optimality by mathematical programming.

The moves of the individual in the network are modeled by an absorbing Markov chain and the development problem is formulated as a mixed‐integer quadratic program, then this program is linearized, and the best developments to make are determined by mixed‐integer linear programming.

First, the approach allows the development problem to be solved to optimality contrary to other methods. Second, the definition of the mathematical program is relatively simple, and its implementation is immediate by using standard, commercially available, software. Third, as it is well known with mixed‐integer linear programming formulation it is possible to add new constraints easily if they are linear (or can be linearized).

With a view to propose a simple and efficient tool to solve a difficult combinatorial optimization problem arising in the improvement of permeability across habitat networks, the approach has been tested on simulated habitat networks. The research does not include the study of some precise species movements in a real network.

The results provide a simple and efficient decision‐aid tool to try to improve the permeability of habitat networks.

The joint use of mathematical programming techniques and Markov chain theory is used to try to lessen the negative effects of landscape fragmentation.

The paper introduces a resource-based linear programming model for resource optimization in small innovative enterprises (SIE).

The model is grounded on resource-based view on the firm and dynamic capabilities approach. Linear programming technique is used to provide the actual framework to the resource-based model.

The paper introduces a new resource-based linear programming model for resource optimization in small innovative enterprises. The conceptual model is grounded on resource-based view (RBV) and dynamic capabilities strategy. The RVB of firm and firm strategy is based on the concept of economic rent. Linear programming technique is used to provide the actual framework to the resource-based model. In developing the versatility concept, study suggests a distinct sight regarding resource fungibility. Study classifies resources into multipliable, rentable and expendable resources to increases adequacy of the model. The developed model includes both tangible and intangible assets such as human capital. The survival rate of SIE in the early stages of life cycle is very low due to the competition among SIEs. In this regard, the greatest advancement of the developed resource-based linear programming model is its simplicity and versatility which is much desirable for the SIE especially in their initial stages of the life cycle. Kelliher and Reinl (2009) argued that micro firms have unique advantage over bigger firms in following term: rate of learning or redeployment of strategy in micro firms is faster than the rate of change in their environment. One very significant feature of the developed resource-based linear programming model is that mathematically the proposed model could easily be transformed into mixed integer or stochastic linear programming models to meet the time variant requirement of small firms especially when it expands its operation.

The survival rate of SIE in the early stages of life cycle is very low due to the competition among SIEs. In this regard, the greatest advancement of the developed resource-based linear programming model is its simplicity and versatility which is much desirable for the SIE especially in their initial stages of the life cycle. Kelliher and Reinl (2009) argued that micro firms have unique advantage over bigger firms in following term: rate of learning or redeployment of strategy in micro firms is faster than the rate of change in their environment. One very significant feature of the developed resource-based linear programming model is that mathematically the proposed model could easily be transformed into mixed integer or stochastic linear programming models to meet the time variant requirement of small firms especially when it expands its operation.

One very significant contribution of the present study is that the study develops a new resource-based model for SIE especially for the SIE in the initial stages of the life cycle, to gain competitive advantages. Furthermore, the present study contributes to the existing literature in strategy at least in three senses as mentioned below: 1. further addition of SIE research based on the RBV and dynamic capabilities in the strategy literature 2. in developing the versatility concept, the study suggests a distinct sight regarding resource fungibility and it classifies resources into three categories as follows: multipliable, rentable and expendable resources to increases adequacy of the model. 3. Finally, the study introduces a new resource-based linear programming model for SIE resources allocation. To the best of author’s knowledge, no such similar model is introduced by any previous studies for small firm. The greatest advancement of the developed resource-based linear programming model is its simplicity and versatility.

This chapter presents the portfolio optimization problem formulated as a multi-criteria mixed integer program. Weighting and lexicographic approach are proposed. The portfolio selection problem considered is based on a single-period model of investment. An extension of the Markowitz portfolio optimization model is considered, in which the variance has been replaced with the Value-at-Risk (VaR). The VaR is a quantile of the return distribution function. In the classical Markowitz approach, future returns are random variables controlled by such parameters as the portfolio efficiency, which is measured by the expectation, whereas risk is calculated by the standard deviation. As a result, the classical problem is formulated as a quadratic program with continuous variables and some side constraints. The objective of the problem considered in this chapter is to allocate wealth on different securities to maximize the weighted difference of the portfolio expected return and the threshold of the probability that the return is less than a required level. The auxiliary objectives are minimization of risk probability of portfolio loss and minimization of the number of security types in portfolio. The four types of decision variables are introduced in the model: a continuous wealth allocation variable that represents the percentage of wealth allocated to each asset, a continuous variable that prevents the probability that return of investment is not less than required level, a binary selection variable that prevents the choice of portfolios whose VaR is below the minimized threshold, and a binary selection variable that represents choice of stocks in which capital should be invested. The results of some computational experiments with the mixed integer programming approach modeled on a real data from the Warsaw Stock Exchange are reported.

This chapter presents selected multiobjective methods for multiperiod portfolio optimization problem. Portfolio models are formulated as multicriteria mixed integer programs. Reference point method together with weighting approach is proposed. The portfolio selection problem considered is based on a multiperiod model of investment, in which the investor buys and sells securities in successive investment periods. The problem objective is to allocate the wealth on different securities to optimize the portfolio expected return, the probability that the return is not less than a required level. Multiobjective methods were used to find tradeoffs between risk, return, and the number of securities in the portfolio. In computational experiments the data set of daily quotations from the Warsaw Stock Exchange were used.

Describes a procedure for modelling the costs of production and distribution between several production facilities with economies of scale and many customers who are widely dispersed. The problem takes the form of a large transportation problem on which is superimposed a cost minimization problem involving variable production quantities. These costs involve fixed costs for initiating production and variable costs with diminishing returns to scale. Models the problem as a non‐linear integer programming problem and then solves it using a recently developed non‐linear integer algorithm. Describes two applications in Australia and New Zealand and illustrates how comparison with a mixed‐integer linear programming formulation shows a significant improvement.

This chapter presents application of multi-criteria mathematical programming models by integer and mixed-integer programming for optimal allocation of workers among supporting services in a hospital. The services include logistics, inventory management, financial management, operations management, medical analysis, etc. The optimality criteria of the problem are minimization of operational costs of supporting services subject to some specific constraints. The constraints represent specific conditions for resource allocation in a hospital. The overall problems are formulated as assignment models, where the decision variables represent the assignment of people to various jobs. Numerical examples are presented. Some computational results modeled on a real data from a hospital in Poland are reported.

This chapter presents two multicriteria optimization models with bi and triple objectives solved with weighted-sum approach. Solved problems are allocation of personnel in a health care institution. To deal with these problems, mixed integer programming formulation has been applied. Results have shown the impact of problem parameter change for importance of the different objectives. Presented problems have been solved using AMPL programming language with solver CPLEX v9.1, with the use of branch and bound method.

Introduction Operations research, i.e. the application of scientific methodology to operational problems in the search for improved understanding and control, can be said to have started with the application of mathematical tools to military problems of supply bombing and strategy, during the Second World War. Post‐war these tools were applied to business problems, particularly production scheduling, inventory control and physical distribution because of the acute shortages of goods and the numerical aspects of these problems.