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This chapter proposes a multiobjective model to design a Closed Loop Supply Chain (CLSC) network. The first objective is to minimize the total cost of the network, while the second objective is to minimize the carbon emission resulting from production, transportation, and disposal processes using carbon cap and carbon tax regularity policies. In the third objective, we maximize the service level of retailers by using maximum covering location as a measure of service level. To model the proposed problem, a physical programming approach is developed. This work contributes to the literature in designing an optimum CLSC network considering the service level objective and product substitution.

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 will detail the development of a multi-objective mathematical programming model for audit sampling of balances for accounts receivable. The nonlinear nature of the model structure will require the use of a nonlinear solution algorithm, such as the GRG or the genetic algorithm embedded in a Solver spreadsheet modeling system, to obtain appropriate results.

This chapter presents two optimization multicriteria models (bi and triple objective) using a lexicographic approach. Solved models are formulated as assignment of workers to different jobs or services of a real hospital, taking into account the available budget and requirements of each job. Presented problems have been solved using AMPL programming language with solver CPLEX v9.1, with the use of branch and bound method for mixed integer mathematical programming.

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.

This chapter presents a multi-criteria portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a single-objective linear program, as a bi-objective linear program, and as a triple-objective mixed integer program. The problem objective is to allocate the wealth on different securities to optimize the portfolio return. The portfolio approach has allowed the two popular financial engineering percentile measures of risk, value-at-risk (VaR) and conditional value-at-risk (CVaR) to be applied. The decision-maker can assess the value of portfolio return, the risk level, and the number of assets, and can decide how to invest in a real-life situation comparing with ideal (optimal) portfolio solutions. The concave efficient frontiers illustrate the trade-off between the conditional value-at-risk and the expected return of the portfolio. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments prove that both proposed linear and mixed integer programming approaches provide the decision-maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

This chapter considers warranty policies involving two attributes, such as the time elapsed since sale of the product and product usage at a given point in time. Examples of such policies are found for automobiles, where warranty may be invoked by the consumer if both time and usage are within specified warranty parameters when a product failure occurs. Here, we assume that usage and product age are related through a random variable, the usage rate, which may have a certain probabilistic distribution as influenced by consumer behavior patterns. Furthermore, product failure rate is influenced by the usage rate and product age as well as research and development expenditures per unit. It is assumed that, in production, there is a learning effect with time. The attained market share of a product will be influenced by the warranty policy parameters of warranty time and usage limit and also by the product price and product quality. An integrated model is developed to address multiobjective goals such as attainment of a specified level of market share and net profit per unit when manufacturing and warranty costs are taken into account. The impact of the goal priorities are investigated on the attained warranty policy parameters.

In this chapter, four bi-objective vehicle routing problems are considered. Weighted-sum approach optimization models are formulated with the use of mixed-integer programming. In presented optimization models, maximization of capacity of truck versus minimization of utilization of fuel, carbon emission, and production of noise are taken into account. The problems deal with real data for green logistics for routes crossing the Western Pyrenees in Navarre, Basque Country, and La Rioja, Spain.

Heterogeneous fleet of trucks is considered. Different types of trucks have not only different capacities, but also require different amounts of fuel for operations. Consequently, the amount of carbon emission and noise vary as well. Modern logistic companies planning delivery routes must consider the trade-off between the financial and environmental aspects of transportation. Efficiency of delivery routes is impacted by truck size and the possibility of dividing long delivery routes into smaller ones. The results of computational experiments modeled after real data from a Spanish food distribution company are reported. Computational results based on formulated optimization models show some balance between fleet size, truck types, and utilization of fuel, carbon emission, and production of noise. As a result, the company could consider a mixture of trucks sizes and divided routes for smaller trucks. Analyses of obtained results could help logistics managers lead the initiative in environmental conservation by saving fuel and consequently minimizing pollution. The computational experiments were performed using the AMPL programming language and the CPLEX solver.