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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 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.

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 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.

Supply chain is an important aspect for all the companies and can affect many aspects of companies. Especially the disruption in supply chain is causing huge impacts and consequences that are difficult to deal with. This chapter presents a review of selected multiple criteria problems used in supply chain optimization. Research analyzed the multiple criteria decision-making methods to tackle the problem of supplier evaluation and selection. It also focuses on the problem of supply chain when a disruption happens and presents strategies to deal with the issue of disruptions in supply chain and how to mitigate the impact of disruptions. Prevention, response, protection, and recovery strategies are explained. Practical part is focused in the risk-averse models to minimize expected worst-case scenario by single sourcing. Computational experiments for practical examples have been solved using CPLEX solver.

Economic incentives, government regulations, and customer perspective on environmental consciousness (EC) are driving more and more companies into product recovery business, which forms the basis for a reverse supply chain. A reverse supply chain consists a series of activities that involves retrieving used products from consumers and remanufacturing (closed-loop) or recycling (open-loop) them to recover their leftover market value. Much work has been done in the areas of designing forward and reverse supply chains; however, not many models deal with the transshipment of products in multiperiods. Linear physical programming (LPP) is a newly developed method whose most significant advantage is that it allows a decision-maker to express his/her preferences for values of criteria for decision-making in terms of ranges of different degrees of desirability but not in traditional form of weights as in techniques such as analytic hierarchy process, which is criticized for its unbalanced scale of judgment and failure to precisely handle the inherent uncertainty and vagueness in carrying out pair-wise comparisons. In this chapter, two multiperiod models are proposed for a remanufacturing system, which is an element of a Reverse Supply Chain (RSC), and illustrated with numerical examples. The first model is solved using mixed integer linear programming (MILP), while the second model is solved using linear physical programming. The proposed models deliver the optimal transportation quantities of remanufactured products for N-periods within the reverse supply chain.

Nowadays, the increase in the capacity of batteries has laid the foundations for a broader diffusion of electric mobility. However, electric mobility is causing a growing electricity demand as well as the need to increase the diffusion of suitable charging stations. Within these last challenges, drawing on the recent literature, this chapter provides a critical and wide-ranging review of papers dealing with the formulation of the problem of the localisation of electric vehicle (EV) charging points. This problem is approached considering the electric charging infrastructure technologies, localisation criteria and related methodologies. This review shows how the ‘electric mobility revolution’ applies the technological innovations provided by the energy supply systems, and the location of these systems within the urban contexts. Since the technological innovations have different options, achieving an international standard of charging systems is still far away. Moreover, as there are several criteria, parameters and methodologies, and some analytical approaches for the localisation of electric vehicle charging points, the formulation of the ‘localisation’ problem should require the application of multi-criteria analysis to be addressed. Finally, the results show that there is no consensus on technologies, criteria, and methodologies to be adopted. Therefore, this wide-ranging analysis of the literature would be useful to support possible benchmarking and systematisation accordingly.