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In this study, we analyze the power of the individual return-to-volatility security performance heuristic (ri/stdi) to simplify the identification of securities to buy and, consequently, to form the optimal no short sales mean–variance portfolios. The heuristic ri/stdi is powerful enough to identify the long and shorts sets. This is due to the positive definiteness of the variance–covariance matrix – the key is to use the heuristic sequentially. At the investor level, the heuristic helps investors to decide what securities to consider first. At the portfolio level, the heuristic may help us find out whether it is a good idea to invest in equity to begin with. Our research may also help to integrate individual security analysis into portfolio optimization through improved security rankings.

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.

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.

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.

Should we include cryptocurrency in risky portfolio investing? Bitcoin, given its status as the leader of cryptocurrencies and a speculative asset due to its non-dividend-paying trait and high volatility as well as high returns, poses an interesting question whether it can also be beneficial in a portfolio of risky assets. In order to find an answer, we revisit the conventional dual objective of minimizing risk and maximizing expected return for risky assets. Various models are tested to analyze the risk-return trade-off of risky portfolios including Bitcoin. Given an initial budget for a finite portfolio, the cumulative filtration yields the expected return and the covariance matrix. With the addition of Bitcoin, we compare the performance of the portfolio generated from the optimization models and technical analysis. The main implications are follows: (1) risk tolerance and diversification constraints are the key factors in portfolio optimization; (2) including cryptocurrency enhances portfolio returns; and (3) the Markowitz model (Kataoka’s and conditional value-at-risk models) recommends to fully weigh (unload) Bitcoin in (from) the portfolio.

Purpose: The present study aims to test the Quadratic Programming model for Optimal Portfolio selection empirically.

Need for the Study: All the investors who buy financial products are motivated to obtain higher profits or, in other words, to maximise their returns. However, the high returns are often accompanied by higher risks, and avoiding such risks has become the primary concern for all investors. There is a great need for such a model to maximise profits and minimise risk, which can help design an investment portfolio with minimum risk and maximum return. The Quadratic Programming model is one such model which can be applied for selected shares to build an optimised portfolio.

Methodology: This study optimises the stock samples using a two-level screening of correlation coefficient and coefficient of variation. The monthly closing prices of the NSE-listed Indian pharmaceutical stocks from December 2019 to January 2022 have been used as sample data. The Lagrange Multiplier method is used to apply the model to achieve the optimal portfolio solution. Based on the market reality, the transaction costs have also been considered. The Quadratic programming model is further optimised to achieve the optimal portfolio for the select stocks.

Findings: The traditional portfolio theory and the modified quadratic model gives similar and consistent results. In other words, the modified quadratic model asserts the accuracy of the conventional portfolio model. The portfolio constructed in the present study gives a return much higher than the return of the benchmark portfolio of Nifty Fifty, indicating the usefulness of applying the Quadratic Programming model.

Practical Implications: The construction of an optimal portfolio using the traditional or modified Quadratic model can help investors make rational investment decisions for better returns with lower risks.

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.