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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 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 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 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 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.

This paper investigates a problem in a reverse logistics (RLs) network to decide whether to dispose of unsold goods in primary stores or re-commercialize them in outlet centers. By deducting the costs associated with each policy from its revenue, this study aims to maximize the profit from managing unsold goods.

A new mixed-integer linear programming model has been developed to address the problem, which considers the selling prices of products in primary and secondary stores and the costs of transportation, cross-docking and returning unwanted items. As a result of uncertain nature of the cost and time parameters, gray numbers are used to deal with it. In addition, an innovative uncertain solution approach for gray programming problems is presented that considers objective function satisfaction level as an indicator of optimism.

According to the results, higher costs, including transportation, cross-docking and return costs, make sending goods to outlet centers unprofitable and more goods are disposed of in primary stores. Prices in primary and secondary stores heavily influence the number of discarded goods. Higher prices in primary stores result in more disposed of goods, while higher prices in secondary stores result in fewer. As a result of the proposed method, the objective function satisfaction level can be viewed as a measure of optimism.

An integral contribution of this study is developing a new mixed-integer linear programming model for selecting the appropriate goods for re-commercialization and choosing the best outlet center based on the products' price and total profit. Another novelty of the proposed model is considering the matching percentage of boxes with secondary stores’ desired product lists and the probability of returning goods due to non-compliance with delivery dates. Moreover, a new uncertain solution approach is developed to solve mathematical programming problems with gray parameters.

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.

This paper aims to present two mathematical models to solve the Energy Management problem of a building microgrid (MG). In particular, it proposes a deterministic mixed integer linear programming (MILP) and non-linear programming (NLP) formulations. This paper focuses on the modelling process and the optimization performances for both approaches regarding optimal operation of near-zero energy buildings connected to an electric MG with a 24-h time horizon.

A general architecture of a MG is detailed, involving energy storage systems, distributed generation and a thermal reduced model of the grid-connected building. A continuous non-linear model is detailed along with linearizations for the mixed-integer liner formulation. Multi-physic, non-linear and non-convex phenomena are detailed, such as ventilation and air quality models.

Results show that both approaches are relevant for solving the energy management problem of the building MG.

Introduction and modelling of the thermal loads within the MG. The resulting linear program handles the mutli-objective trade-off between discomfort and the cost of use taking into account air quality criterion. Linearization and modelling of the ventilation system behaviour, which is generally non-linear and non-convex equality constraints, involving air quality model, heat transfer and ventilation power. Comparison of both MILP and NLP methods on a general use case provides a solution that can be interpreted for implementation.

Supplier selection in the supply chain network (SCN) has strategic importance and involves multiple factors. The multi-criteria nature of the problem coupled with environmental uncertainty requires several procedures and considerations. The issue of decision-making in selecting the best among various qualified suppliers remains the major challenge in the pharmaceutical industry. This study investigated the multi-criteria multi-supplier decision-making process and proposed a model for supplier selection problems based on mixed-integer linear programming.

The concept of principal component analysis (PCA) was used to reduce data dimensionality, and the four best criteria have been considered and selected. The result is subjected to decision-makers’ (DMs’) reliability test using the concept of a triangular fuzzy number (TFN). The importance of each supplier to each measure is established using fuzzy technique for order preference by similarity to an ideal solution approach, and the suppliers have ranked accordingly.

This study proposes a mixed integer linear programming model for supplier selection in a pharmaceutical company. The effectiveness of the proposed model has been demonstrated using a numerical example. The solution shows the model's applicability in making a sound decision in pharmaceutical companies in the space of reality. The model proposed is simple. Readily commercial packages such as LINDO/LINGO and GAMS can solve the model.

This research contributed to the systematic manner of supplier selection considering DMs’ value judgement under a fuzzy environment and is limited to the case study area. However, interested researchers can apply the study in other related manufacturing industries. However, the criteria have to be revisited to suit that system and might require varying ratings based on the experts' opinions in that field.

This work suggests more insights practically by considering a realistic and precise investigation based on a real-life case study of pharmaceutical companies with six primary criteria and twenty-four sub-criteria. The study outcome will assist organizations and managers in conducting the best decision objectively by selecting the best suppliers with their various standards and terms among many available contenders in the manufacturing industry.

In this paper, the authors attempted to identify the most critical attributes to be preserved by the top managers (DMs) while selecting suppliers in pharmaceutical companies. The study proposed an MILP model for supplier selection in the pharmaceutical company using fuzzy TOPSIS.

This paper aims to minimize the mean-risk cost of sustainable and resilient supplier selection, order allocation and production scheduling (SS,OA&PS) problem under uncertainty of disruptions. The authors use conditional value at risk (CVaR) as a risk measure in optimizing the combined objective function of the total expected value and CVaR cost. A sustainable supply chain can create significant competitive advantages for companies through social justice, human rights and environmental progress. To control disruptions, the authors applied (proactive and reactive) resilient strategies. In this study, the authors combine resilience and social responsibility issues that lead to synergy in supply chain activities.

The present paper proposes a risk-averse two-stage mixed-integer stochastic programming model for sustainable and resilient SS,OA&PS problem under supply disruptions. In this decision-making process, determining the primary supplier portfolio according to the minimum sustainable-resilient score establishes the first-stage decisions. The recourse or second-stage decisions are: determining the amount of order allocation and scheduling of parts by each supplier, determining the reactive risk management strategies, determining the amount of order allocation and scheduling by each of reaction strategies and determining the number of products and scheduling of products on the planning time horizon. Uncertain parameters of this study are the start time of disruption, remaining capacity rate of suppliers and lead times associated with each reactive strategy.

In this paper, several numerical examples along with different sensitivity analyses (on risk parameters, minimum sustainable-resilience score of suppliers and shortage costs) were presented to evaluate the applicability of the proposed model. The results showed that the two-stage risk-averse stochastic mixed-integer programming model for designing the SS,OA&PS problem by considering economic and social aspects and resilience strategies is an effective and flexible tool and leads to optimal decisions with the least cost. In addition, the managerial insights obtained from this study are extracted and stated in Section 4.6.

This work proposes a risk-averse stochastic programming approach for a new multi-product sustainable and resilient SS,OA&PS problem. The planning horizon includes three periods before the disruption, during the disruption period and the recovery period. Other contributions of this work are: selecting the main supply portfolio based on the minimum score of sustainable-resilient criteria of suppliers, allocating and scheduling suppliers orders before and after disruptions, considering the balance constraint in receiving parts and using proactive and reactive risk management strategies simultaneously. Also, the scheduling of reactive strategies in different investment modes is applied to this problem.