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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.

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 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.

To propose a new algorithmic platform (minlp_machine) for complex mixed‐integer non‐linear programming (MINLP) problems.

The platform combines features from classical non‐linear optimization methodology with novel innovations in computational techniques. The system constructs discrete search zones around non‐integer discrete‐valued variables of local solutions, which reduces the search process significantly. In complicated problems fast feasibility restoration is achieved through concentrated Hessians. The system is programmed in strict ANSI C and can be run either stand alone or as a support library for other programs. File I/O is designed to recognize possible usage in both single and parallel processor environments.

The system has been tested on Alpha and Sun mainframes and – as a support library for a Genetic Hybrid Algorithm (GHA()) – in Linux and IBM parallel supercomputer environments. The constrained problem can, for example, be solved through a sequence of first order Taylor approximations of the non‐linear constraints and occasional feasibility restoration through Hessian information of the Lagrangian of the MINLP problem, or by invoking a nonlinear solver like SQP directly in the branch and bound tree. The system has been successfully tested on a small sample of representative continuous‐valued non‐linear programming problems.

It is demonstrated that – through zone‐constrained search – minlp_machine() outperforms some recent competing approaches with respect to the number of nodes in the branch and bound tree.

Response surface methodology (RSM) is used for optimality analysis of the cost coefficients in mixed integer linear programming. This optimality analysis goes beyond traditional sensitivity and parametric analysis in allowing investigation of the optimal objective function value response over pre‐specified ranges on multiple problem parameters. Design of experiments and least squares regression are used to indicate which cost coefficients have the greatest impact on the optimal total cost surface over the specified coefficient ranges. The mixed integer linear programming problems of interest are the large‐scale facility location and allocation problems in supply chain optimization. A system that automates this process for supply chain optimization at PFS Logistics Consulting is discussed and an example is presented.

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 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.

To discuss a new parallel algorithmic platform (minlp_machine) for complex mixed‐integer non‐linear programming (MINLP) problems.

The platform combines features from classical non‐linear optimization methodology with novel innovations in computational techniques. The system constructs discrete search zones around noninteger discrete‐valued variables at local solutions, which simplifies the local optimization problems and reduces the search process significantly. In complicated problems fast feasibility restoration may be achieved through concentrated Hessians. The system is programmed in strict ANSI C and can be run either stand alone or as a support library for other programs. File I/O is designed to recognize possible usage in both single and parallel processor environments. The system has been tested on Alpha, Sun and Linux mainframes and parallel IBM and Cray XT4 supercomputer environments. The constrained problem can, for example, be solved through a sequence of first order Taylor approximations of the non‐linear constraints and feasibility restoration utilizing Hessian information of the Lagrangian of the MINLP problem, or by invoking a nonlinear solver like SQP directly in the branch and bound tree. minlp_machine( ) has been tested as a support library to genetic hybrid algorithm (GHA). The GHA(minlp_machine) platform can be used to accelerate the performance of any linear or non‐linear node solver. The paper introduces a novel multicomputer partitioning of the discrete search space of genuine MINLP‐problems.

The system is successfully tested on a small sample of representative MINLP problems. The paper demonstrates that – through concurrent nonlinear branch and bound search – minlp_machine( ) outperforms some recent competing approaches with respect to the number of nodes in the branch and bound tree. Through parallel processing, the computational complexity of the local optimization problems is reduced considerably, an important aspect for practical applications.

This paper shows that binary‐valued MINLP‐problems will reduce to a vector of ordinary non‐linear programming on a suitably sized mesh. Correspondingly, INLP‐ and ILP‐problems will require no quasi‐Newton steps or simplex iterations on a compatible mesh.