Because the problem is **integer** in the first stage, the solution can be achieved with the same basic steps of the SD algorithm developed for **linear** problems; the only difference is that, instead of a **linear** problem, now an **integer** **linear** programming problem has to be solved in order to find the first stage decisions in each of the iterations. Fig. 3 shows a flowsheet of the solution steps. Our computational implementation of the algorithm involves a framework that integrates the Hammersley sequence sampling technique (Kalagnanam & Diwekar, 1997) coded in FORTRAN, the GAMS modeling environment (Brooke, Kendrick, Meeraus, & Raman, 1998), and a C++ manager **program**. The manager **program** generates the appropriate MILP and LP problems for each of the SD iterations, transfers the control of execution and verifies the convergence of the algorithm. Fig. 4 represents the computational implementation of the algorithm.

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In financial practice, the transaction cost has significant effects on portfolio selection. It has been shown in (Arnott and Wagner 1990) that ignoring the transaction cost could result into inefficient portfolios. This has also been justified by experimental studies in (Yoshimoto 1996). If the transaction cost function is **linear**, which leads to a convex optimisation problem, then the problem is generally easy to solve. However, a function which better reflects realistic transaction costs is usually non- convex (Konno and Wijayanayake 2001). The non-convex optimisation problem is more challenging. In this paper, we consider a model that includes a fixed transaction fee plus a **linear** cost, thus leads to a non-convex function shown in Fig. 2, as the cost decreases relatively when the trading amount increases (Yoshimoto 1996, Konno and Wijayanayake 2001, Konno and Wijayanayake 2002). This function is also applied in (Lobo, Fazel et al. 2007). The transaction cost function is given in (4), and shown in Fig. 2, notations given in Table 1. In this work we aggregate the costs occurred in selling and buying an asset, and use a compact transaction cost function i i i b b i , i 0 .

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Jiekang et al. [40] presented a dynamic generation flow plan using the dynamically organizing net head of water in the reservoir and the consumption quantity of water. The results show that this new approach can improve the synthesis generation utilization of cascaded hydropower plants. Xin-Yu [41] composed the multi-objective optimal peak shaving model. It minimizes the maximum remaining loads per energy grid, which is an integral part of distributing the energy of a plant among some energy grids. A case study shows that the solution method is realistic, flexible and strong to get near-optimal results proficiently. Lu et al. [42] suggested a real binary bee colony optimization algorithm that is used to resolve parallel sub-problems of unit commitment and economic load dispatch. The simulation results prove that the suggested approach can obtain top-advantage solutions with shorter computing times and less water consumption. Marchand et al. [43] proposed a proficient model as a **mixed**-**integer** **linear** **program**, which shows a three-phase method based on a cost analysis that produces, rapidly, close optimal solutions to real-world cases. Ellen et al. [44] presented a model for hydropower bidding according to the OGS from a **stochastic** model. Furthermore, they presented a heuristic algorithm for decreasing the bid matrix into a size desired by a market operator. The results show how unchecked inflows may change the bids.

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Recently, a two-stage **stochastic** programming approach was applied by Giarola et al. [7] and Verma et al. [8] to manage uncertainties in single-objective environmental supply chain design. Pishvaee et al. [9] introduced Robust Possibilistic Programming (RPP) as a programming approach to coping with uncer- tain parameters in their bi-objective model, includ- ing minimizing the total cost and maximizing SC Social Responsibility (SR). A computational frame- work has been proposed to quantify the probable role of uncertainty in the environmental damage for the multi-objective optimization of sustainable sup- ply chain in [10]. A multi-objective (economic and environmental factors) facility location model, which investigates the impact of demand and return un- certainties on the SCND by implementing scenario- based **stochastic** programming, has been introduced by Amin and Zhang [11]. Ruiz-Femenia et al. [12] presented a **stochastic** multi-scenario **Mixed**-**Integer** **Linear** **Program** (MILP) in which demand uncertainty was considered for the multi-objective optimization of chemical supply chain and economic and environmental performances were accounted for, simultaneously. The

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In this subsection, we empirically study the impact of strategies for choosing the PH ρ parameter on the convergence of lower bounds in the **mixed** **integer** programming case. Note that in Proposition 1 the minimization is over the convex closure of the constraints, but in these experiments we minimize over the X(ξ) defined in Section 2 or X (β) as defined in Section 3 in the bundling case (i.e., we solve the MIP rather than optimizing over the convex closure). We consider different classes of two-stage **stochastic** **mixed**-**integer** programs. Previous experience [26, 25] indicates that larger values of ρ can accelerate the convergence of the PHA to a primal feasible solution. In [40], the authors study the impact of ρ for obtaining fast primal solutions and give recommendations for choosing ρ for a general class of **stochastic** resource allocation problems. Here, we instead focus on the relationship between ρ and the convergence of lower bounds. We show that as in the primal case, the quality of lower bounds obtained by the PHA in the case of SMIPs is significantly impacted by the choice of ρ. However, the relationship between ρ and bound quality differs in key aspects from the primal case, as we will illustrate below.

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Here, belotti also describes about the tree search method which is one of the widely used method to solve MINLP problems. The tree search method has been classified into single tree and multi-tree method. These two classes of methods solve a problem involving function like the convex type. The classical single tree method uses nonlinear branch and cut method, cutting plane method, and branch and bound method for solving MINLP problems. The conventional multi- tree approach comprises of both outer approximation and benders decomposition methods, in order to obtain global optimal solution for the considered problem. By combining the above approaches, a new hybrid method has been proposed in the literature, for solving convex MINLP problems more effectively, in terms of computation time and the quality of optimal solutions. In [11], Bonami et al. have provided a summary of various convex MINLP algorithms and software used to solve multiple MINLP problems. It is too challenging to solve nonconvex MINLP problems because it consists of nonconvex functions and nonlinear constraints in the objective function. Methods like Spatial branch and bound Piecewise **linear** approximation, and Generic Relaxation Strategies are used to solve nonconvex MINLP problems. In [12], Tawarmalani and Sahinidis N.V have explained about global optimization theory for solving MINLP problems using different algorithms and solvers.

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It is natural that various special cases involving binary variables have received the most attention, given the rich theory that has been developed specifically addressing this case. In the early work of Carøe and Tind (1997), the authors suggested the use of disjunctive programming and lift-and-project cuts when the recourse is **mixed** binary. They first decomposed the feasible set of (DE) into |Ω| subsets, which they sequentially convexified to generate the convex hull of each subset. Sherali and Fraticelli (2002) modified Benders’ method by generating valid inequalities in the subproblems. Assuming the first-stage variables are binary, each such valid inequality can be generated using the reformulation linearization technique (RLT) or lift-and-project. To ensure the generated valid inequality is globally valid, it can be re-expressed as a function of the first-stage variables. Using the dual solution of the optimal subproblem, optimality cuts for the master problem can be generated. In the same vein, Sen and Higle (2005) developed valid inequalities in both stages. The valid inequalities to augment the **linear** relaxation of the second-stage problem were generated so as to be valid for the union of disjunctive sets obtained by a disjunction arising from a fractional second-stage variable. Ntaimo (2010) provided a variation of the latter for problems with fixed T and h. Sherali and Zhu (2006) extended the framework of Sherali and Fraticelli (2002) to accommodate binary variables in the first stage by using decomposition and global branch-and-bound methods. Sherali and Smith (2009) used the RLT to devise a specialization of Benders’ algorithm for two-stage **stochastic** risk management problems with a pure binary first- stage problem.

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We study the probabilistic path properties of the **mixed** Poisson IN SV model, such as ergodic- ity, mixing, covariance structure and existence of moments. Moreover, by construction, the proposed model leads to an intractable likelihood function, as it depends on high-dimensional integrals. Yet, conditional of the intensity parameter, the likelihood function has a closed form and parameter esti- mation can be achieved by MCMC methods. The proposed posterior sampler can be easily modified to accommodate any conditional distribution that belongs to the family of the **mixed** Poisson IN SV pro- cess (or of any IN SV -type process). To demonstrate that, we consider two specific cases of the **mixed** Poisson IN SV specification, the Poisson IN SV model (P-IN SV ) and the negative binomial IN SV model (N B-IN SV ). For both models, the parameters of the autoregression are assigned conjugate priors and are updated from well-defined conditional posterior distributions.

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[15] C. Helmberg & F. Rendl (1998) Solving quadratic (0,1)-programs by semidefinite programs and cutting planes. Math. **Program**., 82, 291–315. [16] C. Helmberg, F. Rendl & R. Weismental (2000) A semidefinite pro- gramming approach to the quadratic knapsack problem. J. Combin. Optim., 4, 197–215.

Abstract. In IACR ePrint 2014/747, a method for constructing **mixed**- **integer** **linear** programming (MILP) models whose feasible regions are exactly the sets of all possible differential (or **linear**) characteristics for a wide range of block ciphers is presented. These models can be used to search for or enumerate differential and **linear** characteristics of a block cipher automatically. However, for the case of SIMON (a lightweight block cipher designed by the U.S. National Security Agency), the method proposed in IACR ePrint 2014/747 is not exact anymore. That is, the feasible region of the MILP model constructed for SIMON contains in- valid differential characteristics due to the dependent input bits of the AND operations, and these invalid characteristics must be filtered out by other methods. This is a very inconvenient process and reduces the level of automation of the framework of MILP-based automatic differen- tial analysis. In this paper, by using quadratic constraints or constraints from the H-representation of a specific convex hull, we give a method for constructing **mixed**-**integer** (non)**linear** programming models whose feasible regions are exactly the sets of all valid differential characteristics for SIMON. The technique presented in this paper may be also useful for other ciphers. How to construct an MILP model whose feasible region is exactly the set of all **linear** characteristics of SIMON is still an open problem.

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Margaretha Gansterer (2015) [12] presented a comprehensive hierarchical production planning (HPP) framework, to investigate the impact of aggregate planning in a make-to-order (MTO) environment. The planning problem is formulated as a **linear** mathematical model and solved to optimality by a standard optimization engine. The performance of the system is evaluated based on service and inventory levels. Real world data coming from the automotive supplier industry is used to define four demand scenarios.

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network, and finally integrated forward/reverse logistics network [1]. A comprehensive review of reverse and integrated logistics can be found in (Fleischmann [2]; Pokharel and Mutha [6]; Govindan [8]). In this paper, we survey specific network design problems for reverse and integrated logistics network design problems. Some authors like Üster et al. [10], Amiri [11], Patia et al. [12], and Aras et al. [13] are those ones, who carried out investigations about the integrated logistics. Keyvanshokooh [1] presented a **mixed**-**integer** **linear** programming to consider dynamic pricing approach for used products, forward/reverse logistics network configuration and inventory. Kamali et al. [14] proposed a single product, multi-echelon, multi-period closed loop supply chain for high-tech products (which have continuous price decrease). Four heuristics-based methods including genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), and artificial bee colony (ABC) are proposed for solving their model. Sahraeian et al. [15] introduced a supply chain network design problem which contains environmental concerns in arcs and nodes of network and there are some routes such as road, rail and etc. in each pair of nodes. Demand uncertainty and uncertainty

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Classification models derived via multivariable optimal discriminant analysis (MultiODA) are **linear** discriminant classifiers which explicitly maximize classification accuracy for a given sample. 1 **Mixed**-**integer** **linear** programming formulations for two-group MultiODA models require estimation of the value of a parameter, M, commonly defined as “a prohibitively large number.” 2 If the estimated value of M is too low then suboptimal solutions may occur, and exces- sively large values of M will decrease computa- tional efficiency and may introduce numerical (round-off) error. 3 We present a goal **program**- ming formulation which establishes a lower bound for M, and then we discuss a sufficient condition for the nonexistence of classification gaps and ambiguous solutions, weighted classi- fication, the use of nonlinear terms, selection of

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Recently, Dash and G¨unl¨uk [11] extended the idea to one dimension higher. They derived the 2-step MIR facet for a three-dimensional simple **mixed** **integer** set and used it to generate valid inequalities, called 2-step MIR inequalities, for the feasible set of a general **integer** programming constraint. The result of this process is what they call the 2-step MIR functions, which are easy tools for generating valid inequalities. Another important property of MIR functions proved by Dash and G¨unl¨uk is that they generate facets for Gomory’s group polyhedra. This shows that 2- step MIR inequality are potentially strong valid inequalities. Additionally, Gomory [23] showed that facets of group polyhedra are sources for generating valid inequalities for **integer** programming problems. Therefore, this property of MIR functions is of great value in generating even more valid inequalities.

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The second category pays attention to integrating RESs with SGs, where the potential role of EVs and other ﬂexible electric loads for accommodating higher levels of renewables are highlighted. Some related literature is found in [14]—[19]. Reference [14] proposed a **stochastic** method, based on a Monte Carlo simulation and particle swarm optimization, for sizing a smart household energy system which takes into consideration the demand uncertainty. Another **stochastic** based optimization that accounts for load shifting is presented in [15]. Regarding the EV utilization in power system application, the effect of different EV control strategies in reducing surplus wind generation and harmful emissions was investigated [16].

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Paydar et al. [13] presented an elaborative mathematical model based on operations sequence, intracellular layout, lot splitting, alternative process routing, duplicate machines, multi-period production planning, system reconfiguration, machine capacity, and material flow between machines, which is called comprehensive **integer** **linear** programming approach. Kia et al. [14] proposed a dynamic nonlinear **mixed** **integer** programming approach to provide a group layout of CMS concerning the production planning decisions. Lim et al. [6] presented multi-objective hybrid algorithms for layouts optimization in multi-robot assembly for solving the cellular manufacturing problems. Rezazadeh and Khiali- Miab [15] developed a novel mathematical model for designing the reliable cellular manufacturing systems that leads to decrease in the manufacturing costs, improved total reliability of the manufacturing system and improved product quality. In addition, a two-layer genetic algorithm is presented to address the complexity of cell formation problems to obtain near optimal solutions. Kia et al. [16] proposed a novel **mixed** **integer** nonlinear mathematical model for elaborating the group layout of unequal area facilities in a dynamic CMS. Sakhaii et al. [3] presented a robust **mixed** **integer** **linear** programming approach with the goal of minimizing the total costs related to machines, production, workers, and part movements. Aalaei and Davoudpour [8] proposed a novel mathematical programming model for a CMS along supply chain design regarding labor assignment with the goal of minimizing the total cost of holding, inter-cell material handling, external transportation, fixed cost for producing each part in each plant, machine and labor salaries.

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This work presents an optimization model to support decisions during production planning and control in the personal protective equipment (PPE) industry (in par- ticular, gloves). A case study was carried out at a Brazilian company with the aim of increasing productivity and improving customer service with respect to meeting deadlines. In this case study, the **mixed** **integer** **linear** programming model of Luche (2009) was revisited. A new model for single-stage lot sizing was applied to the pro- duction scheduling of gloves. Optimizing this scheduling was not a simple task be- cause of the scale of the equipment setup time, the diversity of the products and the deadlines for the orders. The model was implemented in GAMS IDE and solved by CPLEX 12. The model and the associated heuristic produce better solutions than those currently used by the company.

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Abstract In this paper, system planning network is formulated with **mixed**-**integer** programming. Two meta-heuristic procedures are considered for this problem. The cost function of this problem consists of the capital investment cost in discrete form, the cost of transmission losses and the power generation costs. The DC load flow equations for the network are embedded in the constraints of the mathematical model to avoid sub-optimal solutions that can arise if the enforcement of such constraints is done in an indirect way. The solution of the model gives the best line additions, and also provides information regarding the optimal generation at each generation point. This method of solution is demonstrated on the expansion of a 5 bus-bar system to 6 bus-bars.

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Fractional programming has been widely reviewed by many authors (Schaible [3], Nagih and Plateau [4]) and there are entire books and chapters devoted to this subject (Craven [5], Stancu-Minasian [6], Horst et al. [7] and Frenk and Schaible [8]). Schaible [3] has published a comprehensive review of the work in fractional programming, outlining some of its major developments. Stancu-Minasian’s textbook [6] contains the state-of-the-art theory and practice of fractional programming, allowing the reader to quickly become acquainted with what has been done in the field. Different approaches have been proposed in the literature to solve both continuous **linear** fractional programming (LFP) and **integer** **linear** fractional programming (ILFP) problems. These can be divided in studies that have developed solution methods (e.g., [6, 9-13]) and those which concentrated on applications (e.g. [6, 8]).

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separation procedures for framed capacity, strengthened comb, multistar, partial multistar, generalized multistar and hypotour contraints. Their approach is known as branch-and-cut-and-price algorithm. In the second approach which was proposed by Baldacci et al [9] and Baldacci and Mingozzi [10], they use a series of bounding procedures to find near optimal solutions for the LP-relaxation of the problem. They use q-route approach and combine it with the Lagrangian relaxation. Then, a set of routes whose reduced costs are smaller than the gap between upper and lower bounds generated through their method are found and added to the LP-relaxation of the problem. Finally, the resulting problem is solved using an **integer** programming solver. Although both the approaches are capable of solving the same problems and present the same performance, we follow Fukasawa et al.’s method as their method is more flexible and suitable for the framework we propose for the VRPUD in this chapter and the next chapter.

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