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Classification and regression trees, as well as their variants, are off-the-shelf methods in Machine Learning. In this paper, we review recent contributions within the Continuous Optimization and the Mixed-Integer Linear Optimization paradigms to develop novel formulations in this research area. We compare those in terms of the nature of the decision variables and the constraints required, as well as the optimization algorithms proposed. We illustrate how these powerful formulations enhance the flexibility of tree models, being better suited to incorporate desirable properties such as cost-sensitivity, explainability, and fairness, and to deal with complex data, such as functional data.

The canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift–diffusion model (DDM). In this model, a decision variable keeps track of the integrated difference in sensory evidence for two competing alternatives. Here I extend the notion of a drift–diffusion process to multiple alternatives. The competition between n alternatives takes place in a linear subspace of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n-1$\end{document}n−1 dimensions; that is, there are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n-1$\end{document}n−1 decision variables, which are coupled through correlated noise sources. I derive the multiple-alternative DDM starting from a system of coupled, linear firing rate equations. I also show that a Bayesian sequential probability ratio test for multiple alternatives is, in fact, equivalent to these same linear DDMs, but with time-varying thresholds. If the original neuronal system is nonlinear, one can once again derive a model describing a lower-dimensional diffusion process. The dynamics of the nonlinear DDM can be recast as the motion of a particle on a potential, the general form of which is given analytically for an arbitrary number of alternatives.

In a tri-level hierarchical decision problem, each decision entity at one level has its objective, constraints and decision variables affected in part by the decision entities at the other two levels. The choice of values for its variables may allow it to influence the decisions made at other levels, and thereby improve its own objective. We called this a tri-level decision problem. When multiple decision entities are involved at the middle and bottom levels, the top-level entity’s decision will be affected not only by these followers’ individual reactions but also by the relationships among the followers. We call this problem a tri-level multi-follower (TLMF) decision.

Material purchasing and supply is a great insurance to the continuity of the construction process. In this paper, a multi-objective purchasing model is presented for the purchasing and inventory in the perspective of each construction partner in a construction supply chain. The model aims to reduce the purchasing and supply cost as well as the lead time in a construction supply chain. Purchasing quantity and lead time are both considered as controllable decision variables in this model.

In real-world applications, a bi-level decision problem may involve multiple decision entities on the upper level, that is, the bi-level decision problem has multiple leaders. The leaders may have their individual decision variables, objective functions and/or constraint conditions. This kind of bi-level decision problem is called a bi-level multi-leader (BLML) decision problem.

To improve the performance of copper-matte Peirce-Smith Converting (PSC), the influence of local process data to \( \varepsilon \)-support vector regression (SVR) model for converting process is studied. This paper proposes an Error Correction method for \( \varepsilon \)-Support Vector Regression (EC_SVR), in which the influence of local support vector to prediction results is considered. Two EC_SVR models for slag weight and blowing time of S1 period (that is, the first slag producing period of PSC) are developed by the real production data. Simulation results show that EC_SVR model can significantly improve prediction accuracy and generalization of the converting decision variable in S1 period.

This paper formulates a multi-objective linear optimization model of land use surrounding rail transit stations based on TOD theory. With FAR value of different land use types as decision variables, this model is to maximize rail transit ridership, maximize service facility level and maximize total added economic values from land. And Xizhimen rail transit station is selected as a study case. The data used in the case come from investigation on-the-spot, images of Google Earth. At last the model is applied to optimize land use of Xizhimen rail transit station surroundings, and some suggestions for future development are given in this area.

This work investigates a method to search practically desirable solutions expanding the objective space with additional fitness functions associated to particular decision variables. The aim is to find solutions around preferred values of the chosen variables while searching for optimal solutions in the original objective space. Solutions to be practically desirable are constrained to be within a certain distance from the instantaneous Pareto optimal set computed in the original objective space. Our experimental results show that the proposed method can effectively find practically desirable solutions.

In this chapter, the concept of discrete sliding modes (DSMs) is applied to design an efficient supply policy for a class of perturbed processes with delay – goods flow control in supply chain. In the considered systems, the stock used to satisfy the unknown, time-varying demand placed at a goods distribution center; is replenished with delay from a remote supply source. The order quantity is fixed, leaving the time between the consecutive orders as a decision variable, which perfectly suits the switching nature of input signals obtained in DSM control systems. It is shown that under the proposed nonlinear policy, the stock level does not exceed the assigned storage space. Moreover, it is also demonstrated that the stock is never entirely depleted, which guarantees full demand satisfaction and maximum service level.

In this paper we consider a low-carbon inventory control policy planned for creative products sold in hybrid channels, where there is vertical and horizontal competition. We set up a tow-echelon base-stock model and under the assumption of demand transfer we analyze competitive stocking decisions with manufacture and retailer using base-stock inventory control policy. By numerical experiments, we analyze the impact of demand uncertainty and competition factor on manufacturer and retailer’s stocking decision variables and the corresponding performance. Finally we conclude the management enlightenments.

One of the earliest applications of fuzzy logic was fuzzy control. Indeed, early Lofti Zadeh’s background was in systems and control engineering and, hence, his way of thinking was close to that of control engineers, the group of people I belong to.

In particular production environment, e.g. some machining shops and semiconductor test stations, multiple resources are simultaneously needed to perform an operation. To conduct capacity planning with cooperative resources, we propose two methods to construct compact linear programming models, i.e. Generalized Step-Separated Formulation and Generalized Workload Allocation Formulation. These modeling methods are extended from the existing modeling methods for capacity planning only with independent resources. Generalized Workload Allocation Formulation weakens the application scope of the uniformity assumption made in the previous formulation for independent resources. Dimension analysis and comparison shows that Generalized Workload Allocation Formulation generates less decision variables and constraints than Generalized Step-Separated Formulation.

OpenSolver is an open source Excel add-in that allows spreadsheet users to solve their LP/IP models using the COIN-OR CBC solver. OpenSolver is largely compatible with the built-in Excel Solver, allowing most existing LP and IP models to be solved without change. However, OpenSolver has none of the size limitations found in Solver, and thus can solve larger models. Further, the CBC solver is often faster than the built-in Solver, and OpenSolver provides novel model construction and on-sheet visualisation capabilities. This paper describes Open- Solver’s development and features. OpenSolver can be downloaded free of charge at http://www.opensolver.org.

We introduce a model considering the single-leg revenue management problem from the perspective of online algorithms and competitive analysis. In this model, the price and limitation of bookings are both decision variables. Assuming the process of the customers meets the low-before-high manner and the expected value of customer falls in a closed interval, we analyze the upper bound for deterministic online joint pricing and booking policy and propose one optimal policy for each case.

Genetic Algorithms (GAs) are categorized as search heuristics and have been broadly applied to optimization problems. These algorithms have been used for solving problems in many applications, but it has been shown that simple GA is not able to effectively solve complex real world problems. For proper solving of such problems, knowing the relationships between decision variables which is referred to as linkage learning is necessary. In this paper a linkage learning approach is proposed that utilizes the special features of the decomposable problems to solve them. The proposed approach is called Linkage Learner based on Local Optimums and Clustering (LLLC). The LLLC algorithm is capable of identifying the groups of variables which are related to each other (known as linkage groups), no matter if these groups are overlapped or different in size. The proposed algorithm, unlike other linkage learning techniques, is not done along with optimization algorithm; but it is done in a whole separated phase from optimization search. After finding linkage group information by LLLC, an optimization search can use this information to solve the problem. LLLC is tested on some benchmarked decomposable functions. The results show that the algorithm is an efficient alternative to other linkage learning techniques.

Genetic Algorithms (GAs) are categorized as search heuristics and have been broadly applied to optimization problems. These algorithms have been used for solving problems in many applications, but it has been shown that simple GA is not able to effectively solve complex real world problems. For proper solving of such problems, knowing the relationships between decision variables which is referred to as linkage learning is necessary. In this paper a linkage learning approach is proposed that utilizes the special features of the decomposable problems to solve them. The proposed approach is called Local Optimums based Linkage Learner (LOLL). The LOLL algorithm is capable of identifying the groups of variables which are related to each other (known as linkage groups), no matter if these groups are overlapped or different in size. The proposed algorithm, unlike other linkage learning techniques, is not done along with optimization algorithm; but it is done in a whole separated phase from optimization search. After finding linkage group information by LOLL, an optimization search can use this information to solve the problem. LOLL is tested on some benchmarked decomposable functions. The results show that the algorithm is an efficient alternative to other linkage learning techniques.

The use of computational methodologies for the optimization of aesthetic parameters is not frequent mainly due to the fact that these parameters are not quantifiable and are subjective. In this work an interactive methodology based on the use of multi-objective optimization algorithms is proposed. This strategy associates the results of different optimization runs considering the existent quantifiable objectives and different sets of boundary conditions concerning the decision variables, as defined by an expert decision maker. The associated results will serve as initial population of solutions for a final optimization run. The idea is that a more global picture of potential ”good” solutions can be found. At the end this will facilitate the work of the expert decision maker since more solutions are available. The method was applied to a case study and the preliminary results obtained showed the potentially of the strategy adopted., One of the authors acknowledges the financial support received by the Portuguese Science Foundation under grant SFRH/BD/44600/ 2008.

Recently MOEA/D (multi-objective evolutionary algorithm based on decomposition) was proposed as a high-performance EMO (evolutionary multiobjective optimization) algorithm. MOEA/D has high search ability as well as high computational efficiency. Whereas other EMO algorithms usually do not work well on many-objective problems with four or more objectives, MOEA/D can properly handle them. This is because its scalarizing function-based fitness evaluation scheme can generate an appropriate selection pressure toward the Pareto front without severely increasing the computation load. MOEA/D can also search for well-distributed solutions along the Pareto front using a number of weight vectors with different directions in scalarizing functions. Currently MOEA/D seems to be one of the best choices for multi-objective optimization in various application fields. In this paper, we examine its performance on multi-objective problems with highly correlated objectives. Similar objectives to existing ones are added to two-objective test problems in computational experiments. Experimental results on multi-objective knapsack problems show that the inclusion of similar objectives severely degrades the performance of MOEA/D while it has almost no negative effects on NSGA-II and SPEA2. We also visually examine such an undesirable behavior of MOEA/D using manyobjective test problems with two decision variables.

Coal transportation system plays an important role in guaranteeing the safety and stability of China’s coal supply. How to distribute coal resources among provinces in China by railway was discussed. An optimization model was set up to test the rationality of China’s coal distribution. Furthermore, by using the model we analyzed the optimal transportation pattern. The model takes transportation cost as optimization goal and the volumes of inter-provincial coal transportation by railway as decision variables. Findings indicate that the pattern of coal railway transportation among provinces changes in four aspects and the transportation cost and the average transport distance decrease compared with that before optimization. Based on the research, we suggest that six coal transportation channels should be strengthened constructing and transportation rate for coastal cities in southeast China reduced.

Bi-objective portfolio optimization for minimizing risk and maximizing expected return has received considerable attention using evolutionary algorithms. Although the problem is a quadratic programming (QP) problem, the practicalities of investment often make the decision variables discontinuous and introduce other complexities. In such circumstances, usual QP solution methodologies can not always find acceptable solutions. In this paper, we modify a bi-objective evolutionary algorithm (NSGA-II) to develop a customized hybrid NSGA-II procedure for handling situations that are non-conventional for classical QP approaches. By considering large-scale problems, we demonstrate how evolutionary algorithms enable the proposed procedure to find fronts, or portions of fronts, that can be difficult for other methods to obtain.

As quality management being of great importance on many companies increasingly, quality coordination has become the new one of supply chain coordination. When demand is in the linear correlation with quality in supply chain member, the traditional wholesale price has not coordinated the supply chain. In this paper, the incentive penalty contract is put forward under the linear quality demand and the coordination can be achieved through it. Also the decision variables such as price, quantity and quality are obtained.

In this paper we introduce a new method for solving systems of linear inequalities. The algorithm incorporates many state-of-the-art techniques from DPLL-style reasoning. We prove soundness, completeness and termination of the method.

The solution of mixed-integer nonlinear programming (MINLP) problems often suffers from a lack of robustness, reliability, and efficiency due to the combined computational challenges of the discrete nature of the decision variables and the nonlinearity or even nonconvexity of the equations. By means of a continuous reformulation, the discrete decision variables can be replaced by continuous decision variables and the MINLP can then be solved by reliable NLP solvers. In this work, we reformulate 98 representative test problems of the MINLP library MINLPLib with the help of Fischer-Burmeister (FB) NCP-functions and solve the reformulated problems in a series of NLP steps while a relaxation parameter is reduced. The solution properties are compared to the MINLP solution with branch & bound and outer approximation solvers. Since a large portion of the reformulated problems yield local optima of poor quality or cannot even be solved to a discrete solution, we propose a reinitialization and a post-processing procedure. Extended with these procedures, the reformulation achieved a comparable performance to the MINLP solvers SBB and DICOPT for the 98 test problems. Finally, we present a large-scale example from synthesis of distillation systems which we were able to solve more efficiently by continuous reformulation compared to MINLP solvers.

In this paper, we investigate the performance of CMA-ES on large scale non-separable optimisation problems. CMA-ES is a robust local optimiser that has shown great performance on small-scale non-separable optimisation problems. Self-adaptation of a covariance matrix makes it rotational invariant which is a desirable property, especially for solving non-separable problems. The focus of this paper is to compare the performance of CMA-ES with Cooperative Co-evolutionary Algorithms (CCEAs) for large scale global optimisation (on problems with up to 1000 real-valued variables). Since the original CMA-ES is incapable of handling problems with more than several hundreds dimensions, sep-CMA-ES was developed using only the diagonal elements of the covariance matrix. In this paper sep-CMA-ES is compared with several existing CCEAs. Experimental results revealed that the performance of sep-CMA-ES drops significantly when the dimensionality of the problem increases. However, our results suggest that the rotational invariant property of CMA-ES can be utilised in conjunction with a CCEA to further enhance its capability to handle large scale optimisation problems.

The determination of optimal outflow discharge hydrograph in a pre-discharged period is an important component of the reservoir operation. Because of the high dimensional decision variables, lots of constrained conditions and complicated nonlinear relationships, it is usually difficult for the traditional methods and most of the current popular optimization software to handle the problem aforementioned. This research has presented the optimization models with single objective or multi-objective functions, and the optimization software package, 1stOpt, developed by the first author has been employed to solve the optimal problem. The research results have shown: 1) the optimal regulation models can meet various requirements with a universal feature, and thus may be applied in different reservoirs; 2) 1stOpt is a robust, easy-to-use and powerful optimization tool, and can be used widely in various engineering fields.

As Figure 7.1 indicates, Step 3 in our general heuristic decisionmaking procedure is the development of options. The problem-naming in Step 2 provides a basis for the subsequent options. How specific the problem naming is depends on how difficult the problem is and on the capacity of the actor to solve it. There are three possibilities: problem-naming can be so specific that the options are largely predetermined, problem-naming can reveal the decision variables on which the options are based, problem-naming yields no more than very vague clues regarding possible options.

Any sufficiently complex finite-domain constraint modelling language has the ability to express undefined values, for example division by zero, or array index out of bounds. This paper gives the first systematic treatment of undefinedness for finite-domain constraint languages. We present three alternative semantics for undefinedness, and for each of the semantics show how to map models that contain undefined expressions into equivalent models that do not. The resulting models can be implemented using existing constraint solving technology.

In the recent years, there has been an increasing interest on the use of case-based reasoning (CBR) in Medicine. CBR is characterized by four phases: retrieve, reuse, revise and retain. The first and last phases have received a lot of attention by the researchers, while the reuse phase is still in its infancy. The reuse phase involves a multi-facet problem which includes dealing with the closeness to the decision threshold used to determine similar cases, among other issues. In this paper, we propose a new reuse method whose decision variable is based on the similarity ratio. We have applied the method and tested in a breast cancer diagnosis database.

This paper introduces a real, unmodified Many-Objective optimisation problem for use in optimisation algorithm benchmarking. The radar waveform design problem has 9 objectives and an integer decision space that can be scaled from 4 to 12 decision variables. Proprietary radar waveform design software has been encapsulated in a fast and portable form to facilitate research groups in studying high-order optimisation of real engineering problems.

ing from the specific requirements in the RFS-domain the block planning problem can be formulated as follows: Given a fixed planning period 1, ..., T with a set F of transportation tasks/trips with some domain-specific requirements and a non-homogenous set V of vehicle/driver-combinations which are available for serving the trips, find an assignment of the vehicles to the trips forming blocks of trips which can be operated sequentially by a vehicle such that within blocks certain constraints are fulfilled. Here a block b is a sequence (f0, f1, ..., fn) , fi ∈ F , of trips ordered by increasing departure time with the sequence of trips starting and ending in a fixed location, the depot or hub. Given a block b ∈ B we can calculate cb = contribution to profit of block b ∈ B, if it is served by vehicle v ∈ V BT v b = departure time from the depot of block b ∈ B, if served by vehicle v ∈ V ; BT v b ∈ T ET v b = arrival time at the depot of block b ∈ B, if served by vehicle v ∈ V ; ET v b ∈ T γ b = { 1 if block b ∈ B can be served by vehicle v ∈ V 0 else δ b = { 1 if block b ∈ B contains trip f ∈ F 0 else 46 Paul Bartodziej et al. Assume that we have constructed the set B of all possible blocks. Introducing the decision variable xb = { 1 if block b ∈ B is served by vehicle v ∈ V 0 else we can model the BPP as a ”set partition like” mathematical program

We adopt Benders' decomposition algorithm to solve scenario-based Stochastic Constraint Programs (SCPs) with linear recourse. Rather than attempting to solve SCPs via a monolithic model, we show that one can iteratively solve a collection of smaller sub-problems and arrive at a solution to the entire problem. In this approach, decision variables corresponding to the initial stage and linear recourse actions are grouped into two sub-problems. The sub-problem corresponding to the recourse action further decomposes into independent problems, each of which is a representation of a single scenario. Our computational experience on stochastic versions of the well-known template design and warehouse location problems shows that, for linear recourse SCPs, Benders' decomposition algorithm provides a very efficient solution method.

Preferences and uncertainty occur in many real-life problems. The theory of possibility is one non-probabilistic way of dealing with uncertainty, which allows for easy integration with fuzzy preferences. In this paper we consider an existing technique to perform such an integration and, while following the same basic idea, we propose various alternative semantics which allow us to observe both the preference level and the robustness w.r.t. uncertainty of the complete instantiations. We then extend this technique to other classes of soft constraints, proving that certain desirable properties still hold.

We investigate the accuracy of parameters in the Logic Scoring of Preference (LSP) criterion functions for system evaluation. Main parameters are weights and conjunction/disjunction degrees (andness/orness). Weights reflect the level of relative importance of various decision variables. Andness/orness describes a desired level of simultaneity/replaceability in satisfying component criteria. These parameters are assessed by one or more professional evaluators and their values differ from (usually unknown) optimum values. In this paper we identify all potential sources of errors in LSP criterion functions. Our goal is to investigate the distribution of errors, their average values, and the quality of individual evaluators and their teams.

A two-population Genetic Algorithm for constrained optimization is exercised and analyzed. One population consists of feasible candidate solutions evolving toward optimality. Their infeasible but promising offspring are transferred to a second, infeasible population. Four striking features are illustrated by executing challenge problems from the literature. First, both populations evolve essentially optimal solutions. Second, both populations actively exchange offspring. Third, beneficial genetic materials may originate in either population, and typically diffuse into both populations. Fourth, optimization vs. constraint tradeoffs are revealed by the infeasible population.

Constraint models contain a number of common patterns. For example, many constraint models involve an array of decision variables with symmetric rows and/or columns. By documenting such constraint patterns, we can share modelling expertise. Constraint solvers can also be extended to exploit such patterns. For example, we can develop specialized methods like the global lexicographical ordering constraint for breaking such row and column symmetry.

In this thesis we have analyzed both the technical and the economic performance of differently designed inbound call centers using Markov queueing models. We showed that the performance of an inbound call center depends on several parameters, which are related to the customer profile, the agent characteristics, the particular routing policy, and the limitation of the waiting room. For the short-term economic optimization, the allocation of agents and telephone trunks are the major decision variables.

Another major characteristic of inventory systems in a Reverse Logistics context as identified in Chapter 6 concerns the presence of multiple, alternative supply sources, namely recovery of used products versus procurement of new ones. In the previous chapter we assumed that the goods return flow directly affects the serviceable inventory and cannot be influenced. Therefore, procurement orders have, in fact, been the only means to control the system. Let us now turn to a more detailed picture of the inbound channel where the recovery of returned products is also a decision variable (see Figure 6.1). To this end, we assume that returned products are collected in a distinct inventory upon arrival. The serviceable stock can then be replenished alternatively by means of procurement or by processing recoverable stock. In addition to lotsizing and safety stock considerations the choice between both sources then becomes an important issue in achieving efficient system performance.

Among formalisms used to model complex systems and organisations, the GRAI Grid has a special status because it focuses on the decisional aspects of the management of systems. The GRAI grid defines the points where decisions are made (decision centres) and the information relationships among these. Models built using the grid allow the analysis and design of how decisions are co-ordinated and synchronised in the enterprise.

This paper presents Cover-Time Planning, a kind of a reorder “point” system which exploits the information available in the bill of material. Cover-Time Planning contrary to an ordinary reorder point system uses time as a decision variable instead of quantity. Cover-Time Planning has like Kanban a built in restriction of work-in-process. Not until previous production is consumed a new production is initiated.

Negotiations are often characterized by the fact that the participants do not know each others’ goals. This paper describes an interactive approach to computing Pareto optimal agreements under such circumstances in two-party negotiations. The approach is based on so called proposals that link the decision variables together. The proposals also provide a way of enforcing the resulting agreement, meaning that no party has an incentive to break loose from it.

Genetic Algorithms (GA) are robust search methods that mimic basic principles of evolution. They can be employed to optimise the decision variables of simulation models. In this paper, a GA produces better results than two conventional methods when optimising the decision variables of a stochastic inventory simulation. It is also shown that GA are also more robust optimisers when only few simulations of each trial solution are performed. This characteristic may be used to reduce the generally higher CPU-requirements of population-based search methods like GA as opposed to point-based traditional optimisation techniques in stochastic optimisation.

In the paper the deterministic optimization models are summarized first. In these models the main decision variables are the feed, speed and depth of cut. Then we treat tool life as a random parameter and select a probability model that defines the tool life variations. Finally stochastic optimization models are defined for determining machining operations.

We investigate the problem of scheduling n independent jobs on a single machine, in which processing times p i (i = 1,⋯ ,n) of the n jobs are decision variables whose values can be compressed between [l i ,u i ]. Past research mainly concerned with minimizing the sum of a first order function and a cost function for compression. In this paper, we consider a new model with the aim to minimize the sum of CTV (Completion Time Variance) and a linear cost for compression. The CTV problem, which is NP-hard, can be considered as a special case of our problem when l i = u i for all i. Up to now, there is no algorithm, except complete enumeration of all the schedules of the n processing times, to solve this problem. We show, in this paper, that if an agreeable condition is satisfied, then our problem can be solved by a pseudo-polynomial algorithm with time complexity bounded above by \(O({n^2}{u_n}{U^2}),where U = {\Sigma _i}^n{ = _1}{u_i}.\)

The multi-site planning problem has been studied. Two approaches have been used to model and solve this problem. This lead us to compare principles of the two main approaches in term of optimization, autonomy, evolution and dynamicity. Up to now only the centralized approach has been validated on real data from industrial case (the lighting division of Magneti Marelli). A real case with about 30 production units, 5 time periods and 30 orders with 70 parts of final items (with about 10 bill of material levels) has been tested. Thie CPU time to solve these cases with 4000 decision variables and 16000 constraints is about 5 minutes on a SPARC station. The validation of the distributed approach will be done on these data to compare quantitatively both approaches efficiency in term of optimality and solving efficiency.

We propose a new solution principle to Fuzzy Multiobjective Linear Programs (FMLP) of the form maximize $$ \left( {\tilde c_{11} \times _{11} + \ldots + \tilde c_{1n} \times _n , \ldots ,\tilde c_{k1} \times _1 + \ldots + \tilde c_{kn} \times _n } \right) $$ subject to $$ \begin{array}{*{20}c} {\tilde a_{11} \times _1 + \ldots + \tilde a_{1n} \times _n \, \prec \,\tilde b_1 } \\ { } \\ {{\text{ }}\tilde a_{m1} \times _1 + \ldots + \tilde a_{mn} \times _n \, \prec \,\tilde b_m } \\ \end{array} $$ where x∈Rn is a vector of decision variables, all coefficients with tilde are fuzzy quantities, the operations addition and multiplication by scalar of fuzzy sets are defined by Zadeh’s extension principle, and < is a certain fuzzy relation given by the decision-maker.

In the paper, a discrete event system has been considered, consisting of a set of tasks, organized in precedence-relation graphs, to be executed by a set of machines, in the fulfilment of a certain set of constraints. It has been shown that, using minimax algebra formalism, one is able to: i) express the task completion times as analytic functions of the binary decision variables relevant to the alternative choices and of the continuous variables representing the task activation delays with respect to the earliest activation times; ii) represent analytically all the constraints affecting the decision variables and resulting from the structure of the model considered. Thus, we are able to determine the structure, which means, the cost functional plus the constraints imposed, of a mathematical programming problem having the objective of optimizing a cost functional related to task completion times. The resulting optimization problems are of the mixed-integer type. To write the optimization problem, i.e., the cost functional and the relevant constraints, in terms of the decision variables it is not necessary to list explicitly all the possible perturbed semi-schedules.

One usually distinguishes in science between descriptive and normative theories. The former tells us what an entity does under certain conditions and, possibly, why. Cognitive psychology, sociology and demography are examples of the disciplines that provide descriptive aspects of human behavior. In contrast, normative theories tell us what to do in order to accomplish certain goals. Various optimization techniques and methods of Operations Research yield normative theories of well-defined and mathematically formulable tasks.

The Advice Taker/Inquirer is a domain-independent program that is used to construct, fine-tune and oversee the operation of an expert system. It consists of two phases: a learning phase, during which a human expert teaches the system interactively about a domain in terms of principles, high-level examples, rules, and facts; and an operational phase, during which the program monitors the resulting expert system as it is applied to a domain of interest, and continually attempts to improve its performance by hypothesizing new rules and reorganizing existing knowledge.

MCDM is applied to select regular exploration networks for several, serially correlated and anisotrope geological parameters. The relative importance of estimation accuracy for the different parameters is different. Geostatistics is used for calculating the estimation variance for ungaged points, areal averages or blocks as a function of the decision variables: exploration density, distance between exploration points and rotation angle. The statistical criteria refer to the minimimization of the estimation variance. The other objective to be minimized is the exploration effort. A composite form of compromise programming is applied to find a trade-off among the conflicting objectives. The methodology is illustrated by a coal exploration case.