A clone-based graphical modeler and mathematical model generator for optimal production planning in process industries

This paper outlines a visually interactive graphical modeling approach for process type production systems, with hidden generation of complex optimization models for production planning. The proposed system lets the users build a graphical model of the production system with one-to-one clones of its production units through its interactive visual interface, accepts production-specific data for its components, and finally, internally generates and solves its mathematical programming model without any interaction from the user. This “clone-based” modeling approach allows the continued use of optimization models with minimal mathematical programming understanding, as generation of mathematical model by clones is hidden and automatic, therefore maintenance-free: Updating graphical production system models is enough for renewing internal optimization models. The concept is demonstrated in this paper with a linear programming prototype developed for a petroleum refinery.     

Risk-averse dynamic programming for Markov decision processes

We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we develop a risk-averse policy iteration method and we prove its convergence. We also propose a version of the Newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence. Finally, we discuss relations to min–max Markov decision models.     

A revenue management approach to address a truck rental problem

This paper describes an application of revenue management techniques and policies in the field of logistics and distribution. In particular, the problem of transportation operators, who offer products for hire, is considered. A product is a truck of a given capacity, which can be rented for one or several time periods, throughout a multi-period planning horizon. The logistic operator can satisfy the demand of a given product with trucks with a capacity greater than that initially required, that is an ‘upgrade’ can take place. In this context, the logistic operator has to decide whether to accept or reject a request and which type of truck should be used to address it. For this purpose, a dynamic programming (DP) formulation of the problem under consideration is devised. The ‘course of dimensionality’ leads to the necessity of introducing different mathematical programming models to represent the problem. The mathematical models we presented are an extension of the well-known approximations for the DP of traditional network capacity management analysis. Based on these models and exploiting revenue management concepts, primal and dual acceptance policies are developed and compared in a computational study.     

Effective program planning for multiple projects under limited resources

This work deals with an analytical approach to effective planning based upon the relative economics of projects contained in a program. A mathematical model is formulated to maximize the total return over the planning horizon subject to various annual cash flow and resource constraints. Stochastic mixed-integer programming is applied to determine the optimal sequence of project planning activities. To illustrate the use of the proposed procedure, a numerical example is solved using the mathematical programming software package SCICONIC/VM in a VAX-11/750 computer system.     

Models and algorithms for a staff scheduling problem

We present mathematical models and solution algorithms for a family of staff scheduling problems arising in real life applications. In these problems, the daily assignments to be performed are given and the durations (in days) of the working and rest periods for each employee in the planning horizon are specified in advance, whereas the sequence in which these working and rest periods occur, as well as the daily assignment for each working period, have to be determined. The main objective is the minimization of the number of employees needed to perform all daily assignments in the horizon. We decompose the problem into two steps: the definition of the sequence of working and rest periods (called pattern) for each employee, and the definition of the daily assignment to be performed in each working period by each employee. The first step is formulated as a covering problem for which we present alternative ILP models and exact enumerative algorithms based on these models. Practical experience shows that the best approach is based on the model in which variables are associated with feasible patterns and generated either by dynamic programming or by solving another ILP. The second step is stated as a feasibility problem solved heuristically through a sequence of transportation problems. Although in general this procedure may not find a solution (even if one exists), we present sufficient conditions under which our approach is guaranteed to succeed. We also propose an iterative heuristic algorithm to handle the case in which no feasible solution is found in the second step. We present computational results on real life instances associated with an emergency call center. The proposed approach is able to determine the optimal solution of instances involving up to several hundred employees and a working period of up to 6 months. Mathematics Subject Classification (2000): 90B70, 90C10, 90C27, 90C39, 90C57, 90C59     

基于Fuzzy推理的时变系统建模

提出一种基于Fuzzy推理的时变系统建模方法,其基本思想是:对时间维度进行分割,在每个较短的时间间隔内用时不变模型代替时变模型,将这些时不变模型组合在一起,最终获得一个整体非线性时变的微分方程模型.分别研究了输入输出型时变系统和状态空间型时变系统的模型建立方法,除了从理论上保证了所获得的模型对系统的逼近性,还从仿真实验验证了用该方法建立的模型对非线性时变系统有很好的逼近效果.     

Expanded model of the project portfolio selection problem with divisibility,time profile factors and cardinality constraints

This paper develops a new model for project portfolio selection over a planning horizon with multiple time periods. The model considers the divisibility of projects and combines reinvestment, set-up cost, cardinality constraints and precedence relationship in the scheduling, simultaneously. For efficient computation, an equivalent mixed integer linear programming representation is provided. One numerical example with three scenarios is given to highlight the capability and characteristics of the proposed model.     

Probabilistic programming models for traffic incident management operations planning

This paper proposes mathematical programming models with probabilistic constraints in order to address incident response and resource allocation problems for the planning of traffic incident management operations. For the incident response planning, we use the concept of quality of service during a potential incident to give the decision-maker the flexibility to determine the optimal policy in response to various possible situations. An integer programming model with probabilistic constraints is also proposed to address the incident response problem with stochastic resource requirements at the sites of incidents. For the resource allocation planning, we introduce a mathematical model to determine the number of service vehicles allocated to each depot to meet the resource requirements of the incidents by taking into account the stochastic nature of the resource requirement and incident occurrence probabilities. A detailed case study for the incident resource allocation problem is included to demonstrate the use of proposed model in a real-world context. The paper concludes with a summary of results and recommendations for future research.     

Planning and scheduling of multistage multiproduct batch plants operating under production campaigns

When plants are operated under stable conditions during reasonable time periods, operation with campaigns is particularly appropriate. The regular operation of the facilities simplifies the production control, the inventory management, the plant operability, etc. A?campaign includes several batches of different products that are going to be manufactured and the same one is cyclically repeated over the time horizon. In this work, a mixed integer linear programming formulation is proposed for the planning and scheduling of given multiproduct batch plants operating with campaigns. The number and size of batches for each product, the campaign composition, the assignment of batches to units and their sequencing, and the number of times that the campaign is repeated over the time horizon must be determined. Taking into account this scenario, an appropriate performance measure is the minimization of the cycle time. An asynchronous slot-based continuous-time representation for modeling the assignment of batches to units and their sequencing is employed, and a novel rule for determining the maximum number of slots postulated for each unit is proposed.     

A Long-range Selection and Timing Analysis System for Facility Location: Theory

This paper describes the development of a mixed-integer mathematical programming model designed to provide strategic decisions for the selection of a set of proposed facilities to serve distinct geographic regions and, of those selected, their projected timing within a short-time implementation horizon, so that the ratio of net return to total assets is maximized over the entire economic life of the facilities considered.     

A SEQUENTIAL APPROACH TO DERIVATION OF STAND TREATMENT AND FOREST-WIDE HARVEST PRESCRIPTION OR SUPPLY

Current methods for optimization of stand treatment and forest-wide harvest scheduling use mathematical programming that presumes perfect information on production, costs, and revenues over long planning periods. These approaches simultaneously optimize harvest for all periods in the planning horizon. In contrast, the method presented here assumes that stand-level planning and harvest scheduling proceed sequentially rather than simultaneously over every period. A backward-recursion dynamic program is used to determine the discounted net value of a wide range of current harvest strategies for each stand class in the forest inventory on the basis of a projected set of optimal treatments for future harvest and regeneration of each stand. The most highly valued strategy is selected if there are no volume constraints. Otherwise, suboptimal harvest alternatives are ranked in order of increasing opportunity cost for increasing or decreasing harvest; constraints met only up to a maximum opportunity cost is also demonstrated.     

Integrating two-dimensional cutting stock and lot-sizing problems

The two-dimensional cutting stock problem (2DCSP) consists in the minimization of the number of plates used to cut a set of items. In industry, typically, an instance of this problem is considered at the beginning of each planning time period, what may result in solutions of poor quality, that is, excessive waste, when a set of planning periods is considered. To deal with this issue, we consider an integrated problem, in which the 2DCSP is extended from the solution in only a single production planning period to a solution in a set of production planning periods. The main difference of the approach in this work and the ones in the literature is to allow sufficiently large residual plates (leftovers) to be stored and cut in a subsequent period of the planning horizon, which may further help in the minimization of the waste. We propose two integrated integer programming models to optimize the combined two-dimensional cutting stock and lot-sizing problems, minimizing the total cost, which includes material, waste and storage costs. Two heuristics based on the industrial practice to solve the problem were also presented. Computational results for the proposed models and for the heuristics are presented and discussed.     

A parallel algorithm of multi-variant evolutionary synthesis of nonlinear models

A parallel algorithm has been proposed for solving the problem of construction of nonlinear models (mathematical expressions, functions, algorithms, and programs) using given experimental data, set of variables, basic functions and operations. The designed algorithm of multivariant evolutionary synthesis of nonlinear models includes linear representation of a chromosome, modular operations in decoding of a genotype into a phenotype for interpreting a chromosome as a sequence of instructions, and a multi-variant method for presenting a set of models (expressions) using a single chromosome. A sequential version of the algorithm is compared with a standard genetic programming (GP) algorithm and a Cartesian genetic programming (CGP) one. The algorithm proposed was shown to excel the GP and CGP algorithms both in the time required for search for a solution (more than by an order of magnitude in most cases) and in the probability of finding a given function (model). Experiments have been carried out on parallel supercomputer systems, and estimates of the efficiency of the parallel algorithm offered have been obtained; the estimates demonstrate linear acceleration and scalability.     

A two-phase heuristic approach to the bike repositioning problem

An approach to overcome the bike imbalance problem is to transfer excess bikes to branches with bike shortages. This study develops a constrained mathematical model to deal with a multi-vehicle bike-repositioning problem, and aims to minimize the sum of transportation and unmet demand costs over a planning horizon through bike-transfer strategies under a minimum service requirement. A two-phase heuristic based on linear programming was proposed to solve the problem and produce compromising solutions. In the first phase, the paper developed a linear programming model to quickly develop decisions related to bike inventory, unloading, and loading for all stations for each time slot. In the second phase, this paper proposed an iterative approach through two parameter sensitive mathematical models to sequentially reduce the problem scale to develop decisions related to bike transfers. Computational results show that the proposed approach is superior to a CPLEX optimizer and a hybrid heuristic based on a genetic algorithm. The proposed approach was used to analyze the bicycle system in Taiwan. The impacts of various system parameters on the system were also investigated.     

Models and solution techniques for production planning problems with increasing byproducts

We consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete-time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MILP) approximation and relaxation models of this nonconvex MINLP, and we derive modifications that strengthen the linear programming relaxations of these models. We also introduce nonlinear programming formulations to choose piecewise-linear approximations and relaxations of multiple functions that share the same domain and use the same set of break points in the domain. We conclude with computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MILP approximation and relaxation models can be used to obtain provably near-optimal solutions for large instances of this nonconvex MINLP. Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.     

Furniture supply chain tactical planning optimization using a time decomposition approach

We study the supply chain tactical planning problem of an integrated furniture company located in the Province of Québec, Canada. The paper presents a mathematical model for tactical planning of a subset of the supply chain. The decisions concern procurement, inventory, outsourcing and demand allocation policies. The goal is to define manufacturing and logistics policies that will allow the furniture company to have a competitive level of service at minimum cost. We consider planning horizon of 1 year and the time periods are based on weeks. We assume that customer’s demand is known and dynamic over the planning horizon. Supply chain planning is formulated as a large mixed integer programming model. We developed a heuristic using a time decomposition approach in order to obtain good solutions within reasonable time limit for large size problems. Computational results of the heuristic are reported. We also present the quantitative and qualitative results of the application of the mathematical model to a real industrial case.     

Interval efficiency measures in data envelopment analysis with imprecise data

The conventional data envelopment analysis (DEA) measures the relative efficiencies of a set of decision making units (DMUs) with exact values of inputs and outputs. For imprecise data, i.e., mixtures of interval data and ordinal data, some methods have been developed to calculate the upper bound of the efficiency scores. This paper constructs a pair of two-level mathematical programming models, whose objective values represent the lower bound and upper bound of the efficiency scores, respectively. Based on the concept of productive efficiency and the application of a variable substitution technique, the pair of two-level nonlinear programs is transformed to a pair of ordinary one-level linear programs. Solving the associated pairs of linear programs produces the efficiency intervals of all DMUs. An illustrative example verifies the idea of this paper. A real case is also provided to give some interpretation of the interval efficiency. Interval efficiency not only describes the real situation in better detail; psychologically, it also eases the tension of the DMUs being evaluated as well as the persons conducting the evaluation.     

Neuro-fuzzy based constraint programming

Constraint programming models appear in many sciences including mathematics, engineering and physics. These problems aim at optimizing a cost function joint with some constraints. Fuzzy constraint programming has been developed for treating uncertainty in the setting of optimization problems with vague constraints. In this paper, a new method is presented into creation fuzzy concept for set of constraints. Unlike to existing methods, instead of constraints with fuzzy inequalities or fuzzy coefficients or fuzzy numbers, vague nature of constraints set is modeled using learning scheme with adaptive neural-fuzzy inference system (ANFIS). In the proposed approach, constraints are not limited to differentiability, continuity, linearity; also the importance degree of each constraint can be easily applied. Unsatisfaction of each weighted constraint reduces membership of certainty for set of constraints. Monte-Carlo simulations are used for generating feature vector samples and outputs for construction of necessary data for ANFIS. The experimental results show the ability of the proposed approach for modeling constrains and solving parametric programming problems.     

Multi-objective optimization in uncertain random environments

Uncertain random variables are used to describe the phenomenon of simultaneous appearance of both uncertainty and randomness in a complex system. For modeling multi-objective decision-making problems with uncertain random parameters, a class of uncertain random optimization is suggested for decision systems in this paper, called the uncertain random multi-objective programming. For solving the uncertain random programming, some notions of the Pareto solutions and the compromise solutions as well as two compromise models are defined. Subsequently, some properties of these models are investigated, and then two equivalent deterministic mathematical programming models under some particular conditions are presented. Some numerical examples are also given for illustration.     

Synchronizing production and air transportation scheduling using mathematical programming models

Traditional scheduling problems assume that there are always infinitely many resources for delivering finished jobs to their destinations, and no time is needed for their transportation, so that finished products can be transported to customers without delay. So, for coordination of these two different activities in the implementation of a supply chain solution, we studied the problem of synchronizing production and air transportation scheduling using mathematical programming models. The overall problem is decomposed into two sub-problems, which consists of air transportation allocation problem and a single machine scheduling problem which they are considered together. We have taken into consideration different constraints and assumptions in our modeling such as special flights, delivery tardiness and no delivery tardiness. For these purposes, a variety of models have been proposed to minimize supply chain total cost which encompass transportation, makespan, delivery earliness tardiness and departure time earliness tardiness costs.