On lexicographic optimal solutions in transportation problems

Lexicographic versions of the cost minimizing transportation problem (CMTP) and the time minimizing transportation problem (TMTP) are presented in this paper. In addition to minimizing the quantity sent on the costliest routes in a cost minimizing transportation problem. an attempt is made to minimize the quantity transported on the second-costliest routes. if the shipment on the costliest routes is as small as possible and the quantity shipped on the third-costliest routes, if the shipments on the costliest and the second- costliest routes are as small as possible. and so on. In a lexicographic time minimizing transportation problem one is not only interested in minimizing the transportation cost on the routes of the longest duration but also on the routes of second longest, third-longest duration and so on. For finding lexicographic optimal solutions (LOS) of lexicographic cost minimizing and time minimizing transportation problems a standard cost minimizing transportation problem is formulated whose optimal solution is shown to provide the answer. Some extensions are also discussed     

Capacitated transportation problem with bounds on RIM conditions

Motivated by dead-mileage problem assessed in terms of running empty buses from various depots to starting points, we consider a class of the capacitated transportation problems with bounds on total availabilities at sources and total destination requirements. It is often difficult to solve such problems and the present paper establishes their equivalence with a balanced capacitated transportation problem which can be easily solved by existing methods. Sometimes, total flow in transportation problem is also specified by some external decision maker because of budget/political consideration and optimal solution of such problem is of practical interest to the decision maker and has motivated us to discuss such problem. Various situations arising in unbalanced capacitated transportation problems have been discussed in the present paper as a particular case of original problem. In addition, we have discussed paradoxical situation in a balanced capacitated transportation problem and have obtained the paradoxical solution by solving one of the unbalanced problems. Numerical illustrations are included in support of theory.     

现金押运路线问题——基于多类型的风险

以人民币现金押运为研究背景,考虑了一种基于多类型风险的现金押运路线问题,以在途风险成本、库存现金风险成本以及运输成本为优化目标,建立了混合整数线性规划模型,并提出了一种基于多样化策略和改进邻域搜索的混合遗传算法,其中遗传算法对押运路线进行选择,贪心算法用来求解各类风险指标。数值实验分别对问题特性和算法性能进行了分析。实验结果表明:1)混合遗传算法能求解更大规模的问题,得到较好的解,并很好地平衡了运行时间和求解质量;2)多类型风险影响了行驶路线;3)客户的期望需求影响了库存现金风险。     

An interactive fuzzy satisficing method for multiobjective nonconvex programming problems through floating point genetic algorithms

In this paper, we focus on multiobjective nonconvex nonlinear programming problems and present an interactive fuzzy satisficing method through floating point genetic algorithms. After determining the fuzzy goals of the decision maker, if the decision maker specifies the reference membership values, the corresponding Pareto optimal solution can be obtained by solving the augmented minimax problems for which the floating point genetic algorithm, called GENOCOP III, is applicable. In order to overcome the drawbacks of GENOCOP III, we propose the revised GENOCOP III by introducing a method for generating an initial feasible point and a bisection method for generating a new feasible point efficiently. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from a Pareto optimal solution set is presented together with an illustrative numerical example.     

条件延期付款下一体化可控提前期模型研究

在一体化决策的供应链中,假设提前期与提前期成本服从幂函数关系,生产商允许零售商延期付款,但需要在订货时预付部分货款。首先,在延迟付款期给定时,通过建立该问题的数学模型,证明了不考虑决策变量整数约束时,系统存在唯一最优生产量、提前期和生产商每个生产周期内的运输批次,使得供应链总成本最小,并设计求解算法对问题进行了求解。以此为基础,假设延迟付款期是零售商订货量的增函数,结合模型的解析性质,给出了新的求解算法。最后通过数值算例和敏感性分析,说明了所得结论及其管理启示。     

An interactive method for multiple-objective mathematical programming problems

An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.     

改进的蝗虫优化算法在双目标应急物资中心选址问题中的应用

近年来世界各地频发灾情疫情等紧急事件,严重影响人民的生活物资保障。在这种情况下,急需建立应急物资中心来缓解燃眉之急。该类问题通常面临资源稀缺并且时间相对紧迫的处境,因此需要在短时间内获得合理的应急设施选址方案来提升服务的质量和效率。本文对应急物资中心选址问题展开研究,提出一种考虑后续运输成本以及有概率发生紧急事件而导致无法正常运送物资的双目标离散选址模型,并为此设计一种二进制多目标蝗虫优化算法。该算法采用模糊关联熵系数来引导迭代更新,同时为其添加外部档案,最优解选择机制和竞争决策机制来提升算法性能。多次数值实验表明该算法的计算效率和求解质量较高,可作为应急物资中心选址问题的一种可行且有效的算法。     

Development of the tree-based link labeling algorithm for optimal path-finding in urban transportation networks

The optimal path-finding algorithm which is an important module in developing route guidance systems and traffic control systems has to provide correct paths to consider U-turns, P-turns, and no-left-turns in urban transportation networks.Traditional methods which have been used to consider those regulations on urban transportation networks can be categorized into network representation and algorithmic methods like the vine-building algorithm. First, network representation methods use traditional optimal path-finding algorithms with modifications to the network structure: for example, just adding dummy nodes and links to the existing network allows constraint-search in the network. This method which creates large networks is hard to implement and introduces considerable difficulties in network coding. With the increased number of nodes and links, the memory requirement tremendously increases, which causes the processing speed to slow down. For these reasons, the method has not been widely accepted for incorporating turning regulations in optimal path-finding problems in transportation networks. Second, algorithmic methods, as they are mainly based on the vine-building algorithm, have been suggested for determining optimal path for networks with turn penalties and prohibitions. However, the algorithms, although they nicely reflect the characteristics of urban transportation networks, frequently provide infeasible or suboptimal solutions.The algorithm to be suggested in this research is a method which is basically based on Dijkstra's algorithm [1] and the tree-building algorithm used to construct optimal paths. Unlike the traditional node labeling algorithms which label each node with minimum estimated cost, this algorithm labels each link with minimum estimated cost.Comparison with the vine-building algorithm shows that the solution of the link-labeling algorithm is better than that of the vine-building algorithm which very frequently provides suboptimal solutions. As a result, the algorithm allows turning regulations, while providing an optimal solution within a reasonable time limit.     

A Stochastic Bottleneck Transportation Problem

In this paper, a stochastic bottleneck transportation problem, which aims at minimizing the transportation time target subject to a chance constraint, is formulated and an algorithm based on a parametric programming approach is developed to solve it. Further, assuming the transportation costs to be deterministic, a trade-off analysis between the transportation time target and the total cost is given. In addition, methods are developed which give the whole spectrum of optimal solutions to the problems mentioned above. The algorithms are illustrated by numerical examples. The computational complexity of the algorithms is also discussed.     

运输问题的合作对策求解方法

产地间或销地间往往存在竞争,在这种情况下,使用运输问题最优化方法是不合理的。因此,从个体理性的视角提出运输问题的合作对策求解方法,方法将运输问题看作是一个博弈问题,各个产地或销地是博弈的局中人,求解其纳什均衡与纳什讨价还价解。在此基础上,说明了运输问题的非合作形式是一个指派问题,并证明指派问题的最优解是一个纳什均衡点。接着,通过实验验证运输问题的最优解是一个纳什讨价还价解,满足产地或销地的自身利益。在此基础上,针对纳什讨价还价解不唯一的问题,从决策者的视角给出最大可能激励成本的计算方法。最后,为弥补纳什讨价还价解不唯一及纳什讨价还价解不允许出现子联盟的缺陷,给出运输收益分配或成本分摊的Shapely值计算方法。     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.     

An interactive approach for multiobjective decision making

We develop an interactive approach for multiobjective decision-making problems, where the solution space is defined by a set of constraints. We first reduce the solution space by eliminating some undesirable regions. We generate solutions (partition ideals) that dominate portions of the efficient frontier and the decision maker (DM) compares these with feasible solutions. Whenever the decision maker prefers a feasible solution, we eliminate the region dominated by the partition ideal. We then employ an interactive search method on the reduced solution space to help the DM further converge toward a highly preferred solution. We demonstrate our approach and discuss some variations.     

An adaptive decision maker for constrained evolutionary optimization

An adaptive decision maker (ADM) is proposed for constrained evolutionary optimization. This decision maker, which is designed in the form of an adaptive penalty function, is used to decide which solution candidate prevails in the Pareto optimal set and to choose the individuals to be replaced. By integrating the ADM with a model of a population-based algorithm-generator, a novel generic constrained optimization evolutionary algorithm is derived. The performance of the new method is evaluated by 13 well-known benchmark test functions. It is shown that the ADM has powerful ability to balance the objective function and the constraint violations, and the results obtained are very competitive to other state-of-the-art techniques referred to in this paper in terms of the quality of the resulting solutions.     

Design of progressively censored group sampling plans for Weibull distributions: An optimization problem

Optimization algorithms provides efficient solutions to many statistical problems. Essentially, the design of sampling plans for lot acceptance purposes is an optimization problem with several constraints, usually related to the quality levels required by the producer and the consumer. An optimal acceptance sampling plan is developed in this paper for the Weibull distribution with unknown scale parameter. The proposed plan combines grouping of items, sudden death testing in each group and progressive group removals, and its decision criterion is based on the uniformly most powerful life test. A mixed integer programming problem is first solved for determining the minimum number of failures required and the corresponding acceptance constant. The optimal number of groups is then obtained by minimizing a balanced estimation of the expected test cost. Excellent approximately optimal solutions are also provided in closed-forms. The sampling plan is considerably flexible and allows to save experimental time and cost. In general, our methodology achieves solutions that are quite robust to small variations in the Weibull shape parameter. A numerical example about a manufacturing process of gyroscopes is included for illustration.     

Single- and multi-objective defensive location problems on a network

This paper considers a new optimal location problem, called defensive location problem (DLP). In the DLPs, a decision maker locates defensive facilities in order to prevent her/his enemies from reaching an important site, called a core; for example, “a government of a country locates self-defense bases in order to prevent her/his aggressors from reaching the capital of the country.” It is assumed that the region where the decision maker locates her/his defensive facilities is represented as a network and the core is a vertex in the network, and that the facility locater and her/his enemy are an upper and a lower level of decision maker, respectively. Then the DLPs are formulated as bilevel 0-1 programming problems to find Stackelberg solutions. In order to solve the DLPs efficiently, a solving algorithm for the DLPs based upon tabu search methods is proposed. The efficiency of the proposed solving methods is shown by applying to examples of the DLPs. Moreover, the DLPs are extended to multi-objective DLPs that the decision maker needs to defend several cores simultaneously. Such DLPs are formulated as multi-objective programming problems. In order to find a satisfying solution of the decision maker for the multi-objective DLP, an interactive fuzzy satisfying method is proposed, and the results of applying the method to examples of the multi-objective DLPs are shown.     

Planning and consolidating shipments from a warehouse

A typical warehouse or distribution centre ships material to various customer locations across the country, using various modes of transportation. Each mode has different constraints on size of shipment, different cost structures and different transportation times. Typically, for a given warehouse there are certain customer locations that receive frequent shipments of material. It is often possible, therefore, for the warehouse to consolidate different orders for the same customer location into a single shipment. The transportation mode and the day of shipment must be chosen such that the consolidated shipment meets the size constraints and arrives within an agreed-upon ‘delivery window’. In preparing a warehouse distribution plan, a planner seeks to achieve transportation economies of scale (by consolidating two or more orders into fewer shipments) while levelling the workload on warehouse resources and ensuring that material arrives at a customer location during the acceptable delivery window.The problem of deciding what shipments to make daily can be formulated as a set partitioning problem with side constraints. This paper describes a heuristic solution approach for this problem. Computational experiments using actual warehouse select activity indicate that, for moderate-size problems, the heuristic produces solutions with transportation costs that are within a few percent of optimal. Larger problems found in practice are generally too large to be solved by optimal algorithms; the heuristic easily handles such problems. The heuristic has been integrated into the transportation planning system of a leading distributor of telecommunications products.     

A multi-objective mathematical model for the industrial hazardous waste location-routing problem

Industrial hazardous waste management involves the collection, transportation, treatment, recycling and disposal of industrial hazardous materials that pose risk to their surroundings. In this paper, a new multi-objective location-routing model is developed, and implemented in the Marmara region of Turkey. The aim of the model is to help decision makers decide on locations of treatment centers utilizing different technologies, routing different types of industrial hazardous wastes to compatible treatment centers, locations of recycling centers and routing hazardous waste and waste residues to those centers, and locations of disposal centers and routing waste residues there. In the mathematical model, three criteria are considered: minimizing total cost, which includes total transportation cost of hazardous materials and waste residues and fixed cost of establishing treatment, disposal and recycling centers; minimizing total transportation risk related to the population exposure along transportation routes of hazardous materials and waste residues; and minimizing total risk for the population around treatment and disposal centers, also called site risk. A lexicographic weighted Tchebycheff formulation is developed and computed with CPLEX software to find representative efficient solutions to the problem. Data related to the Marmara region is obtained by utilizing Arcview 9.3 GIS software and Marmara region geographical database.     

Resolution approach for multi-objective problems with uncertain demands

The job-shop scheduling problem (JSP) is one of the hardest problems (NP-complete problem). In a lot of cases, the combination of goals and resource exponentially increases search space. The objective of resolution of such a problem is generally, to maximize the production with a lower cost and makespan. In this paper, we explain how to modify the objective function of genetic algorithms to treat the multi-objective problem and to generate a set of diversified “optimal” solutions in order to help decision maker. We are interested in one of the problems occurring in the production workshops where the list of demands is split into firm (certain) jobs and predicted jobs. One wishes to maximize the produced quantity, while minimizing as well as possible the makespan and the production costs. Genetic algorithms are used to find the scheduling solution of the firm jobs because they are well adapted to the treatment of the multi-objective optimization problems. The predicted jobs will be inserted in the real solutions (given by genetic algorithms). The solutions proposed by our approach are compared to the lower bound of the cost and makespan in order to prove the quality and robustness of our proposed approach.     

Max-min sum minimization transportation problem

A non-convex optimization problem involving minimization of the sum of max and min concave functions over a transportation polytope is studied in this paper. Based upon solving at most (g+1)(< p) cost minimizing transportation problems with m sources and n destinations, a polynomial time algorithm is proposed which minimizes the concave objective function where, p is the number of pairwise disjoint entries in the m× n time matrix {t _ij } sorted decreasingly and T ^g is the minimum value of the max concave function. An exact global minimizer is obtained in a finite number of iterations. A numerical illustration and computational experience on the proposed algorithm is also included. We are thankful to Prof. S. N. Kabadi, University of New Brunswick-Fredericton, Canada, who initiated us to the type of problem discussed in this paper. We are also thankful to Mr. Ankit Khandelwal, Ms. Neha Gupta and Ms. Anuradha Beniwal, who greatly helped us in the implementation of the proposed algorithm.     

Genetic algorithms applied to the solution of hybrid optimal control problems in astrodynamics

Many space mission planning problems may be formulated as hybrid optimal control problems, i.e. problems that include both continuous-valued variables and categorical (binary) variables. There may be thousands to millions of possible solutions; a current practice is to pre-prune the categorical state space to limit the number of possible missions to a number that may be evaluated via total enumeration. Of course this risks pruning away the optimal solution. The method developed here avoids the need for pre-pruning by incorporating a new solution approach using nested genetic algorithms; an outer-loop genetic algorithm that optimizes the categorical variable sequence and an inner-loop genetic algorithm that can use either a shape-based approximation or a Lambert problem solver to quickly locate near-optimal solutions and return the cost to the outer-loop genetic algorithm. This solution technique is tested on three asteroid tour missions of increasing complexity and is shown to yield near-optimal, and possibly optimal, missions in many fewer evaluations than total enumeration would require.