Sequence Independent Lifting in Mixed Integer Programming

We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive since it involves the solution of an optimization problem to compute a lifting coefficient for each variable. To relieve this computational burden, we study sequence independent lifting, which only involves the solution of one optimization problem. We show that if a certain lifting function is superadditive, then the lifting coefficients are independent of the lifting sequence. We introduce the idea of valid superadditive lifting functions to obtain good aproximations to maximum lifting. We apply these results to strengthen Balas' lifting theorem for cover inequalities and to produce lifted flow cover inequalities for a single node flow problem.     

Shifting strategy for geometric graphs without?geometry

We give a simple framework which is an alternative to the celebrated and widely used shifting strategy of Hochbaum and Maass (J. ACM 32(1):103?C136, 1985) which has yielded efficient algorithms with good approximation bounds for numerous optimization problems in low-dimensional Euclidean space. Our framework does not require the input graph/metric to have a geometric realization??it only requires that the input graph satisfy some weak property referred to as growth boundedness. Growth bounded graphs form an important graph class that has been used to model wireless networks. We show how to apply the framework to obtain a polynomial time approximation scheme (PTAS) for the maximum (weighted) independent set problem on this important graph class; the problem is W[1]-complete. Via a more sophisticated application of our framework, we show how to obtain a PTAS for the maximum (weighted) independent set for intersection graphs of (low-dimensional) fat objects that are expressed without geometry. Erlebach et al. (SIAM J. Comput. 34(6):1302?C1323, 2005) and Chan (J. Algorithms 46(2):178?C189, 2003) independently gave a PTAS for maximum weighted independent set problem for intersection graphs of fat geometric objects, say ball graphs, which required a geometric representation of the input. Our result gives a positive answer to a question of Erlebach et al. (SIAM J. Comput. 34(6):1302?C1323, 2005) who asked if a PTAS for this problem can be obtained without access to a geometric representation.     

A note on anti-coordination and social interactions

This note confirms a conjecture of (Bramoullé in Games Econ Behav 58:30–49, 2007). The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than n ^1?? , where n is the number of nodes, and ? arbitrarily small, unless $\mathrm{P} = \mathrm{NP}$ . For the rather special case where each node has a degree of at most four, the problem is still APX-hard.     

Locating independent facilities with maximum weight: Greedy heuristics

The problem is to locate a maximum-weight set of facilities such that no two are closer than a given distance from each other. The unweighted version is equivalent to the maximum independent set problem in graph theory. This paper presents four greedy heuristics and shows that they all have bad worst-case behavior. Empirically, however, these heuristics perform quite well in the relatively large test problems generated randomly.     

Combinatorial design of a minimum cost transfer line

The problem of equipment selection for a production line is considered. Each piece of equipment, also called unit or block, performs a set of operations. All necessary operations of the line and all available blocks with their costs are known. The difficulty is to choose the most appropriate blocks and group them into (work)stations. There are some constraints that restrict the assignment of different blocks to the same station. Two combinatorial approaches for solving this problem are suggested. Both are based on a novel concept of locally feasible stations. The first approach combinatorially enumerates all feasible solutions, and the second reduces the problem to search for a maximum weight clique. A boolean linear program based on a set packing formulation is presented. Computer experiments with benchmark data are described. Their results show that the set packing model is competitive and can be used to solve real-life problems.     

A refined algorithm for maximum independent set in degree-4 graphs

The maximum independent set problem is one of the most important problems in theoretical analysis on time and space complexities of exact algorithms. Theoretical improvement on upper bounds on time complexity to solve this problem in low-degree graphs can lead to an improvement on that to the problem in general graphs. In this paper, we derive an upper bound \(O^*(1.1376^n)\) on the time complexity of a polynomial-space algorithm that solves the maximum independent set problem in an n-vertex graph with degree bounded by 4, improving all previous upper bounds on the time complexity of exact algorithms to this problem. Our algorithm is a branch-and-reduce algorithm and analyzed by using the measure-and-conquer method. To make an amortized analysis of the running time bound, we use an idea of “shift” to save some decrease of the measure from good branches to bad branches. Our algorithm first deals with small vertex cuts and vertices of degree \({\ge }5\), which may be created in our algorithm even if the input graph has maximum degree 4, then eliminates cycles of length 3 and 4 containing degree-4 vertices, and finally branches on degree-4 vertices. We invoke an exact algorithm for this problem in graphs with maximum degree 3 directly when the graph has no vertices of degree \({\ge }4\). Branching on degree-4 vertices on special local structures will be the bottleneck case, and we carefully design rules of choosing degree-4 vertices to branch on so that the resulting instances after branching decrease the measure effectively in the next step.     

Inventory Control and Scheduling of Multiple Products in a Common Facility*

Consider a set of chemical products to be produced in a single facility. Each product has its own unique reaction time (which is assumed to be independent of its batch size), as well as other cost and demand values. In this paper, we address the problem of determining the optimal number of batches, batch sizes, and an accompanying production schedule for these products in the single facility that will minimize the total cost. Two different algorithms have been developed for this problem, the performances of which are contrasted with classical cyclic production schedules. Finally, some guidelines for the application of these methods to real-life problems are outlined.     

Optimality conditions for a bilevel matroid problem

In bilevel programming there are two decision makers, the leader and the follower, who act in a hierarchy. In this paper we deal with a weighted matroid problem where each of the decision makers has a different set of weights. The independent set of the matroid that is chosen by the follower determines the payoff to both the leader and the follower according to their different weights. The leader can increase his payoff by changing the weights of the follower, thus influencing the follower’s decision, but he has to pay a penalty for this. We want to find an optimum strategy for the leader. This is a bilevel programming problem with continuous variables in the upper level and a parametric weighted matroid problem in the lower level. We analyze the structure of the lower level problem. We use this structure to develop local optimality criteria for the bilevel problem that can be verified in polynomial time.     

Tour Scheduling and Task Assignment of a Heterogeneous Work Force: A Heuristic Approach

The dual problem of work tour scheduling and task assignment involving workers who differ in their times of availability and task qualifications is examined in this paper. The problem is presented in the context of a fast food restaurant, but applies equally well to a diverse set of service operations. Developing a week-long labor schedule is a nontrivial problem, in terms of complexity and importance, which a manager spends as much as a full workday solving. The primary scheduling objective (the manager's concern) is the minimization of overstaffing in the face of significant hourly and daily fluctuations in minimum staffing requirements. The secondary objective (the workers’ concern) is the minimization of the sum of the squared differences between the number of work hours scheduled and the number targeted for each employee. Contributing to scheduling complexity are constraints on the structure of work tours, including minimum and maximum shift lengths and a maximum number of workdays. A goal programming formulation of a representative problem is shown to be too large, for all practical purposes, to be solved optimally. Existing heuristic procedures related to this research possess inherent limitations which render them inadequate for our purposes. Subsequently, we propose and demonstrate a computerized heuristic procedure capable of producing a labor schedule requiring at most minor refinement by a manager.     

An adaptive multistart tabu search approach to solve the maximum clique problem

Given an undirected graph G=(V,E) with vertex set V={1,…,n} and edge set E?V×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.     

计算模糊综合评价逆问题的一种方法

计算模糊综合评价的逆问题,有助于总结评价经验,具有普遍的应用价值。目前常用的方法是经验性枚举选优法,其计算结果与计算者的经验和枚举的次数有关。为此,该文中把该问题等价于一个以贴近度为目标函数、权重模糊集为优化变量、含有最大最小运算的非线性优化问题,并提出用加速遗传算法(AGA)来计算该问题的新方法。实例的结果说明,AGA简便、有效且具有通用性,其计算精度高于枚举选优法的相应结果,在模糊综合评价的理论与实践中具有一定价值。     

Analyzing and Modeling the Maximum Diversity Problem by Zero-One Programming*

The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new model is proposed in which the concept of diversity is quantifiable and measurable. A quadratic zero-one model is formulated for diversity maximization. Based upon the formulation, it is shown that the maximum diversity problem is NP-hard. Two equivalent linear integer programs are then presented that offer progressively greater computational efficiency. Another formulation is also introduced which involves a different diversity objective. An example is given to illustrate how additional considerations can be incorporated into the maximum diversity model.     

On the fair optimization of cost and customer service level in a supply chain under disruption risks

This paper presents a new decision-making problem of a fair optimization with respect to the two equally important conflicting objective functions: cost and customer service level, in the presence of supply chain disruption risks. Given a set of customer orders for products, the decision maker needs to select suppliers of parts required to complete the orders, allocate the demand for parts among the selected suppliers, and schedule the orders over the planning horizon, to equitably optimize expected cost and expected customer service level. The supplies of parts are subject to independent random local and regional disruptions. The fair decision-making aims at achieving the normalized expected cost and customer service level values as much close to each other as possible. The obtained combinatorial stochastic optimization problem is formulated as a stochastic mixed integer program with the ordered weighted averaging aggregation of the two conflicting objective functions. Numerical examples and computational results, in particular comparison with the weighted-sum aggregation of the two objective functions are presented and some managerial insights are reported. The findings indicate that for the minimum cost objective the cheapest supplier is usually selected, and for the maximum service level objective a subset of most reliable and most expensive suppliers is usually chosen, whereas the equitably efficient supply portfolio usually combines the most reliable and the cheapest suppliers. While the minimum cost objective function leads to the largest expected unfulfilled demand and the expected production schedule for the maximum service level follows the customer demand with the smallest expected unfulfilled demand, the equitably efficient solution ensures a reasonable value of expected unfulfilled demand.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

Transforming an MRP plan into a short-term schedule

A new approach for transforming MRP orders, planned periodically, e.g. on a weekly base, into a detailed sequence of jobs is presented. In this model for a single machine environment, the jobs are partitioned into families and a family specific set-up time is required at the start of each period and of each batch, where a batch is a maximal set of jobs in the same family, that are processed consecutively. An integer program is formulated for both the problem of minimizing the number of overloaded periods and the problem of minimizing the total overtime. These programs generate benchmark results for the heuristic approach. A heuristic model is developed that constructs a schedule in which overloaded periods are relieved and set-up time is saved. In this approach, the job sequence is constructed by repeatedly solving a knapsack problem. The weights used in this knapsack problem relate to the preferred priorities of the jobs not yet scheduled and determine the quality of the final sequence. The different features of the heuristic model are compared using a large set of test problems. The results show that the quality of the final sequence depends on an appropriate choice for the weights.     

Maximizing Revenue Through Two‐Dimensional Shelf‐Space Allocation

We consider the problem of optimally allocating contiguous rectangular presentation spaces in order to maximize revenues. Such problems are encountered in the arrangement of products in retail shelf‐space and in the design of feature advertising displays or webpages. Specifically, we allow (i) the shape of a product's presentation to have a vertical as well as a horizontal component and (ii) displays to extend across multiple shelves for in‐store presentations. Since the vertical location of the shelf on which a product is displayed affects its sales, each vertical location is assigned its own effectiveness with regard to revenue generation. The problem of maximizing the total weighted revenue of a display is strongly NP‐hard. Therefore, we decompose it into two subproblems. The first consists of allocating products to different cabinets. In the second, within each cabinet, each product's units are arranged in a contiguous rectangle and assigned a location. These subproblems are solved using an innovative approach that uses a combination of integer programming and an algorithm for the maximum‐weight independent set problem. Based on computational studies on both real‐world and simulated data, we demonstrate the efficiency and effectiveness of our approach. Specifically, the revenue generated by this scheme is within 1% of the optimum for actual data and within 5% for simulated data.     

Allocating a fixed cost across the decision making units with two-stage network structures

A prominent issue in many organizations involves the fair allocation of a total fixed cost among a group of entities. This paper extends the traditional fixed cost allocation problem to situations where the decision making units (DMUs) have a two-stage network structure. To this end, this paper first uses the data envelopment analysis (DEA) methodology to determine the relative efficiency while taking the internal structure and possible allocated costs into account. It shows that each DMU can separately maximize its relative efficiency to one through determining a series of allocations and selecting a set of relative weights. Next, we demonstrate that there exists an efficient allocation set based on a set of common weights, using which all DMUs and their two sub-stages can be simultaneously efficient. However, there are alternative allocation plans in the efficient allocation set. According to this non-uniqueness problem, we further optimize the allocation plans by taking the size of operation units into account, such that the allocation result is proportional to current input usages and output productions from a size point of view. In addition, we suggest a min–max model and a feasible computation algorithm for it to generate the final allocation plan in a way that minimizes the deviation between the efficient allocations and size allocations. More importantly, by repeatedly minimizing the maximum deviation, our proposed method can guarantee a unique allocation plan for all DMUs and sub-stages. Finally, both a numerical example modified from previous literature and an empirical application of bank activities are used to demonstrate the efficacy and usefulness of the proposed approach.     

同时集散货物的开放式车辆路径问题研究

本文研究了车辆工作时间限制下同时集散货物的多配送中心开放式车辆路径问题,以车辆数和运输里程最小为目标,建立了多目标规划模型,提出了基于拉格朗日松弛技术和禁忌搜索算法的混合求解算法。 该算法首先求出最优解的最大下界,然后采用客户点的分配和调整策略实现解的可行化,其中禁忌搜索引入了4种领域搜索方法,采用了随机变领域搜索方法和重起策略。算例分析表明,该算法能有效地找到满意解,且采用开放式安排路线比闭合式安排路线更加经济合理。     

Fitting Hierarchical Holographic Modeling into the Theory of Scenario Structuring and a Resulting Refinement to the Quantitative Definition of Risk

A point of view is suggested from which the Hierarchical Holographic Modeling (HHM) method can be seen as one more method within the Theory of Scenario Structuring (TSS), which is that part of Quantitative Risk Assessment having to do with the task of identifying the set of risk scenarios. Seen in this way, HHM brings strongly to our attention the fact that different methods within TSS can result in different sets of risk scenarios for the same underlying problem. Although this is not a problem practically, it is a bit awkward conceptually from the standpoint of the "set of triplets" definition of risk, in which the scenario set is part of the definition. Accordingly, the present article suggests a refinement to the set of triplets definition, which removes the specific set of scenarios, found by any of the TSS methods, from the definition of risk and casts it, instead, as an approximation to the "true" set of scenarios that is native to the problem at hand and not affected by the TSS method used.