Separating Type-I Odd-Cycle Inequalities for a Binary-Encoded Edge-Coloring Formulation

In this note, we describe an efficient algorithm for separating a class of inequalities that includes the type-I odd-cycle inequalities for a binary-encoded edge-coloring formulation.     

Facets of an Assignment Problem with 0–1 Side Constraint

We show that the problem of finding a perfect matching satisfying a single equality constraint with a 0–1 coefficients in an n × n incomplete bipartite graph, polynomially reduces to a special case of the same peoblem called the partitioned case. Finding a solution matching for the partitioned case in the incomlpete bipartite graph, is equivalent to minimizing a partial sum of the variables over = the convex hull of incidence vectors of solution matchings for the partitioned case in the complete bipartite graph. An important strategy to solve this minimization problem is to develop a polyhedral characterization of . Towards this effort, we present two large classes of valid inequalities for , which are proved to be facet inducing using a facet lifting scheme.     

Composition of Graphs and the Triangle-Free Subgraph Polytope

In this paper, we study a composition (decomposition) technique for the triangle-free subgraph polytope in graphs which are decomposable by means of 3-sums satisfying some property. If a graph G decomposes into two graphs G ₁ and G ₂, we show that the triangle-free subgraph polytope of G can be described from two linear systems related to G ₁ and G ₂. This gives a way to characterize this polytope on graphs that can be recursively decomposed. This also gives a procedure to derive new facets for this polytope. We also show that, if the systems associated with G ₁ and G ₂ are TDI, then the system characterizing the polytope for G is TDI. This generalizes previous results in R. Euler and A.R. Mahjoub (Journal of Comb. Theory series B, vol. 53, no. 2, pp. 235–259, 1991) and A.R. Mahjoub (Discrete Applied Math., vol. 62, pp. 209–219, 1995).     

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.     

Reducing Boarding in a Post‐Anesthesia Care Unit

When operating room schedules in hospitals are produced, the constraints and preferences of surgeons and hospital workers are a primary consideration. The downstream impact on post‐operative bed availability is often ignored. This can lead to the boarding of patients overnight in the post‐anesthesia care unit (PACU) because intensive care unit beds are unavailable. In this paper, we apply integer programming and simulation to develop improved surgical scheduling assignments. We want to balance new surgeries with hospital discharges in order to reduce the variability of occupied beds from one day to the next and, as a result, to reduce boarding in the PACU.     

Forecasting lead times with economic data

Procurement lead time estimates are critical factors in production planning; however, they are generally based on a buyer's experience. The Westinghouse Corporation developed a linear model which successfully predicted procurement lead times for hot and cold rolled steel. This paper discusses nine models which use utilization and inventory data to predict lead time planning factors for aluminum, magnesium, steel and titanium products. The Lockheed-Georgia Company, which produces aerospace products, provided data for this research. The functional form of the models is significant and suggests that a nonlinear transformation of utilization based upon simple queueing models is a significant predictor of lead time duration.     

On Approximating a Scheduling Problem

Given a set of communication tasks (best described in terms of a weighted bipartite graph where one set of nodes denotes the senders, the other set the receivers, edges are communication tasks, and the weight of an edge is the time required for transmission), we wish to minimize the total time required for the completion of all communication tasks assuming that tasks can be preempted (that is, each edge can be subdivided into many edges with weights adding up to the edge's original weight) and that preemption comes with a cost. In this paper, we first prove that one cannot approximate this problem within a factor smaller than unless P=NP. It is known that a simple approximation algorithm achieves within a ratio of two (H. Choi and S.L. Hakimi, Algorithmica, vol. 3, pp. 223–245, 1988). However, our experimental results show that its performance is worse than the originally proposed heuristic algorithm (I.S. Gopal and C.K. Wong, IEEE Transactions on Communications, vol. 33, pp. 497–501, 1985). We devise a more sophisticated algorithm, called the potential function algorithm which, on the one hand, achieves a provable approximation ratio of two, and on the other hand, shows very good experimental performance. Moreover, the way in which our more sophisticated algorithm derives from the simple one, suggests a hierarchy of algorithms, all of which have a worst-case performance at most two, but which we suspect to have increasingly better performance, both in worst case and with actual instances.     

Strengthened 0-1 linear formulation for the daily satellite mission planning

In this paper, we compare several 0-1 linear programs for solving the satellite mission planning problem. We prove that one of them presents a smaller integrality gap. Our explanation is based on stable set polytope formulations for perfect graphs.     

Simplicial Pivoting Algorithms for a Tractable Class of Integer Programs

We present a tractable class of integer feasibility problems. This class of max-closed IP problems was studied in somewhat restricted form by Glover, Pnueli, Hochbaum and Chandrasekaran and has a logic counterpart known as the class of Horn formulas. First we modify the existing algorithms in order to avoid the related recognition problem. Then we show that in order to solve these max-closed IP problems, simplicial path following methods can be used. This is important because these methods are flexible with respect to starting conditions, which make them more suitable than the top-down truncation algorithms that have been suggested.     

Scheduling professionals in the retail pharmacy chain

To control operating costs, a zero-one integer programming model is developed to assist pharmacy staff scheduling decisions. Variable scheduling needs are met by the assignment of relief (mobile) pharmacists to help or temporarily replace full-time pharmacists. Assignments of relief pharmacists over a two-week planning horizon are determined with consideration given to variations in wage rates and travel costs together with the underlying corporate, contractual and operating constraints. The developed model has been applied with considerable success using data collected from a business district in the US located in northern Louisiana related to a national retail chain pharmacy. Forecasting the number of chain retail outlets in the near future has been also performed and the results obtained argue in favor of adopting the model by the entire chain.     

Multilocation plant sizing and timing problem: a study of spatial—temporal decomposition procedure

An attempt has been made to develop a spatial-temporal decomposition procedure to solve the multilocation plant sizing and timing problem (MLPSTP). MLPSTP involves the determination of site, location, timing and utilization of the plants to meet the demands of geographically distributed customers. These decisions are assumed to interact through a common objective of minimizing the net present value (NPV) of capital costs and streams of operating and transportation costs. A solution procedure for MLPSTP is proposed using aggregation and disaggregation methodology. In the aggregation phase, MLPSTP is considered to be a single-period problem. This problem is decomposed over different geographical areas to obtain a set of feasible sites and projects. These projects are then sequenced over a finite planning horizon in the disaggregation phase. A multiple criteria based evaluation of spatial decomposition, temporal decomposition, and spatial-temporal decomposition is provided.     

Comments on the Paper: Attacking the Market Split Problem with Lattice Point Enumeration

We present a few comments on the paper Attacking the market split problem with lattice point enumeration by A. Wasserman, published in Journal of Combinatorial Optimization, vol. 6, pp. 5–16, 2002.     

A Parametric Approach for a Nonlinear Discrete Location Problem

A discrete location problem is formulated for the design of a postal service network. The cost objective of this problem includes a nonlinear concave component. A parametric integer programming algorithm is developed to find an approximate solution to the problem. The algorithm reduces the problem into a sequence of p-median problems and deals with the nonlinear cost by a node-replacement scheme. Preliminary computational results are presented.     

Product mix in the TFT-LCD industry

As operational costs and equipment depreciation in the TFT-LCD (thin film transistor-liquid crystal display) industry are a high percentage of the total cost, most manufacturers usually fully utilise their production capacity to reduce the average unit cost. However, when the market demand is less than the supply the stock of panels increases; this forces manufacturers to instigate a price war to reduce levels of stock and results in a wide fluctuation in panel prices. Inventory stocks of panels could be decreased by optimising the product mix. This will help manufacturers to reduce the risk of holding stocks, increase profit, and improve competitive advantage. This study uses mixed integer linear programming (MILP) to construct a product mix for the TFT-LCD industry given the conditions of profit, productivity, raw materials supply, and market demand. A case study shows that this model is proven to be effective in generating product mix for the TFT-LCD industry while improving profit. The product mix generated by this model can provide a reference for the sales department for orders and shipping, for the production department for the order quantity, and for master production scheduling for each product.     

Zero-Lifting for Integer Block Structured Problems

This paper deals with the relations between the polyhedron described by the inequalities of a block structured problem and the polyhedra described by the inequalities of the single blocks. In particular, classes of block structured problems are described for which zero-lifting of facet inducing inequalities of a single block yields facet inducing inequalities for the whole problem. Some applications are discussed.     

An extension of the relaxation algorithm for solving a special case of capacitated arc routing problems

In this paper, we propose an exact method for solving a special integer program associated with the classical capacitated arc routing problems (CARPs) called split demand arc routing problems (SDARP). This method is developed in the context of monotropic programming theory and bases a promising foundation for developing specialized algorithms in order to solve general integer programming problems. In particular, the proposed algorithm generalizes the relaxation algorithm developed by Tseng and Bertsekas (Math. Oper. Res. 12(4):569–596, 1987) for solving linear programming problems. This method can also be viewed as an alternative for the subgradient method for solving Lagrangian relaxed problems. Computational experiments show its high potential in terms of efficiency and goodness of solutions on standard test problems.     

Two Novel Evolutionary Formulations of the Graph Coloring Problem

We introduce two novel evolutionary formulations of the problem of coloring the nodes of a graph. The first formulation is based on the relationship that exists between a graph's chromatic number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The second formulation, unlike the first one, does not tackle one graph at a time, but rather aims at evolving a program to color all graphs belonging to a class whose members all have the same number of nodes and other common attributes. The heuristics that result from these formulations have been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and have been found to be competitive when compared to the several other heuristics that have also been tested on those graphs.     

Shift scheduling for tank trucks

In this paper we deal with shift scheduling of tank trucks for a small oil company. Given are a set of tank trucks with different characteristics and a set of drivers with different skills. The objective is to assign a feasible driver to every shift of the tank trucks such that legal and safety restrictions are satisfied, the total working times of the drivers are within desired intervals, requested vacation of the drivers is respected and the trucks are assigned to more favored drivers. We propose a two-phase solution algorithm which is based on a mixed integer linear programming formulation and an improvement procedure. Computational results are reported showing that the algorithm is able to generate feasible schedules in a small amount of time.     

Resolution Search and Dynamic Branch-and-Bound

A novel approach to pure 0-1 integer programming problems called Resolution Search has been proposed by Chvatal (Discrete Applied Mathematics, vol. 73, pp. 81–99, 1997) as an alternative to implicit enumeration, with a demonstration that the method can yield more effective branching strategies. We show that an earlier method called Dynamic Branch-and-Bound (B&B) yields the same branching strategies as Resolution Search, and other strategic alternatives in addition. Moreover, Dynamic B&B is not restricted to pure 0-1 problems, but applies to general mixed integer programs containing both general integer and continuous variables.We provide examples comparing Resolution Search to enhanced variants. We also show the relation of these approaches to Dynamic B&B, suggesting the value of further study of this neglected approach.