Note: A note on the production and inventory systems taking account of time value

This paper deals with the optimal inventory decisions taking account of time value by applying the concept of the present value method, and modifies the bounds for the optimal cycle time described in Chung et al. (Production Planning & Control, 1998, 9, 580–584). A modified algorithm to compute the optimal cycle time is developed to improve the paper of Chung et al. Finally, a numerical example is given to illustrate the algorithm discussed in this paper.     

Decision support system for the batching problems of steelmaking and continuous-casting production

This paper investigates two batching problems for steelmaking and continuous-casting (SCC) production in an integrated iron and steel enterprise. The tasks of the problems are to make the decisions as how to consolidate ordered slabs into charges, and then how to group charges into casts. The effective decisions on these batching problems can help to balance the requirements of materials in downstream production lines, improve the customer satisfaction levels, and reduce production costs (including reduction of open ordered slabs, less slabs quality upgrading, reduction of steel-grade changeovers, and reduction of inefficient utilization of tundishes lives). We first formulate the problems as integer-programming models by consider practical constraints and requirements, and then develop the two heuristic algorithms for the corresponding batching problems. By embedding above models and algorithms, we develop decision support system (DSS) software with interactive planning editor. The DSS has been tested by using practical data set collected from the steelmaking plant in Baosteel which is one of the most advanced iron and steel enterprises in China. Computational experiments demonstrate that the models and algorithms developed can generate the satisfactory solutions when they work together with the planning editor in the DSS.     

Optimal algorithms for online time series search and one-way trading with interrelated prices

The basic models of online time series search and one-way trading are introduced by El-Yaniv et al. in Algorithmica 30(1), 101–139 (2001) where it is assumed that the prices are bounded within interval [m,M] (0<m<M). In this paper, we consider another case where every two consecutive prices are interrelated, that is, the variation range of each price depends on its preceding price. We present optimal deterministic online algorithms for the two problems, respectively. According to one conclusion in Algorithmica 30(1), 101–139 (2001), we further point out that for the case we considered, an optimal deterministic algorithm for the one-way trading problem can be regarded as an optimal randomized one for the time series search problem, and randomization is useless for the one-way trading problem.     

Replenishment Policies for Multi‐Product Stochastic Inventory Systems with Correlated Demand and Joint‐Replenishment Costs

This study analyzes optimal replenishment policies that minimize expected discounted cost of multi‐product stochastic inventory systems. The distinguishing feature of the multi‐product inventory system that we analyze is the existence of correlated demand and joint‐replenishment costs across multiple products. Our objective is to understand the structure of the optimal policy and use this structure to construct a heuristic method that can solve problems set in real‐world sizes/dimensions. Using an MDP formulation we first compute the optimal policy. The optimal policy can only be computed for problems with a small number of product types due to the curse of dimensionality. Hence, using the insight gained from the optimal policy, we propose a class of policies that captures the impact of demand correlation on the structure of the optimal policy. We call this class (s, c, d, S)‐policies, and also develop an algorithm to compute good policies in this class, for large multi‐product problems. Finally using an exhaustive set of computational examples we show that policies in this class very closely approximate the optimal policy and can outperform policies analyzed in prior literature which assume independent demand. We have also included examples that illustrate performance under the average cost objective.     

Polynomial time approximation schemes for minimum disk cover problems

The following planar minimum disk cover problem is considered in this paper: given a set D\mathcal{D} of n disks and a set ℘ of m points in the Euclidean plane, where each disk covers a subset of points in ℘, to compute a subset of disks with minimum cardinality covering ℘. This problem is known to be NP-hard and an algorithm which approximates the optimal disk cover within a factor of (1+ε) in O(mn^{O(\frac1e2log2\frac1e)})\mathcal{O}(mn^{\mathcal{O}(\frac{1}{\epsilon^{2}}\log^{2}\frac{1}{\epsilon})}) time is proposed in this paper. This work presents the first polynomial time approximation scheme for the minimum disk cover problem where the best known algorithm can approximate the optimal solution with a large constant factor. Further, several variants of the minimum disk cover problem such as the incongruent disk cover problem and the weighted disk cover problem are considered and approximation schemes are designed.     

Fast On-Line/Off-Line Algorithms for Optimal Reinforcement of a Network and its Connections with Principal Partition

The problem of computing the strength and performing optimal reinforcement for an edge-weighted graph G(V, E, w) is well-studied. In this paper, we present fast (sequential linear time and parallel logarithmic time) on-line algorithms for optimally reinforcing the graph when the reinforcement material is available continuously on-line. These are the first on-line algorithms for this problem. We invest O(|V|³|E|log|V|) time (equivalent to (|V|) invocations of the fastest known algorithms for optimal reinforcement) in preprocessing the graph before the start of our algorithms. It is shown that the output of our on-line algorithms is as good as that of the off-line algorithms. Thus our algorithms are better than the fastest off-line algorithms in situations when a sequence of more than (|V|) reinforcement problems need to be solved. The key idea is to make use of ideas underlying the theory of Principal Partition of a Graph. Our ideas are easily generalized to the general setting of polymatroid functions. We also present a new efficient algorithm for computation of the Principal Sequence of a graph.     

The bandpass problem: combinatorial optimization and library of problems

A combinatorial optimization problem, called the Bandpass Problem, is introduced. Given a rectangular matrix A of binary elements {0,1} and a positive integer B called the Bandpass Number, a set of B consecutive non-zero elements in any column is called a Bandpass. No two bandpasses in the same column can have common rows. The Bandpass problem consists of finding an optimal permutation of rows of the matrix, which produces the maximum total number of bandpasses having the same given bandpass number in all columns. This combinatorial problem arises in considering the optimal packing of information flows on different wavelengths into groups to obtain the highest available cost reduction in design and operating the optical communication networks using wavelength division multiplexing technology. Integer programming models of two versions of the bandpass problems are developed. For a matrix A with three or more columns the Bandpass problem is proved to be NP-hard. For matrices with two or one column a polynomial algorithm solving the problem to optimality is presented. For the general case fast performing heuristic polynomial algorithms are presented, which provide near optimal solutions, acceptable for applications. High quality of the generated heuristic solutions has been confirmed in the extensive computational experiments. As an NP-hard combinatorial optimization problem with important applications the Bandpass problem offers a challenge for researchers to develop efficient computational solution methods. To encourage the further research a Library of Bandpass Problems has been developed. The Library is open to public and consists of 90 problems of different sizes (numbers of rows, columns and density of non-zero elements of matrix A and bandpass number B), half of them with known optimal solutions and the second half, without.     

Medial Axis and Optimal Locations for Min-Max Sphere Packing

We study the following min-max sphere packing problem originated from radiosurgical treatment planning using gamma knife (Bourland and Wu, 1996; Wu, 1996). Given an input (R, V), where R is a 3-dimensional (3D) bounded region and V a positive integer, find a packing of R using the minimum number of spheres (spheres may not be identical) such that the covered volume is at least V, and the number of points on the boundary of R touched by spheres is maximized. Bourland and Wu (1996) and Wu (1996), devised a greedy algorithm to solve the problem based on medial axis analysis. In particular, the algorithm places the center of each sphere on the medial axis of each subsequent region starting from R. While this approach has met with certain success, we show that medial axis does not always provide optimal locations for min-max sphere packing.     

Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colorability

Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAX-CUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of . This is an improvement over the ratio of attributed to another local improvement heuristic for MAX-CUT (C. Papadimitriou, Computational Complexity, Addison Wesley, 1994). In fact we provide a bound of for this problem, where k 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques (M. Goemans and D.P. Williamson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994a, pp. 422–431) appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both (M. Goemans and D.P. Williamson, SIAM J. Disc. Math, vol. 7, pp. 656–666, 1994b) yield a bound of for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of for this problem.     

Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u _ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector [^(u)]\hat{u} for the arc set A such that a given feasible flow [^(x)]\hat{x} is optimal with respect to the modified capacities. Among all capacity vectors [^(u)]\hat{u} satisfying this condition, we would like to find one with minimum ||[^(u)]-u||\|\hat{u}-u\| value. We consider two distance measures for ||[^(u)]-u||\|\hat{u}-u\| , rectilinear (L ₁) and Chebyshev (L _∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP\mathcal{NP} -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.     

Efficient algorithms for supergraph query processing on graph databases

We study the problem of processing supergraph queries on graph databases. A graph database D is a large set of graphs. A supergraph query q on D is to retrieve all the graphs in D such that q is a supergraph of them. The large number of graphs in databases and the NP-completeness of subgraph isomorphism testing make it challenging to efficiently processing supergraph queries. In this paper, a new approach to processing supergraph queries is proposed. Specifically, a method for compactly organizing graph databases is first presented. Common subgraphs of the graphs in a database are stored only once in the compact organization of the database, in order to reduce the overall cost of subgraph isomorphism testings from the stored graphs to queries during query processing. Then, an exact algorithm and an approximate algorithm for generating the significant feature set with optimal order are proposed, followed by the algorithms for indices construction on graph databases. The optimal order on the feature set is to reduce the number of subgraph isomorphism testings during query processing. Based on the compact organization of graph databases, a novel algorithm for testing subgraph isomorphisms from multiple graphs to one graph is presented. Finally, based on all the above techniques, a query processing method is proposed. Analytical and experimental results show that the proposed algorithms outperform the existing similar algorithms by one to two orders of magnitude.     

Online coupon consumption problem

Nowadays, it is popular that the dealer makes profits by selling a kind of discount coupons, which can be used as money to purchase commodities with total cost less than or equal to the face value of the coupon. We can purchase a coupon at a price of 0<s≤1 times its face value and the number of potential purchasable coupons is a given integer l. The customer has the option to buy the goods by cash completely or by a discount coupon. However, each piece of goods can only use one coupon and the coupon used must have enough balance for the goods. The objective is to minimize the total cost for purchasing all the goods. In this paper, we reduce the problem to a special bin packing model. We consider the online problems for all 0<s≤1 and 1≤l≤∞. We present optimal online algorithms for all 0<s≤1 when l=∞ and l=1. For 2≤l<∞, we give both a lower bound and an algorithm, and show the algorithm is optimal for l=2. A preliminary version of this paper appeared in the proceedings of the 1st International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, Lecture Notes in Computer Science. Y. Jiang supported by Natural Science Foundation of Zhejiang Province (Y605316). Z. Tan supported by Natural Science Foundation of China (10671177, 60021201).     

Strong Duality for a Multiple‐Good Monopolist

We characterize optimal mechanisms for the multiple‐good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure μ derived from the buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand‐bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for n independent uniform items each supported on [c,c+1] is a grand‐bundling mechanism, as long as c is sufficiently large, extending Pavlov's result for two items Pavlov, 2011. At the same time, our characterization also implies that, for all c and for all sufficiently large n, the optimal mechanism for n independent uniform items supported on [c,c+1] is not a grand‐bundling mechanism.     

Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the SteinLib library.     

Flattening topologically spherical surface

The problem of optimal surface flattening in 3-D finds many applications in engineering and manufacturing. However, previous algorithms for this problem are all heuristics without any quality guarantee and the computational complexity of the problem was not well understood. In this paper, we prove that the optimal surface flattening problem is NP-hard. Further, we show that the problem of flattening a topologically spherical surface admits a PTAS and can be solved by a (1+ε)-approximation algorithm in O(nlog n) time for any constant ε>0, where n is the input size of the problem.     

A hybrid beam search looking-ahead algorithm for the circular packing problem

In this paper, we study the circular packing problem. Its objective is to pack a set of n circular pieces into a rectangular plate R of fixed dimensions L×W. Each piece’s type i, i=1,…,m, is characterized by its radius r _i and its demand b _i . The objective is to determine the packing pattern corresponding to the minimum unused area of R for the circular pieces placed. This problem is solved by using a hybrid algorithm that adopts beam search and a looking-ahead strategy. A node at a level ℓ of the beam-search tree contains a partial solution corresponding to the circles already placed inside R. Each node is then evaluated using a looking-ahead strategy, based on the minimum local-distance heuristic, by computing the corresponding complete solution. The nodes leading to the best solutions at level ℓ are then chosen for branching. A multi-start strategy is also considered in order to diversify the search space. The computational results show, on a set of benchmark instances, the effectiveness of the proposed algorithm.     

Constructing the Maximum Consensus Tree from Rooted Triples

We investigated the problem of constructing the maximum consensus tree from rooted triples. We showed the NP-hardness of the problem and developed exact and heuristic algorithms. The exact algorithm is based on the dynamic programming strategy and runs in O((m + n ²)3 ⁿ ) time and O(2 ⁿ ) space. The heuristic algorithms run in polynomial time and their performances are tested and shown by comparing with the optimal solutions. In the tests, the worst and average relative error ratios are 1.200 and 1.072 respectively. We also implemented the two heuristic algorithms proposed by Gasieniec et al. The experimental result shows that our heuristic algorithm is better than theirs in most of the tests.     

Optimal all-to-all personalized exchange in d-nary banyan multistage interconnection networks

All-to-all personalized exchange occurs in many important applications in parallel processing. In the past two decades, algorithms for all-to-all personalized exchange were mainly proposed for hypercubes, meshes, and tori. Recently, Yang and Wang (IEEE Trans Parallel Distrib Syst 11:261–274, 2000) proposed an optimal all-to-all personalized exchange algorithm for binary (each switch is of size 2×2) banyan multistage interconnection networks. It was pointed out in Massini (Discret Appl Math 128:435–446, 2003) that the algorithm in Yang, Wang (IEEE Trans Parallel Distrib Syst 11:261–274, 2000) depends on the network topologies and requires pre-computation and memory allocation for a Latin square. Thus in (Discret Appl Math 128:435–446, 2003), Massini proposed a new optimal algorithm, which is independent of the network topologies and does not require pre-computation or memory allocation for a Latin square. Unfortunately, Massini’s algorithm has a flaw and does not realize all-to-all personalized exchange. In this paper, we will correct the flaw and generalize Massini’s algorithm to be applicable to d-nary (each switch is of size d×d) banyan multistage interconnection networks. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This research was partially supported by the National Science Council of the Republic of China under the grant NSC94-2115-M-009-006.     

Class‐Based Storage with a Finite Number of Items: Using More Classes is not Always Better

Class‐based storage is widely studied in the literature and applied in practice. It divides all stored items into a number of classes according to their turnover. A class of items with higher turnover is allocated to a region closer to the warehouse depot. In the literature, it has been shown that the use of more storage classes leads to a shorter travel time for storing and retrieving items. A basic assumption in this literature is that the required storage space for all items equals their average inventory level, which is valid only if an infinite number of items can be stored in each storage region. This study revisits class‐based storage by considering each storage space to contain only a finite number of items. We develop a travel time model and an algorithm that can be used for determining the optimal number and boundaries of storage classes in warehouses. Different from the conventional research, our findings illustrate that commonly a small number of classes is optimal. In addition, we find the travel time is fairly insensitive to the number of storage classes in a wide range around the optimum. This suggests that a manager can select a near‐optimal number of storage classes in an easy way and need not be worried about the impact of storage‐class reconfigurations. We validate our findings for various cases, including different ABC‐demand curves, space‐sharing factors, number of items, storage rack shapes, discrete storage locations, and stochastic item demand.