The min-Knapsack problem with compactness constraints and applications in statistics

In the min-Knapsack problem, one is given a set of items, each having a certain cost and weight. The objective is to select a subset with minimum cost, such that the sum of the weights is not smaller than a given constant. In this paper, we introduce an extension of the min-Knapsack problem with additional “compactness constraints” (mKPC), stating that selected items cannot lie too far apart. This extension has applications in statistics, including in algorithms for change-point detection in time series. We propose three solution methods for the mKPC. The first two methods use the same Mixed-Integer Programming (MIP) formulation but with two different approaches: passing the complete model with a quadratic number of constraints to a black-box MIP solver or dynamically separating the constraints using a branch-and-cut algorithm. Numerical experiments highlight the advantages of this dynamic separation. The third approach is a dynamic programming labelling algorithm. Finally, we focus on the particular case of the unit-cost mKPC (1c-mKPC), which has a specific interpretation in the context of the statistical applications mentioned above. We prove that the 1c-mKPC is solvable in polynomial time with a different ad-hoc dynamic programming algorithm. Experimental results show that this algorithm vastly outperforms both generic approaches for the mKPC and a simple greedy heuristic from the literature.     

Stronger upper and lower bounds for a hard batching problem to feed assembly lines

In this paper we present an Integer Programming reformulation for a hard batching problem encountered in feeding assembly lines. The study was motivated by the real process to feed the production flow through the shop floor in a leading automobile industry in Brazil. The problem consists of deciding the assignment of items to containers and the frequency of moves from the storage area to the line in order to meet demands with minimum cost. Better lower and upper bounds were obtained by a branch-and-bound algorithm based on the proposed reformulation. We also present valid inequalities that may improve such algorithm even further.     

Linear programming system identification

We define a version of the Inverse Linear Programming problem that we call Linear Programming System Identification. This version of the problem seeks to identify both the objective function coefficient vector and the constraint matrix of a linear programming problem that best fits a set of observed vector pairs. One vector is that of actual decisions that we call outputs. These are regarded as approximations of optimal decision vectors. The other vector consists of the inputs or resources actually used to produce the corresponding outputs. We propose an algorithm for approximating the maximum likelihood solution. The major limitation of the method is the computation of exact volumes of convex polytopes. A numerical illustration is given for simulated data.     

On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

In the paper, we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided.     

A Constraint Programming model for fast optimal stowage of container vessel bays

Container vessel stowage planning is a hard combinatorial optimization problem with both high economic and environmental impact. We have developed an approach that often is able to generate near-optimal plans for large container vessels within a few minutes. It decomposes the problem into a master planning phase that distributes the containers to bay sections and a slot planning phase that assigns containers of each bay section to slots. In this paper, we focus on the slot planning phase of this approach and present a Constraint Programming and Integer Programming model for stowing a set of containers in a single bay section. This so-called slot planning problem is NP-hard and often involves stowing several hundred containers. Using state-of-the-art constraint solvers and modeling techniques, however, we were able to solve 90% of 236 real instances from our industrial collaborator to optimality within 1 second. Thus, somewhat to our surprise, it is possible to solve most of these problems optimally within the time required for practical application.     

Discrete optimization methods for group model selection in compressed sensing

Mathematical Programming - In this article we study the problem of signal recovery for group models. More precisely for a given set of groups, each containing a small subset of indices, and for...     

An Integrated Inventory-Routing System for Multi-item Joint Replenishment with Limited Vehicle Capacity

In this paper, we develop a mathematical programming approach for coordinating inventory and transportation decisions in an inbound commodity collection system. In particular, we consider a system that consists of a set of geographically dispersed suppliers that manufacture one or more non-identical items, and a central warehouse that stocks these items. The warehouse faces a constant and deterministic demand for the items from outside retailers. The items are collected by a fleet of vehicles that are dispatched from the central warehouse. The vehicles are capacitated, and must also satisfy a frequency constraint. Adopting a policy in which each vehicle always collects the same set of items, we formulate the inventory-routing problem of minimizing the long-run average inventory and transportation costs as a set partitioning problem. We employ a column generation approach to determine a lower bound on the total costs, and develop a branch-and-price algorithm that finds the optimal assignment of items to vehicles. We also propose greedy constructive heuristics, and develop a very large-scale neighborhood (VLSN) search algorithm to find near-optimal solutions for the problem. Computational tests are performed on a set of randomly generated problem instances.The work of this author was supported by a scholarship of the Faculty of Engineering of Ubonratchathani University, Ubonratchathani, Thailand., The work of this author was supported in part by the National Science Foundation under Grant No. DMI-0085682.     

Heuristics for container loading of furniture

We consider a container loading problem that occurs at a typical furniture manufacturer. Each furniture item has an associated profit. Given container dimensions and a set of furniture items, the problem is to determine a subset of items with maximal profit sum that is loadable in the container. In the studied company, the problem arises hundreds of times daily during transport planning. Instances may contain more than one hundred different items with irregular shapes. To solve this complex problem we apply a set of heuristics successively that each solve one part of the problem. Large items are combined in specific structures to ensure proper protection of the items during transportation and to simplify the problem. The solutions generated by the heuristic has an average loading utilization of 91.3% for the most general instances with average running times around 100 seconds.     

Batch sizing and job sequencing on a single machine

We study a single-machine scheduling problem in which the items to be processed have to be batched as well as sequenced. Since processed items become available in batches, flow times are defined to be the same for all items in the same batch. A constant set-up delay is incurred between consecutive batches. For any fixed, but arbitrary item sequence, we present an algorithm that finds a sequence of batches such that the total flow time of the items is minimized; we prove that for a set ofn items, the algorithm runs inO(n) time. We show that, among all sequences, the one leading to the minimum flow time has the items in non-decreasing order of running times. Thus, the optimal algorithm for the combined problem, called thebatch-sizing problem, runs inO(n logn) time. We also prove that this algorithm yields an improved solution to a scheduling problem recently studied by Baker [1].     

Implicit optimality criterion for convex SIP problem with box constrained index set

We consider a convex problem of Semi-Infinite Programming (SIP) with a multidimensional index set defined by a finite number of box constraints. In study of this problem we apply the approach suggested in Kostyukova et al. (Int. J. Math. Stat. 13(J08):13–33, 2008) for convex SIP problems with one-dimensional index sets and based on the notions of immobile indices and their immobility orders. For the problem under consideration we formulate optimality conditions that are explicit and have the form of criterion. We compare this criterion with other known optimality conditions for SIP and show its efficiency in the convex case.     

Unifications,deunifications, and their complexity

The execution of a Prolog program can be viewed as a sequence of unifications and backtracks over unifications. We study the time requirement of executing a sequence of such operations (the unify-deunify problem). It is shown that the well-known set union problem is reducible to this problem, even in the case when no function symbols are allowed (the Datalog unify-deunify problem). As the set union problem requires nonlinear time on a large class of algorithms, the same holds for the unify-deunify problem. Thus the linearity of single unifications does not give a complete picture of the time complexity of Prolog primitives. We discuss the methods for executing sequences of Datalog unifications used in Prolog interpreters and show that some of them require even quadratic time in the worst case. Complementing these results, we show that if the number of variables occurring in one clause is bounded by a constant, then the Datalog unify-deunify problem can be solved in linear time.A preliminary version of this paper appeared in the Third International Conference on Logic Programming, London, July 1986. This work was supported by the Academy of Finland and by TEKES.     

Runway sequencing with holding patterns

We study a scheduling problem, motivated by air-traffic control. When aircraft reach the final descent in the “Terminal Radar Approach CONontrol” area (tracon), a set of disjoint time windows in which the landing is possible, can be automatically assigned to each aircraft. The objective is then to determine landing times, within these time windows, which maximize the minimum time elapsed between consecutive landings. We study the complexity of the problem and describe several special cases that can be solved in polynomial time. We also provide a compact Mixed Integer Programming formulation that allows us to solve large instances of the general problem when all time windows have the same size. Finally, we introduce a general hybrid branch and cut framework to solve the problem with arbitrary time windows. Experimental results show that our approach outperforms earlier formulation of the problem.     

MIP models for two-dimensional non-guillotine cutting problems with usable leftovers

In this study we deal with the two-dimensional non-guillotine cutting problem of how to cut a set of larger rectangular objects to a set of smaller rectangular items in exactly a demanded number of pieces. We are concerned with the special case of the problem in which the non-used material of the cutting patterns (objects leftovers) may be used in the future, for example if it is large enough to fulfill future item demands. Therefore, the problem is seen as a two-dimensional non-guillotine cutting/packing problem with usable leftovers, also known in the literature as a two-dimensional residual bin-packing problem. We use multilevel mathematical programming models to represent the problem appropriately, which basically consists of cutting the ordered items using a set of objects of minimum cost, among all possible solutions of minimum cost, choosing one that maximizes the value of the usable leftovers, and, among them, selecting one that minimizes the number of usable leftovers. Because of special characteristics of these multilevel models, they can be reformulated as one-level mixed integer programming (MIP) models. Illustrative numerical examples are presented and analysed.     

Linear programming models with planned lead times for supply chain operations planning

This paper contributes to the development of models for capacity constrained Supply Chain Operations Planning (SCOP). We focus on production environments with arbitrary supply chain structures. The demand for the end items is assumed to be exogenously determined. We solve the SCOP problem with Linear Programming models using planned lead times with multi-period capacity consumption. Using planned lead times increases the reliability of the communication between SCOP and Scheduling with regard to the feasibility of the planning. Planned lead times also reduce the nervousness in the system. We model capacity constraints on the quantity of items that can be assembled within a time interval. In particular, items can be assigned to multiple resources. We discuss two LP approaches which plan the production of items so that a sum of inventory costs and costs due to backordering is minimized.     

Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.     

Optimal allocation of stock levels and stochastic customer demands to a capacitated resource

This paper considers a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers’ demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. We provide an exact solution method for this mixed integer nonlinear optimization problem using a Generalized Benders Decomposition approach. This decomposition approach uses Lagrangian relaxation to solve a constrained multi-item newsvendor subproblem and uses CPLEX to solve a mixed-integer linear master problem. We generate Benders cuts for the master problem by obtaining a series of subgradients of the subproblem’s convex objective function. In addition, we present a family of heuristic solution approaches and compare our methods with several MINLP (Mixed-Integer Nonlinear Programming) commercial solvers in order to benchmark their efficiency and quality.     

Tree search for the stacking problem

The stacking problem is a hard combinatorial optimization problem with high practical interest in, for example, steel storage or container port operations. In this problem, a set of items is stored in a warehouse for a period of time, and a crane is used to place them in a limited number of stacks. Since the entrance and exit of items occurs in an arbitrary order, items may have to be relocated in order to reach and deliver other items below them. The objective of the problem is to find a feasible sequence of movements that delivers all items, while minimizing the total number of movements. We study the scalability of an exact approach to this problem, and propose two heuristic methods to solve it approximately. The two heuristic approaches are a multiple simulation algorithm using semi-greedy construction heuristics, and a stochastic best-first tree search algorithm. The two methods are compared in a set of challenging instances, revealing a superior performance of the tree search approach in most cases.     

Maximum edge-disjoint paths in planar graphs with congestion 2

Mathematical Programming - We study the maximum edge-disjoint path problem (medp) in planar graphs $$G=(V,E)$$ with edge capacities u(e). We are given a set of terminal pairs $$s_it_i$$ , $$i=1,2...     

Critical solutions of nonlinear equations: stability issues

Mathematical Programming - It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain...     

价格控制问题的基本性质

价格控制问题是一类重要的二层规划问题。最近,文[3,4]讨论了价格控制问题的最优性条件和解集的性质。本文先用反例说明[3]中关于价格控制问题的可行解的充分必要条件的一个命题是不确切的,并对[4]中基于该命题证得的不可靠结论在适当条件下进行了证明。