Categorized Assignment Scheduling: a Tabu Search Approach

The assignment problem (AP) and bottleneck assignment problem (BAP) are well studied in operational research literature. In this paper we consider two related problems which simultaneously generalize both AP and BAP. Unlike AP and BAP, these generalizations are strongly NP-complete. We propose two heuristics to solve these generalized problems: one based on a greedy principle and the other based on tabu search. Computational results are presented which show that these heuristics, when used together, produce good quality solutions. Our adaptation of tabu search also gives some new insight into the application of tabu search on permutation problems.     

Optimal mapping in direct mapped cache environments

In this paper we study positioning strategies for improving the performance of a memory system with a direct mapped cache. A positioning technique determines for every program item, (instruction or data), its address in main memory.Assuming the Independent Reference Model, we break the general positioning problem into two, the collision minimization and the grouping problems and show optimal algorithms for both problems. Using these algorithms we derive an optimal algorithm for the general positioning problem.Since the optimal positioning is of very special structure we consider other, less restricted, positionings. We show that the quality of a class of natural assignments that distribute the items almost arbitrarily is good as long as the optimal hit ratio is sufficiently large. Another possible requirement is that the items should be distributed as evenly as possible. We find an optimal assignment for the special case of the pair assignment.In addition we look at the expected performance gain of two frequently suggested cache features. The cache bypass feature supports the access of items in memory without loading the item into the cache. We show an assignment with best possible hit ratio. Also it is shown that a cache which employs a random assignment policy, i.e., the assignment of an item is determined randomly, does not improve the expected hit ratio.     

Scheduling of continuous and discontinuous material flows with intermediate storage restrictions

This paper deals with scheduling batch (i.e., discontinuous), continuous, and semicontinuous production in process industries (e.g., chemical, pharmaceutical, or metal casting industries) where intermediate storage facilities and renewable resources (processing units and manpower) of limited capacity have to be observed. First, different storage configurations typical of process industries are discussed. Second, a basic scheduling problem covering the three above production modes is presented. Third, (exact and truncated) branch-and-bound methods for the basic scheduling problem and the special case of batch scheduling are proposed and subjected to an experimental performance analysis. The solution approach presented is flexible and in principle simple, and it can (approximately) solve relatively large problem instances with sufficient accuracy.     

Single machine common flow allowance scheduling with deteriorating jobs and a rate-modifying activity

We study a single machine slack due date assignment (usually referred to as SLK) scheduling problem with deteriorating jobs and a rate-modifying activity. The deterioration effect manifest such that the job processing time is a function of its starting time in a sequence. The rate-modifying activity is an activity that changes the processing rate of machine, i.e., the machine performs a rate-modifying activity. Hence the actual processing time of a job is a variable, which depends not only on its starting time in a sequence but also on whether it is scheduled before or after a rate-modifying activity. The goal is to schedule the rate-modifying activity, the optimal common flow allowance and the sequence of jobs to minimize the total earliness, the total tardiness and the common flow allowance cost. We show that the problem remains polynomially solvable under the proposed model.     

Formulating and solving production planning problems

Production planning problems frequently involve the assignment of jobs or operations to machines. The simplest model of this problem is the well known assignment problem (AP). However, due to simplifying assumptions this model does not provide implementable solutions for many actual production planning problems. Extensions of the simple assignment model known as the generalized assignment problem (GAP) and the multi-resource generalized assignment problem (MRGAP) have been developed to overcome this difficulty. This paper presents an extension of the (MRGAP) to allow splitting individual batches across multiple machines, while considering the effect of setup times and setup costs. The extension is important for many actual production planning problems, including ones in the injection molding industry and in the metal cutting industry. We formulate models which are logical extensions of previous models which ignored batch splitting for the problem we address. We then give different formulations and suggest adaptations of a genetic algorithm (GA) and simulated annealing (SA). A systematic evaluation of these algorithms, as well as a Lagrangian relaxation (LR) approach, is presented.     

MineLib: a library of open pit mining problems

Similar to the mixed-integer programming library (MIPLIB), we present a library of publicly available test problem instances for three classical types of open pit mining problems: the ultimate pit limit problem and two variants of open pit production scheduling problems. The ultimate pit limit problem determines a set of notional three-dimensional blocks containing ore and/or waste material to extract to maximize value subject to geospatial precedence constraints. Open pit production scheduling problems seek to determine when, if ever, a block is extracted from an open pit mine. A typical objective is to maximize the net present value of the extracted ore; constraints include precedence and upper bounds on operational resource usage. Extensions of this problem can include (i) lower bounds on operational resource usage, (ii) the determination of whether a block is sent to a waste dump, i.e., discarded, or to a processing plant, i.e., to a facility that derives salable mineral from the block, (iii) average grade constraints at the processing plant, and (iv) inventories of extracted but unprocessed material. Although open pit mining problems have appeared in academic literature dating back to the 1960s, no standard representations exist, and there are no commonly available corresponding data sets. We describe some representative open pit mining problems, briefly mention related literature, and provide a library consisting of mathematical models and sets of instances, available on the Internet. We conclude with directions for use of this newly established mining library. The library serves not only as a suggestion of standard expressions of and available data for open pit mining problems, but also as encouragement for the development of increasingly sophisticated algorithms.     

Learning effects and the phenomenon of moving bottlenecks in a two-stage production system

This paper studies the effects of learning and forgetting on a two-stage production system and the position of a potential bottleneck in the system. We start with developing a model for a two-stage serial production system where semi-finished items are fed by the first stage to the second stage, which, in turn, processes the items to their final state. The finished items are transferred either to a subsequent stage or to customers. The paper assumes that both stages of the production system considered are subject to learning and forgetting effects. Learning quickens the production rate as more experience is gained (i.e., when the number of repetitions increases), while forgetting has the opposite effect when production is intermittent (i.e., experience is lost over production breaks). The paper studies how different values of the learning and forgetting parameters influence the ratio of the production rates of both stages and the flow of material in the system. The results of the paper indicate that learning may cause a bottleneck to shift its position in a production system. This happens when an initially slower stage overtakes a previously faster stage over time due to a higher learning rate. The paper thus contributes to the literature on moving bottlenecks and provides practitioners with a model that helps predicting where bottlenecks may arise in the production system, and which enables the system to flexibly react to moving bottlenecks.     

A heuristic file reorganization algorithm based on record clustering

The problem of file organization which we consider involves altering the placement of records on pages of a secondary storage device. In addition, we want this reorganization to be done in-place, i.e., using the file's original storage space for the newly reorganized file. The motivation for such a physical change is to improve the database system's performance. For example, by placing frequently and jointly accessed records on the same page or pages, we can try to minimize the number of page accesses made in answering a set of queeries. The optimal assignment (or reassignment) of records to clusters is exactly what record clustering algorithms attempt to do. However, record clustering algorithms usually do not solve the entire problem, i.e., they do not specify how to efficiently reorganize the file to reflect the clustering assignment which they determine. Our algorithm is a companion to general record clustering algorithms since it actually transforms the file. The problem of optimal file reorganization isNP-hard. Consequently, our reorganization algorithm is based on heuristics. The algorithm's time and space requirements are reasonable and its solution is near optimal. In addition, the reorganization problem which we consider in this paper is similar to the problem of join processing when indexes are used.The research of this author was partially supported by the National Science Foundation under grant IST-8696157.     

Sensitivity analysis of the optimal assignment

This paper concentrates on sensitivity analysis of the optimal solution for the assignment problem (AP). Due to the high degeneracy of the AP, traditional sensitivity analysis, which determines the range in which the current optimal basis remains optimal, is impractical. Thus, changing the optimal basis does not ensure that the optimal assignment will be changed. Herein we investigate the properties of the AP and then propose several lemmas to determine two other types of sensitivity range. The first type is used to determine the range in which the current optimal assignment remains optimal. We further discuss what is the new optimal assignment when the changes surpass the range. The second type of sensitivity range is to determine those values of assignment model parameters for which the rate of change of optimal value function remains constant. An example is presented in order to demonstrate that the approaches are useful in practice.     

Minimizing fleet operating costs for a container transportation company

This paper focuses on a fleet management problem that arises in container trucking industry. From the container transportation company perspective, the present and future operating costs to minimize can be divided in three components: the routing costs, the resource (i.e., driver and truck) assignment costs and the container repositioning costs (i.e., the costs of restoring a given container fleet distribution over the serviced territory, as requested by the shippers that own the containers).This real-world problem has been modeled as an integer programming problem. The proposed solution approach is based on the decomposition of this problem in three simpler sub-problems associated to each of the costs considered above.Numerical experiments on randomly generated instances, as well as on a real-world data set of an Italian container trucking company, are presented.     

Multiple bottleneck assignment problem

We consider the multiple bottleneck assignment problem which subsumes the well known min-sum and bottleneck assignment problems. The problem arises in the context of flexible manufacturing systems, where the objective is to maximize the throughput of a production system with several flow shops, running in parallel, to produce a product. The problem is known to be strongly NP-hard. We propose a new algorithm for obtaining sharp lower bounds to the optimal objective value. Computational experiments are conducted to show the improvement over existing methods.     

Minimizing weighted earliness–tardiness and due-date cost with unit processing-time jobs

The classical single-machine scheduling and due-date assignment problem of Panwalker et al. [Panwalker, S.S., Smith, M.L., Seidmann, A., 1982. Common due date assignment to minimize total penalty for the one machine scheduling problem. Operations Research 30(2) (1982) 391–399] is the following: All n jobs share a common due-date, which is to be determined. Jobs completed prior to or after the due-date are penalized according to a cost function which is linear and job-independent. The objective is to minimize the total earliness–tardiness and due-date cost. We study a generalized version of this problem in which: (i) the earliness and tardiness costs are allowed to be job dependent and asymmetric and (ii) jobs are processed on parallel identical machines. We focus on the case of unit processing-time jobs. The problem is shown to be solved in polynomial (O(n⁴)) time. Then we study the special case with no due-date cost (a classical problem known in the literature as TWET). We introduce an O(n³) solution for this case. Finally, we study the minmax version of the problem, (i.e., the objective is to minimize the largest cost incurred by any of the jobs), which is shown to be solved in polynomial time as well.     

Flight gate scheduling with respect to a reference schedule

This paper considers the problem of assigning flights to airport gates. We examine the general case in which an aircraft serving a flight may be assigned to different gates for arrival and departure processing and for optional intermediate parking. Restrictions to this assignment include gate closures and shadow restrictions, i.e., the situation where certain gate assignments may cause blocking of neighboring gates. The objectives include maximization of the total assignment preference score, a minimal number of unassigned flights during overload periods, minimization of the number of tows, maximization of a robustness measure as well as a minimal deviation from a given reference schedule. We show that in case of a one period time horizon this objective can easily be integrated into our existing model based on the Clique Partitioning Problem. Furthermore we present a heuristic algorithm to solve the problem for multiple periods.     

The assignment game I: The core

The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in thecore of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.     

一类自由作业供应链排序的研究

本文研究一类集成工件生产和发送的排序模型.在该模型中,供应链的上游首先将工件安排在自由作业机器上加工,然后把加工完毕的工件分批发送给下游.问题是寻找生产和发送相连的排序,使得生产排序费用和发送费用总和最少.这里,生产排序费用是以工件带权送到时间和表示;发送费用由固定费用和与运输路径有关的变化费用组成.在指出问题的NP困难性后,本文用动态规划算法构造了一致条件下的多项式时间近似算法,并分析算法的性能比.本文最后还讨论了该问题的其它情形.     

Freight transportation in railway networks with automated terminals: A mathematical model and MIP heuristic approaches

In this paper we propose a planning procedure for serving freight transportation requests in a railway network with fast transfer equipment at terminals. We consider a transportation system where different customers make their requests (orders) for moving boxes, i.e., either containers or swap bodies, between different origins and destinations, with specific requirements on delivery times. The decisions to be taken concern the route (and the corresponding sequence of trains) that each box follows in the network and the assignment of boxes to train wagons, taking into account that boxes can change more than one train and that train timetables are fixed.The planning procedure includes a pre-analysis step to determine all the possible sequences of trains for serving each order, followed by the solution of a 0-1 linear programming problem to find the optimal assignment of each box to a train sequence and to a specific wagon for each train in the sequence. This latter is a generalized assignment problem which is NP-hard. Hence, in order to find good solutions in acceptable computation times, two MIP heuristic approaches are proposed and tested through an experimental analysis considering realistic problem instances.     

Multi-reservoir production optimization

When a large oil or gas field is produced, several reservoirs often share the same processing facility. This facility is typically capable of processing only a limited amount of commodities per unit of time. In order to satisfy these processing limitations, the production needs to be choked, i.e., scaled down by a suitable choke factor. A production strategy is defined as a vector valued function defined for all points of time representing the choke factors applied to reservoirs at any given time. In the present paper we consider the problem of optimizing such production strategies with respect to various types of objective functions. A general framework for handling this problem is developed. A crucial assumption in our approach is that the potential production rate from a reservoir can be expressed as a function of the remaining recoverable volume. The solution to the optimization problem depends on certain key properties, e.g., convexity or concavity, of the objective function and of the potential production rate functions. Using these properties several important special cases can be solved. An admissible production strategy is a strategy where the total processing capacity is fully utilized throughout a plateau phase. This phase lasts until the total potential production rate falls below the processing capacity, and after this all the reservoirs are produced without any choking. Under mild restrictions on the objective function the performance of an admissible strategy is uniquely characterized by the state of the reservoirs at the end of the plateau phase. Thus, finding an optimal admissible production strategy, is essentially equivalent to finding the optimal state at the end of the plateau phase. Given the optimal state a backtracking algorithm can then used to derive an optimal production strategy. We will illustrate this on a specific example.     

Inventory model with stochastic lead-time and price dependent demand incorporating advance payment

Inventory model for an item is developed in stochastic environment with price-dependent demand over a finite time horizon. Here, probabilistic lead-time is considered and shortages are allowed (if occurs). In any business, placement of an order is normally connected with the advance payment (AP). Again, depending upon the amount of AP, unit price is quoted, i.e., price discount is allowed. Till now, this realistic factor is overlooked by the researchers. In this model, unit price is inversely related with the AP amount. Against this financial benefit, the management has to incur an expenditure paying interest against AP. Taking these into account, mathematical expression is derived for the expected average profit of the system. A closed form solution to maximize the expected average profit is obtained when demand is constant. In other cases model is solved using generalized reduced gradient (GRG) technique and stochastic search genetic algorithm (GA). Moreover, results of the models without and with advance payment are presented and solved. The numerical examples are presented to illustrate the model and the results for two models obtained from two methods are compared in different cases. Also, some parametric studies and sensitivity analyses have been carried out to illustrate the behavior of the proposed model. It is observed that advance payment has positive effect on the system.     

Some inverse optimization problems under the Hamming distance

Given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph, the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal. The aim is to optimize the difference between the initial cost vector and the disturbed one. This difference can be measured in several ways. We consider the Hamming distance measuring in how many components two vectors are different, where weights are associated to the components. General algorithms for the bottleneck or minimax criterion are described and (after modification) applied to the inverse minimum spanning tree problem, the inverse shortest path tree problem and the linear assignment problem.     

Weight of a link in a shortest path tree and the Dedekind Eta function

The weight of a randomly chosen link in the shortest path tree on the complete graph with exponential i.i.d. link weights is studied. The corresponding exact probability generating function and the asymptotic law are derived. As a remarkable coincidence, this asymptotic law is precisely the same as the distribution of the cost of one “job” in the random assignment problem. We also show that the asymptotic (scaled) maximum interattachment time to that shortest path tree, which is a uniform recursive tree, equals the square of the Dedekind Eta function, a central function in modular forms, elliptic functions, and the theory of partitions. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010