Tabu search for a class of scheduling problems

Scheduling problems are often modeled as resourceconstrained problems in which critical resource assignments to tasks are known and the best assignment of resource time must be made subject to these constraints. Generalization toresource scheduling, where resource assignments are chosen concurrently with times results is a problem which is much more difficult. A simplified model of the general resource scheduling model is possible, however, in which tasks must be assigned a singleprimary resource, subject to constraints resulting from preassignment ofsecondary, or auxiliary, resources. This paper describes extensions and enhancements of tabu search for the special case of the resource scheduling problem described above. The class of problems is further restricted to those where it is reasonable to enumerate both feasible time and primary resource assignments. Potential applications include shift oriented production and manpower scheduling problems as well as course scheduling where classrooms (instructors) are primary and instructors (rooms) and students are secondary resources. The underlying model is a type of quadratic multiple choice problem which we call multiple choice quadratic vertex packing (MCQVP). Results for strategic oscillation and biased candidate sampling strategies are shown for reasonably sized real and randomly generated, synthetic, problem instances. The strategies are compared with other variations using consistent measures of solution time and quality developed for this study.     

Multiple criteria sorting with a set of additive value functions

We present a new multiple criteria sorting method that aims at assigning actions evaluated on multiple criteria to p pre-defined and ordered classes. The preference information supplied by the decision maker (DM) is a set of assignment examples on a subset of actions relatively well known to the DM. These actions are called reference actions. Each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes. The set of assignment examples is used to build a preference model of the DM represented by a set of general additive value functions compatible with the assignment examples. For each action a, the method computes two kinds of assignments to classes, concordant with the DM’s preference model: the necessary assignment and the possible assignment. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. The compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs.     

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _ij and incurs a cost ofc _ij ; each machinei is available forT _i time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _i time units.We also extend this result to a variant of the problem where, instead of a fixed processing timep _ij , there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T. Research partially supported by an NSF PYI award CCR-89-96272 with matching support from UPS, and Sun Microsystems, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.Research supported in part by a Packard Fellowship, a Sloan Fellowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

一类广义指派问题的有效解法

提出一类广义指派问题,这类问题研究的是m个人执行n项任务,每个人执行的任务数、执行每项任务的人数以及总的指派人项数均有限制,要求最优指派.对这类广义指派问题建立了数学模型,并找到一种转换方法,将这类问题转换为平衡指派问题,从而用传统方法,如匈牙利法求解.最后用一个箅例来说明这种转换方法的简便和有效性.     

A Special Multi-Objective Assignment Problem

The following assignment problem is considered. There are n activities to be assigned to n personnel. The cost of assigning activity i to person j is c _ij . It is required to find all the efficient assignments, i.e. those for which there exists no other assignment which has at least as small costs for each person and strictly smaller costs for at least one person. The main results are as follows. In Theorem 1 it is shown that whereas, for many integer problems, the standard scalar weighting factor approach will not produce all the efficient solutions, in this case it will. In Theorem 2 it is shown that when each efficient vector is determined by a single assignment solution, the efficient set is identical to the set of efficient vertices of the convex hull of the assignment solution set.     

Random channel assignment in the plane

In the model for random radio channel assignment considered here, points corresponding to transmitters are thrown down independently at random in the plane, and we must assign a radio channel to each point but avoid interference. In the most basic version of the model, we assume that there is a threshold d such that, in order to avoid interference, points within distance less than d must be assigned distinct channels. Thus we wish to color the nodes of a corresponding scaled unit disk graph. We consider the first n random points, and we are interested in particular in the behavior of the ratio of the chromatic number to the clique number. We show that, as n → ∞, in probability this ratio tends to 1 in the “sparse” case [when d = d(n) is such that the average degree grows more slowly than ln n] and tends to ～ 1.103 in the “dense” case (when the average degree grows faster than ln n). We also consider related graph invariants, and the more general channel assignment model when assignments must satisfy “frequency‐distance” constraints. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 187–212, 2003     

Lagrangean/surrogate relaxation for generalized assignment problems

This work presents Lagrangean/surrogate relaxation to the problem of maximum profit assignment of n tasks to m agents (n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. The Lagrangean/surrogate relaxation combines usual Lagrangean and surrogate relaxations relaxing first a set of constraints in the surrogate way. Then, the Lagrangean relaxation of the surrogate constraint is obtained and approximately optimized (one-dimensional dual). The Lagrangean/surrogate is compared with the usual Lagrangean relaxation on a computational study using a large set of instances. The dual bounds are the same for both relaxations, but the Lagrangean/surrogate can give improved local bounds at the application of a subgradient method, resulting in less computational times. Three relaxations are derived for the problem. The first relaxation considers a vector of multipliers for the capacity constraints, the second for the assignment constraints and the other for the Lagrangean decomposition constraints. Relaxation multipliers are used with efficient constructive heuristics to find good feasible solutions. The application of a Lagrangean/surrogate approach seems very promising for large scale problems.     

Relaxation heuristics for a generalized assignment problem

We propose relaxation heuristics for the problem of maximum profit assignment of n tasks to m agents (n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. Using Lagrangian or surrogate relaxation, the heuristics perform a subgradient search obtaining feasible solutions. Relaxation considers a vector of multipliers for the capacity constraints. The resolution of the Lagrangian is then immediate. For the surrogate, the resulting problem is a multiple choice knapsack that is again relaxed for continuous values of the variables, and solved in polynomial time. Relaxation multipliers are used with an improved heuristic of Martello and Toth or a new constructive heuristic to find good feasible solutions. Six heuristics are tested with problems of the literature and random generated problems. Best results are less than 0.5% from the optimal, with reasonable computational times for an AT/386 computer. It seems promising even for problems with correlated coefficients.     

A new characterization of matrices with the consecutive ones property

We consider the following constraint satisfaction problem: Given a set F of subsets of a finite set S of cardinality n, and an assignment of intervals of the discrete set {1,…,n} to each of the subsets, does there exist a bijection f:S→{1,…,n} such that for each element of F, its image under f is same as the interval assigned to it. An interval assignment to a given set of subsets is called feasible if there exists such a bijection. In this paper, we characterize feasible interval assignments to a given set of subsets. We then use this result to characterize matrices with the Consecutive Ones Property (COP), and to characterize matrices for which there is a permutation of the rows such that the columns are all sorted in ascending order. We also present a characterization of set systems which have a feasible interval assignment.     

An extended assignment problem considering multiple inputs and outputs

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit incurred by each possible assignment. However, in real applications, various inputs and outputs are usually concerned in an assignment problem, such as a general decision-making problem. This paper develops a procedure for resolving assignment problems with multiple incommensurate inputs and outputs for each possible assignment. The concept of the relative efficiency in using various resources, instead of cost or profit, is adopted for each possible assignment of the problem. Data envelopment analysis (DEA) is employed in this paper to measure the efficiency of one assignment relative to that of the others according to a set of decision-making units. A composite efficiency index, consisting of two kinds of relative efficiencies under different comparison bases, is defined to serve as the performance measurement of each possible assignment in the problem formulation. A mathematical programming model for the extended assignment problem is proposed, which is then expressed as a classical integer linear programming model to determine the assignments with the maximum efficiency. A numerical example is used to demonstrate the approach.     

The elastic generalized assignment problem

The generalized assignment problem (GAP) has been studied by numerous researchers over the past 30 years or so. Simply stated, one must find a minimum-cost assignment of tasks to agents such that each task is assigned to exactly one agent and such that each agent's resource capacity is honoured. The problem is known to be NP-hard. In this paper, we study the elastic generalized assignment problem (EGAP). The elastic version of GAP allows agent resource capacity to be violated at additional cost. Another version allows undertime costs to be assessed as well if an agent's resource capacity is not used to its full extent. The EGAP is also NP-hard. We describe a special-purpose branch-and-bound algorithm that utilizes linear programming cuts, feasible solution generators, Lagrangean relaxation and subgradient optimization. We present computational results on a large collection of randomly generated ‘hard’ problems with up to 4000 binary variables.     

Stable Marriage and Genetic Algorithms: A Fertile Union

We describe a pair of genetic algorithms for solving two stable matching problems. Both stable matching problems we will consider involve a set of applicants for positions and a set of employers. Each applicant and each employer prepares a rank order list of a subset of the actors in the other set. The goal is to find an assignment of applicants to employers in which if applicant a is not assigned to employer b then either a prefers his assignment to b or b prefers its assignment toa . In other words, no applicant/employer pair can both improve their situations by dropping their current assignments in favor of each other. Our goal will be to enumerate the stable matchings. One of the problems we will consider is the well-known stable marriage problem, in which neither applicant nor employer preference lists are linked. In the other problem, we will allow pairs of applicants who form a couple to submit joint rank order lists of ordered pairs of employers.     

Decentralized list scheduling

Classical list scheduling is a very popular and efficient technique for scheduling jobs for parallel and distributed platforms. It is inherently centralized. However, with the increasing number of processors, the cost for managing a single centralized list becomes too prohibitive. A suitable approach to reduce the contention is to distribute the list among the computational units: each processor only has a local view of the work to execute. Thus, the scheduler is no longer greedy and standard performance guarantees are lost. The objective of this work is to study the extra cost that must be paid when the list is distributed among the computational units. We first present a general methodology for computing the expected makespan based on the analysis of an adequate potential function which represents the load imbalance between the local lists. We obtain an equation giving the evolution of the potential by computing its expected decrease in one step of the schedule. Our main theorem shows how to solve such equations to bound the makespan. Then, we apply this method to several scheduling problems, namely, for unit independent tasks, for weighted independent tasks and for tasks with precedence constraints. More precisely, we prove that the time for scheduling a global workload W composed of independent unit tasks on m processors is equal to W/m plus an additional term proportional to log₂ W. We provide a lower bound which shows that this is optimal up to a constant. This result is extended to the case of weighted independent tasks. In the last setting, precedence task graphs, our analysis leads to an improvement on the bound of Arora et al. (Theory Comput. Syst. 34(2):115–144, 2001). We end with some experiments using a simulator. The distribution of the makespan is shown to fit existing probability laws. Moreover, the simulations give a better insight into the additive term whose value is shown to be around 3log₂ W confirming the precision of our analysis.     

Generalized cover facet inequalities for the generalized assignment problem

The generalized assignment problem is that of finding an optimal assignment of agents to tasks, where each agent may be assigned multiple tasks and each task is performed exactly once. This is an NP-complete problem. Algorithms that employ information about the polyhedral structure of the associated polytope are typically more effective for large instances than those that ignore the structure. A class of generalized cover facet-defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from generalized cover inequalities.     

Asymptotic analysis for multi-objective sequential stochastic assignment problems

ABSTRACT

We provide an asymptotic analysis of multi-objective sequential stochastic assignment problems (MOSSAP). In MOSSAP, a fixed number of tasks arrive sequentially, with an n-dimensional value vector revealed upon arrival. Each task is assigned to one of a group of known workers immediately upon arrival, with the reward given by an n-dimensional product-form vector. The objective is to maximize each component of the expected reward vector. We provide expressions for the asymptotic expected reward per task for each component of the reward vector and compare the convergence rates for three classes of Pareto optimal policies.     

A buffer minimization problem for the design of embedded systems

We consider a set of tasks, each of them is composed by a set of sequential operations, and a set of buffers. Each buffer b is defined between two tasks T_i and T_j, and is managed as a FIFO structure. Some operations from T_i write data to the buffer b, others from T_j get data from b.The writings and readings generate precedence constraints between the operations. The limitation of the size of the buffers generates another set of precedence constraints between them and circuits in the precedence graph may appear. In this case, there is no feasible schedule for the operations. The aim is to determine the size of each buffer such that the global surface of the buffers is minimized and there is no circuit in the precedence graph.We prove that this problem is polynomial for two tasks using a flow algorithm. We also prove that it is NP-complete in the strong sense for three tasks.     

Teaching Probability in Intermediate Grades∗;†

This paper discusses three alternative models for structuring homework assignments in college level mathematics. A distributive model which incorporates both early and late review of previously learned material into the daily assignment schedule is explicated in detail. The relative efficacy of the conventional method for assigning homework and the distributive method in promoting achievement and favourable attitudes toward mathematics in a first semester course in calculus was investigated in a controlled experiment. Significant (p < 0.03) interactions between mathematical ability and assignment model were found on three of four unit tests. In each case, pupils with the weaker pre‐calculus background profited more from the distributive assignment schedule. There were no significant interactions or differences on a comprehensive post‐test or on attitudes toward mathematics. Implications for research and practice in college level mathematics, physics, and engineering courses are suggested.     

A constant factor approximation algorithm for the fault-tolerant facility location problem

We consider a generalization of the classical facility location problem, where we require the solution to be fault-tolerant. In this generalization, every demand point j must be served by r_j facilities instead of just one. The facilities other than the closest one are “backup” facilities for that demand, and any such facility will be used only if all closer facilities (or the links to them) fail. Hence, for any demand point, we can assign nonincreasing weights to the routing costs to farther facilities. The cost of assignment for demand j is the weighted linear combination of the assignment costs to its r_j closest open facilities. We wish to minimize the sum of the cost of opening the facilities and the assignment cost of each demand j. We obtain a factor 4 approximation to this problem through the application of various rounding techniques to the linear relaxation of an integer program formulation. We further improve the approximation ratio to 3.16 using randomization and to 2.41 using greedy local-search type techniques.     

Scheduling unit — time tasks on flow — shops under resource constraints

We consider the problem of scheduling tasks on flow shops when each task may also require the use of additional resources. It is assumed that all operations have unit lengths, the resource requirements are of 0–1 type and there is one type of the additional resource in the system. It is proved that when the number of machines is arbitrary, the problem of minimizing schedule length is NP-hard, even when only one unit of the additional resource is available in the system. On the other hand, when the number of machines is fixed, then the problem is solvable in polynomial time, even for an arbitrary number of resource units available. For the two machine case anO(n log ₂ ² n) algorithm minimizing maximum lateness is also given. The presented results are also of importance in some message transmission systems.     

Dynamic Capacity Acquisition and Assignment under Uncertainty

Given a set of m resources and n tasks, the dynamic capacity acquisition and assignment problem seeks a minimum cost schedule of capacity acquisitions for the resources and the assignment of resources to tasks, over a given planning horizon of T periods. This problem arises, for example, in the integrated planning of locations and capacities of distribution centers (DCs), and the assignment of customers to the DCs, in supply chain applications. We consider the dynamic capacity acquisition and assignment problem in an environment where the assignment costs and the processing requirements for the tasks are uncertain. Using a scenario based approach, we develop a stochastic integer programming model for this problem. The highly non-convex nature of this model prevents the application of standard stochastic programming decomposition algorithms. We use a recently developed decomposition based branch-and-bound strategy for the problem. Encouraging preliminary computational results are provided.