The equilibrium generalized assignment problem and genetic algorithm

The well-known generalized assignment problem (GAP) is to minimize the costs of assigning n jobs to m capacity constrained agents (or machines) such that each job is assigned to exactly one agent. This problem is known to be NP-hard and it is hard from a computational point of view as well. In this paper, follows from practical point of view in real systems, the GAP is extended to the equilibrium generalized assignment problem (EGAP) and the equilibrium constrained generalized assignment problem (ECGAP). A heuristic equilibrium strategy based genetic algorithm (GA) is designed for solving the proposed EGAP. Finally, to verify the computational efficiency of the designed GA, some numerical experiments are performed on some known benchmarks. The test results show that the designed GA is very valid for solving EGAP.     

Complexity of Workforce Scheduling in Transfer Lines

Consider a production system that consists of m assembly stations arranged in series. All jobs enter the assembly line at station 1 and proceed with subsequent stations in the same order as in a flow shop. Each job spends a fixed amount of time c in each station, known as the production cycle. This production system is synchronous or paced because jobs move one station forward synchronously, every c time units. To ensure that all required work is performed in precisely c periods, the appropriate number of workers is assumed to be known for every task in each station. Hence, each job is specified by an m-tuple of workforce requirements. We are interested in ``level' workforce schedules where workforce size fluctuations are minimal during the production horizon. In this article we define level workforce scheduling objectives and analyze the complexity status of the associated problems. We find that most of these problems are NP-complete even when m=2.     

Complexity analysis of an assignment problem with controllable assignment costs and its applications in scheduling

We extend the classical linear assignment problem to the case where the cost of assigning agent j to task i is a multiplication of task i’s cost parameter by a cost function of agent j. The cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for our assignment problem is defined by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The first criterion is the total assignment cost and the second is the total weighted resource consumption. We address these criteria via four different problem variations. We prove that our assignment problem is NP-hard for three of the four variations, even if all the resource consumption weights are equal. However, and somewhat surprisingly, we find that the fourth variation is solvable in polynomial time. In addition, we find that our assignment problem is equivalent to a large set of important scheduling problems whose complexity has been an open question until now, for three of the four variations.     

Asymptotical optimality of WSEPT for stochastic online scheduling on uniform machines

We study the stochastic online scheduling on m uniform machines with the objective to minimize the expected value of total weighted completion times of a set of jobs that arrive over time. For each job, the processing time is a random variable, and the distribution of processing time is unknown in advance. The actual processing time could be known only when the job is completed. For the problem, we propose a policy which is proved to be asymptotically optimal when the processing times and weights are uniformly bounded, i.e. the relative error of the solution achieved by our policy approaches zero as the number of jobs increases to infinity.     

A multi-period machine assignment problem

In this paper, a multi-period assignment problem is studied that arises as part of a weekly planning problem at mail processing and distribution centers. These facilities contain a wide variety of automation equipment that is used to cancel, sort, and sequence the mail. The input to the problem is an equipment schedule that indicates the number of machines required for each job or operation during the day. This result is then post-processed by solving a multi-period assignment problem to determine the sequence of operations for each machine. Two criteria are used for this purpose. The first is to minimize the number of startups, and the second is to minimize the number of machines used per operation.The problem is modeled as a 0–1 integer program that can be solved in polynomial time when only the first criterion is considered. To find solutions in general, a two-stage heuristic is developed that always obtains the minimum number of startups, but not necessarily the minimum number of machines per operation. In a comparative study, high quality solutions were routinely provided by the heuristic in negligible time when compared to a commercial branch-and-bound code (Xpress). For most hard instances, the branch-and-bound code was not able to even find continuous solutions within acceptable time limits.     

Multi-criteria assignment problem with incompatibility and capacity constraints

The considered assignment problem generalizes its classical counterpart by the existence of some incompatibility constraints limiting the assignment of tasks to processing units within groups of mutually exclusive tasks. The groups are defined for each processing unit and the constraints allow at most one task from each group to be assigned to the corresponding processing unit. The processing units can normally process a certain number of tasks without any cost; this capacity can be extended, however, at some extra marginal cost that is non-decreasing with the number of additional tasks. Each task has to be assigned to exactly one processing unit and has some preference for the assignment; it is expressed for each pair ‘task-processing unit’ by a dissatisfaction degree. The quality of feasible assignments is evaluated by three criteria: g ₁-the maximum dissatisfaction of tasks, g ₂-the total dissatisfaction of tasks, g ₃-the total cost of processing units. If there is no feasible assignment, tasks and processing units creating a blocking configuration are identified and all actions of unblocking are proposed. Formal properties of blocking configurations and unblocking actions are proven, and an interactive procedure for exploring the set of non-dominated assignments is described together with illustrative examples processed by special software.     

A faster algorithm for a due date assignment problem with tardy jobs

The single-machine due date assignment problem with the weighted number of tardy jobs objective, (the TWNTD problem), and its generalization with resource allocation decisions and controllable job processing times have been solved in O(n⁴) time by formulating and solving a series of assignment problems. In this note, a faster O(n²) dynamic programming algorithm is proposed for the TWNTD problem and for its controllable processing times generalization in the case of a convex resource consumption function.     

Scheduling with batch setup times and earliness-tardiness penalties

We consider a scheduling model in which several batches of jobs need to be processed by a single machine. During processing, a setup time is incurred whenever there is a switch from processing a job in one batch to a job in another batch. All the jobs in the same batch have a common due date that is either externally given as an input data or internally determined as a decision variable. Two problems are investigated. One problem is to minimize the total earliness and tardiness penalties provided that each due date is externally given. We show that this problem is NP-hard even when there are only two batches of jobs and the two due dates are unrestrictively large. The other problem is to minimize the total earliness and tardiness penalties plus the total due date penalty provided that each due date is a decision variable. We give some optimality properties for this problem with the general case and propose a polynomial dynamic programming algorithm for solving this problem with two batches of jobs. We also consider a special case for both of the problems when the common due dates for different batches are all equal. Under this special case, we give a dynamic programming algorithm for solving the first problem with an unrestrictively large due date and for solving the second problem. This algorithm has a running time polynomial in the number of jobs but exponential in the number of batches.     

Max-min matching problems with multiple assignments

In job assignment and matching problems, we may sometimes need to assign several jobs to one processor or several processors to one job with some limit on the number of permissible assignments. Some examples include the assignment of courses to faculty, consultants to projects, etc. In terms of objectives, we may wish to maximize profits or minimize costs, or maximize the minimal value (max-min criterion) of an attribute such as the performance rating of a processor in the matching, or combine the two goals into one composite objective function entailing time-cost tradeoffs. The regular bipartite matching algorithms cannot solve the matching problem, when upper and lower bounds are imposed on the number of assignments. In this paper, we present a method, referred to as the node-splitting method, that transforms the given problem into an assignment problem solvable by the Hungarian method.     

An algebraic geometry algorithm for scheduling in presence of setups and correlated demands

We study here a problem of schedulingn job types onm parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance constrained integer program.Methods of solution currently available—in integer programming and stochastic programming—are not sufficient to solve this model exactly. We develop and introduce here a new approach, based on a geometric interpretation of some recent results in Gröbner basis theory, to provide a solution method applicable to a general class of chance constrained integer programming problems.Out algorithm is conceptually simple and easy to implement. Starting from a (possibly) infeasible solution, we move from one lattice point to another in a monotone manner regularly querying a membership oracle for feasibility until the optimal solution is found. We illustrate this methodology by solving a problem based on a real system.Corresponding author.     

An Optimization Model for Stochastic Project Networks with Cash Flows

Project networks – or PERT networks – can be characterized by random completion times of activities and positive or negative cash flows throughout the project. In these cases the decision maker’s problem consists of determining a feasible activities schedule, to maximize the project financial value, where the financial value is measured by the net present value (npv) of cash flows.The analysis of these networks is a difficult computational task for the following reason. First, suppose that a schedule is fixed using a heuristic rule. Then the expected npv is calculated. But, due to stochastic job completion times, this problem belongs to the ♯-P complete difficulty class, e.g. problems that involve finding all the Hamiltonian cycles in a network. The problem is such that evaluating one project alone is not sufficient, but the optimal one has to be selected. This involves a further increase in computational time.This paper proposes a stochastic optimization model to determine a heuristic scheduling rule, that provides an approximate solution to finding the optimal project npv. A feature of this approach is that the scheduling rule is completely deterministic and defined when the project begins. Therefore an upper bound of the expected npv, that is an optimistic estimate, can be calculated through linear programming and a lower bound, that is a pessimistic estimate, can be calculated using simulation before the project begins.     

Single machine scheduling problems with general position-dependent processing times and past-sequence-dependent delivery times

This paper considers single machine scheduling with past-sequence-dependent (psd) delivery times, in which the processing time of a job depends on its position in a sequence. We provide a unified model for solving single machine scheduling problems with psd delivery times. We first show how this unified model can be useful in solving scheduling problems with due date assignment considerations. We analyze the problem with four different due date assignment methods, the objective function includes costs for earliness, tardiness and due date assignment. We then consider scheduling problems which do not involve due date assignment decisions. The objective function is to minimize makespan, total completion time and total absolute variation in completion times. We show that each of the problems can be reduced to a special case of our unified model and solved in O(n ³) time. In addition, we also show that each of the problems can be solved in O(nlogn) time for the spacial case with job-independent positional function.     

An algebraic approach to assignment problems

For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.     

Group Scheduling with Controllable Setup and Processing Times: Minimizing Total Weighted Completion Time

The following single machine scheduling problem is studied. A partition of a set of n jobs into g groups on the basis of group technology is given. The machine processes jobs of the same group contiguously, with a sequence independent setup time preceding the processing of each group. The setup times and the job processing times are controllable through the allocation of a continuously divisible or discrete resource to them. Each job uses the same amount of the resource. Each setup also uses the same amount of resource, which may be different from that for the jobs. Polynomial-time algorithms are constructed for variants of the problem of finding an optimal job sequence and resource values so as to minimize the total weighted job completion time, subject to given restrictions on resource consumption. The algorithms are based on a polynomial enumeration of the candidates for an optimal job sequence and solving the problem with a fixed job sequence by linear programming. This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grants Council of Hong Kong under grant number PolyU 5191/01E. In addition, the research of M.Y. Kovalyov was supported by INTAS under grant number 00-217.     

Second-order scenario approximation and refinement in optimization under uncertainty

When solving scenario-based stochastic programming problems, it is imperative that the employed solution methodology be based on some form of problem decomposition: mathematical, stochastic, or scenario decomposition. In particular, the scenario decomposition resulting from scenario approximations has perhaps the least tendency to be computationally tedious due to increases in the number of scenarios. Scenario approximations discussed in this paper utilize the second-moment information of the given scenarios to iteratively construct a (relatively) small number of representative scenarios that are used to derive bounding approximations on the stochastic program. While the sizes of these approximations grow only linearly in the number of random parameters, their refinement is performed by exploiting the behavior of the value function in the most effective manner. The implementation SMART discussed here demonstrates the aptness of the scheme for solving two-stage stochastic programs described with a large number of scenarios.This paper was presented at the IFIP Workshop onStochastic Programming: Algorithms and Models, Lillehammer, Norway, January 1994.     

Complexity of shop-scheduling problems with fixed number of jobs: a survey

The paper surveys the complexity results for job shop, flow shop, open shop and mixed shop scheduling problems when the number n of jobs is fixed while the number r of operations per job is not restricted. In such cases, the asymptotical complexity of scheduling algorithms depends on the number m of machines for a flow shop and an open shop problem, and on the numbers m and r for a job shop problem. It is shown that almost all shop-scheduling problems with two jobs can be solved in polynomial time for any regular criterion, while those with three jobs are NP-hard. The only exceptions are the two-job, m-machine mixed shop problem without operation preemptions (which is NP-hard for any non-trivial regular criterion) and the n-job, m-machine open shop problem with allowed operation preemptions (which is polynomially solvable for minimizing makespan).     

A large population job search game with discrete time

A job search problem is considered, in which there is a large population of jobs initially available and a large population of searchers. The ratio of the number of searchers to the number of jobs is α. Each job has an associated value from a known distribution. At each of N moments the searchers observe a job, whose value comes from the distribution of the values of currently available jobs. If a searcher accepts a job, s/he ceases searching and the job becomes unavailable. Hence, the distribution of the values of available jobs changes over time. Also, the ratio of the number of those still searching to the number of available jobs changes. The model is presented and Nash equilibrium strategies for such problems are considered. By definition, when all the population use a Nash equilibrium strategy, the optimal response of an individual is to use the same strategy. Conditions are given that ensure the existence of a unique Nash equilibrium strategy. Examples are given to illustrate the model and present different approaches to solving such problems.     

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem’s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution—VSS—and the expected value of perfect information—EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

预知工件大小上界的平行机排序问题

平行机排序问题广泛出现并应用于各领域,如通讯网信道分配的负载均衡,大型计算中的并行计算,柔性制造系统的任务编排等等.研究了预知工件大小上界的半在线平行机排序问题.考察了仅预知工件大小上界和既预知工件大小上界又预知最优目标值的两类半在线模型.基于资源分配公平性和提高服务质量的考虑,针对每类模型都分别考察了两个目标:C_(max)(极小化机器最大负载makespan)和C_(min)(极大化机器最小负载).在不同的目标下,针对m台平行机的一般情况均给出了问题的下界并设计了半在线算法,某些情况下设计的算法是最优算法.