Dimitri Golenko-Ginzburg

Stochastic network project scheduling with non-consumable limited resources

Abstract This paper presents a newly developed resource constrained scheduling model for a PERT type project. Several non-consumable activity related resources, such as machines or manpower, are imbedded in the model. Each activity in a project requires resources of various types with fixed capacities. Each type of resource is in limited supply with a resource limit that is fixed at the same level throughout the project duration. For each activity, its duration is a random variable with given density function. The problem is to determine starting time values S ij for each activity ( i , j ) entering the project, i.e., the timing of feeding-in resources for that activity. Values S ij are not calculated beforehand and are random values conditional on our decisions. The models objective is to minimize the expected project duration. Determination of values S ij is carried out at decision points when at least one activity is ready to be operated and there are free available resources. If, at a certain point of time, more than one activity is ready to be operated but the available amount of resources is limited, a competition among the activities is carried out in order to choose those activities which can be supplied by the resources and which have to be operated first. We suggest carrying out the competition by solving a zero-one integer programming problem to maximize the total contribution of the accepted activities to the expected project duration. For each activity, its contribution is the product of the average duration of the activity and its probability of being on the critical path in the course of the projects realization. Those probability values are calculated via simulation. Solving a zero-one integer programming problem at each decision point results in the following policy: the project management takes all measures to first operate those activities that, being realized, have the greatest effect of decreasing the expected project duration. Only afterwards, does the management take care of other activities. A heuristic algorithm for resource constrained project scheduling is developed. A numerical example is presented.

A heuristic for network project scheduling with random activity durations depending on the resource allocation

Abstract This paper presents a heuristic for resource-constrained network project scheduling. A PERT-type project, where activities require resources of various types with variable capacities, is considered. Each activity is of random duration depending on the resource amounts assigned to that activity. The problem is to minimize the expected project duration by determining for each activity its starting time and the assigned resource capacities. The resource project scheduling model is an NP-complete knapsack resource reallocation problem. To obtain a precise solution, a lookover algorithm has been developed. For the case when a lookover requires mmuch computational time, an approximate heuristic algorithm is suggested. The problem has to be solved at each decision point, when at least more than one activities are ready to be operated but the available amount of resources is limited. The heuristic model is illustrated by a numerical example.

Optimal job-shop scheduling with random operations and cost objectives

Abstract We consider a job-shop manufacturing cell of n jobs (orders), Ji, 1⩽i⩽n, and m machines Mk, 1⩽k⩽m. Each job-operation Oil (the lth operation of job i) has a random time duration til with the average value t il and the variance Vil. Each job Ji has its due date Di and the penalty cost Ci* for not delivering the job on time (to be paid once to the customer). An additional penalty C i ∗∗ has to be paid for each time unit of delay, i.e., when waiting for the jobs delivery after the due date. If job Ji is accomplished before Di it has to be stored until the due date with the expenses C i ∗∗∗ per time unit. The problem is to determine optimal earliest start times Si of jobs Ji, 1⩽i⩽n, in order to minimize the average value of total penalty and storage expenses. Three basic principles are incorporated in the model: 1. At each time moment when several jobs are ready to be served on one and the same machine, a competition among them is introduced. It is based on the newly developed heuristic decision-making rule with cost objectives. 2. A simulation model of manufacturing the job-shop and comprising decision-making for each competitive situation, is developed. 3. Optimization is carried out by applying to the simulation model the coordinate descent search method. The variables to be optimized are the earliest start times Si. A numerical example of a simulation run is presented to clarify the decision-making rule. The optimization model is verified via extensive simulation.

A two-level decision-making model for controlling stochastic projects

Abstract We consider projects with a high level of uncertainty comprising; activities of random time duration and both random and deterministic alternative outcomes in key nodes. A two-level control model is developed. On the upper level decision-making centers on choosing an optimal outcome direction at every deterministic alternative node which is reached in the course of the projects realization. On the lower level (between two adjacent decision-making nodes) the problem of controlling a project is, in essence: (a) Calculating periodically and examining the probability of meeting the projects due date on time; (b) Redistributing, if necessary, the projects budget among the activities to minimize the projects average remaining time duration. Heuristic procedures to solve this stochastic optimization problem are presented. A numerical example is given.

Industrial job-shop scheduling with random operations and different priorities

Abstract A classical job-shop scheduling problem with n jobs (orders) and m machines is considered. Each job-operation Oil (the l− th operation of job i, l = 1,…,m, i = 1, 2,…,n) has a random time duration til with the average value t il and the variance Vil. Each job Ji has its due date Di and its priority index ϱi. Given p i ∗ , the desired probability for job Ji to be accomplished on time, and p i ∗ ∗ , the least permissible probability for the job to meet its due date on time, the problem is to determine starting time values Sil for each job-operation Oil. Those values are not calculated beforehand and are values conditioned on our decisions. Decision-making, i.e., determining values Sil is carried out at the moments when at least one of the machines is free for service and at least one job is ready to be processed on that machine. If at a certain moment t more than one job is ready to be processed, these jobs are compared pairwise. The winner of the first pair will be compared with the third job, etc., until only one job will be left. The latter has to be chosen for the machine. The competition is carried out by calculating the jobs delivery performance, i.e., the probability for a certain job to meet its due date on time. Such a calculation is carried out by determining the probability to meet the deadline for the chain of random operations. Two different heuristics for choosing a job from the line will be imbedded in the problem. The first one is based on examining delivery performance values together with priority indices ϱi. The second one deals with examining confidence possibilities p i ∗ and p i ∗ ∗ and does not take into account priority indices. A numerical example is presented. Both heuristics are examined via extensive simulation in order to evaluate their comparative efficiency for practical industrial problems.

Resource constrained scheduling simulation model for alternative stochastic network projects

The paper presents a heuristic for resource constrained network project scheduling. A network project comprising both alternative deterministic decision nodes and alternative branching nodes with probabilistic outcomes is considered. Several renewable activity related resources, such as machines and manpower, are imbedded in the model. Each type of resource is in limited supply with a resource limit that is fixed at the same level throughout the project duration. Each activity in the project requires resources of various types with fixed capacities. The activity duration is a random variable with given density function.The problem is to minimize the expected project duration by determining for each activity, which will be realized within the projects realization, its starting time (decision variable), i.e. the time of feeding-in resources. The resource delivery schedule is not calculated in advance and is based on decision-making in the course of monitoring the project. The suggested heuristic algorithm is performed in real time via simulation. Decision-making is carried out: • at alternative deterministic decision nodes, to single out all the alternative sub-networks (joint variants) in order to choose the one with the minimal average duration; • at other essential moments when at least one activity is ready to be operated but the available amount of resources is limited. A competition among those activities is carried out to determine the subset of activities which have to be operated first and can be supplied by available resources. Such a competition is realized by a combination of a knapsack resource reallocation model and a subsidiary simulation algorithm.

Hierarchical decision-making model for planning and controlling stochastic projects

Abstract We will consider a three-level decision-making model for controlling stochastic network projects. The upper level (the company level) is faced with the problem of optimal budget allocation among several projects. We will consider network projects with random activity durations and alternative outcomes in key nodes. There are two different types of alternative events. The first one reflects stochastic (uncontrolled) branching of the project development. The alternative event of the second type is of a deterministic nature, i.e., the projects manager chooses the outcome direction. At the medium level (project level) the management determines on the basis of the available budget an optimal joint variant together with an optimal outcome direction at every decision node which is reached in the course of the project realization. At the subnetwork level the optimal joint variant is controlled by reallocating the remaining budget among the remaining activities to raise the confidence probability of meeting the projects due date on time. This paper is a further development of our current publications. The main goal of the paper is to develop a unified three-level decision-making model and to indicate planning and control actions and optimization problems for all levels. A numerical example is presented.

Hierarchical control of semi-automated production systems

Abstract An integrated control model of a hierarchical production system is presented where the output can be measured only at preset control points as it is impossible or costly to measure it continuously. Three levels are considered—company, section, production unit—each level faces stochastic optimization problems. Each unit produces a given target amount by a given due date lcommon to all unitsr and has several possible speeds, which are subject to disturbances. On the unit level, at each control point, decision-making centres on determining both the next control point and the speed to proceed with up to that point. The section level is faced with problems of either reallocating resources among the sections units or reassigning the remaining target amounts among the units so that the faster one will help the slower one. The company level is faced with similar problems, i.e. reallocating resources or reassigning target amounts among the sections. Two different cases are cons...

High performance heuristic algorithm for controlling stochastic network projects

Abstract An activity-on-arc network project of PERT type with random activity durations is considered. The progress of the project cannot be inspected and measured continuously, but only at preset inspection points. An on-line control model has to determine both inspection points and control actions to be introduced at those points to alter the progress of the project in the desired direction. On-line control is carried out to minimize the number of inspection points needed to meet the target, subject to the chance constraint. In the recently developed control models, determining the next inspection point is carried out via extensive simulation with a constant time step. This determination is based on sequential statistical analysis at each intermediate point to maximize the time span between two adjacent control points. The main shortcoming of the control algorithm is its long computational time due to the need to make numerous decisions. In this paper we present a newly developed heuristic control algorithm in which the timing of inspection points does not comprise intermediate decision making. Given a routine inspection point t i , the adjacent point t i +1 is determined so that even if the project develops most unfavorably in the interval [ t i , t i +1 ], introducing proper control action at moment t i +1 enables the project to meet its target on time, subject to the chance constraint. The newly developed control algorithm is essentially more efficient than the step-by-step control procedures. The computational time is reduced by a factor of 25–30 while the algorithm provides better solutions than would be attained by using on-line sequential statistical analysis. Extensive experimentation has been undertaken to illustrate the comparative efficiency of the presented algorithm.

A generalized control model for man-machine production systems with disturbances

A control model for a two-level man-machine production system is considered. The system comprises a section and several production units. Within the planning horizon the section is faced with manufacturing several different products with planned target amounts. Each unit can manufacture all kinds of products. In the course of manufacturing, each unit utilizes different types of non-consumable resources which may be reallocated among the units. Each production unit can manufacture a product at several possible speeds which correspond to one and the same resource capacities. Those speeds depend only on the degree of intensity of manufacturing and are subject to random disturbances. To carry out the process of manufacturing, the products have to be rescheduled among the units. This means that for each unit and for each product assigned to that unit the corresponding planned amount and the planning horizon have to be determined. Controlling the system is carried out at two levels: the section level and the unit level. At the unit level all production units are controlled separately. For each unit and for each product manufactured by that unit decision-making centers on determining: (i) control points to observe the products output; (ii) the speeds to manufacture the product. If at a routine control point it is anticipated that a unit is unable to meet its deadline on time, emergency is called. The section level is then faced with the problem of both resource and target amount reallocation among the units. New resource capacities and target amounts for each product and each production unit are decision variables to be determined. The objective is to maximize the probability of the slowest unit to accomplish the planned amounts of its products by the due date. The problem is too difficult to obtain a precise or even an approximate solution. Heuristic algorithms are outlined on both levels. The models performance is verified via extensive simulation.