Study of the Constrained Resources to Projects Based on the Random Networks

In this paper, a number of concurrent network projects with random activities are studied, and the proposed model enables people to control various resources such as rare and expensive resources transported from outside in a short time. For each project, the schedule date and the admissible confidence probability of timely completion of the project were defined. The initial instants of realization of all projects, the required total power of each of the reproducible resources, the delivery schedule of rare reproducible resources that are not at the disposal of the project management system, and the schedule of beginning all activities were determined. The minimal nonoperational costs, such as penalty provisions for failure to execute the project in time, fines for idling of the rare external resources, costs of resource lease were used as the target function. The developed model can be used to plan and supervise one or more random PERT-like network projects. It can significantly reduce the cost of project implementation.


Introduction
There are numerous papers on optimization of allocation and delivery of the resources for network projects with deterministic operations. Recently, there exist some resource models for projects with random durations of operations. These models are more sophisticated mostly compared with the deterministic models because of the difficulties involved in constructing the schedule of resource delivery and consumption. At the same time, the majorities of projects are carried out in the random environment and constrained resources. Therefore, development of more generalized resource models for network projects seems quite topical both from the scientific and applied standpoints.

Problem Descriptions
The resources involved into the network projects can be classified with two groups [1]: Expensive and rare resources are Class A resources. The A-resources must be strictly controlled in the sense that their delivery must be scheduled in advance. The Aresource delivery graph must be deterministic, because one cannot envisage even approximately when any activity will be ready for execution.
Reproducible resources are class B-resources. Bresources can be used in one or more project activities at any time t, if available. Therefore, the instants of delivering the B-resources to the project activities are random and defined in the course of project execution.
In the paper, the models of planning and supervising the network projects [2] not only include the project schedule dates but also include supervision by probability, that is, the model parameters must include the lower bound on the confidence probability of executing each project by the schedule date. The instants of beginning each project also are among the optimized parameters. This is necessary because the lease of the reproducible Bresources is related with the initial instant of supplying them to the project store. The leased resources are stored from the initial instant (deterministic variable to be optimized) and to the instant of project completion (random variable). The model developed below includes numerous cost parameters. The operational costs such as the cost of raw materials, electric power, the A-resources and B-resources in the course of project execution, and so on, which are independent of the model of planning or supervision, were not included in the model intentionally. The target function of the model includes the following nonoperational costs.
Costs of leasing and maintaining the B-resources during the time of project realization.
Costs of project storage for each time unit of storage. Fines on the project management system for idling of the A-resources.
Fine paid once to the customer for failure to execute the project in time.
Fine paid for each project for each unit of the outstanding project time.

Problem Formalization
Before starting the projects, we produce a e-th (e is the number of network projects) random PERT network project G e (N, A); the deterministic optimal values of S e (the instant of starting the eth project), R ke (the total power of the kth resource leased for the eth project) and

Heuristic Algorithms
In the heuristic algorithm, different projects are assumed independently of each other, which eliminate the possibility of using a unique central store of resources. Under this assumption, the problem (1)-(7) can be solved independently for each project. The resulting optimal mean values of the nonoperational costs e C for each eth project can be summed to obtain the desired optimal value C for the project system as a whole. Here, the values of S e , R ke , and obtained for individual projects are the solution of the global problem (1)- (7).
Two hierarchical levels, external and internal, are introduced in the model. At the upper level, we solve for one project the following problem (called below P1). In order to minimize the nonoperational conditional costs with regard for the confidence probability p*, it is required to determine the optimal values of S, {Rk}, 1≤ k≤ m, and ) , ( is established at the lower hierarchical level by means of the simulation model combined at the upper level with the extremum-seeking model;  p which is determined by simulation is the statistical frequency of the estimate of the probability P r {F <D} with a fixed set of values S, R k , and ) , ( At the lower hierarchical level, the simulation project model with a built in optimizer, the optimal problem P2, is realized. The current search point X in the space of 1 + m + n A variables S, R k , and ) , ( is the input parameter of the model. The simulation model determines the random instants of beginning all project operations S ij with regard for constraints (3)- (5). As for the optimizer, it allocates the free B-resources to the ready project activities which consume these resources. The free resources are allocated at the decision points provided that there are activities queued for the resources at the project store.
Execution of the project must be preceded by solving problem P1 at the upper hierarchical level, that is, determining the optimal set of values of S, Rk, and ) , ( the resulting optimal set of parameters is accepted as the plan, then the real project can be supervised, the Bresources leased, and the A-resources supplied. Therefore, solution of the problem P1 provides the optimal project schedule. As for estimation of the random realization of S ij , they are determined by a single run of the simulation model [4].

Simulation Model Of Consumming The B-Resources
As was noted above, the optimal values of S, R k , and ) , (

1)
Determines the instants t of making decisions about the allocation of the available B-resources to the ready project activities (if the available B-resources can be supplied to at least one activity. 2) Queues the ready activities.

3) Supplies the A-resources to the ready activities in compliance with constraints (3), (4).
4) Supplies the available B-resources to the queued activities if the total need for resource of all queued activities does not exceed the free stored resources. 5) Models durations tij of all project activities (i, j) upon supplying the A-resources or B-resources for their execution. Therefore, the simulation submodel determines both S ij and F ij = S ij + t ij .

7) Determines (after simulation of t ij and F ij ) the instants of all project events i to estimate T(i) in constraint (4) by the formula
where D(j) is the set of events i that immediately precedes the event .
8) Determines by simulation the auxiliary parameters p(i B ; j B ) for the problem P2.

Optimal Submodel
It includes P2 and at the instants t of decision making allocates the available B-resources to the queued project activities. If at the instant t more than one activity is queued and the free B-resources are limited, then one must conduct `competition' between the activities to establish which of them must be provided with resources in the first place. The problem P2 is formulated in mathematical terms as follows: let at the instant t there be q<n B queued activities be valid at least for one of the subscripts k, that is, the B-resources of the kth type are insufficient to supply all queued activities [5].
In the case of fixed powers

Example
The paper considers a PERT-like network project (Fig. 1) consisting of twenty activities to estimate the efficiency of the developed heuristic algorithm. Its source information is condensed in Table 1. The two emphasized activities consume A-resources; the remaining 18 activities consume the B-resources of two different types. Therefore, n = 20, n A = 2, n B = 18, m = 2. The boundary resource values are R 1min = 30, R 1max = 80, R 2min = 27, R 2max = 80, the schedule date D = 550, the confidence probability p** = 0:9, and the fine and project storage estimates on the whole are C* = 1000, C** = 200, C*** = 150. Values of r ijk are represented in Table 1 except for activities (4,6) and (7,10). The penalty provisions for these activities for idling of the A-resources and the cost of lease and maintenance of the B-resources are condensed in Table 2. We note that the parameters of Table 2 were varied in the course of experiment. Sixteen different combinations were included in the experiment (see Table 2). Consideration was given to the uniform distribution over the interval [aij; b ij ] and the normal distribution with the parameters u ij = 0:5(a ij + b ij ) and variance V ij = (b ij -a ij ) 2 /36 as the distributions of t ij . In the problem P1, therefore, the coordinate wise cyclic search of the extremum was carried out in five coordinates [6].
The following output parameters were established experimentally for each of the 16 combinations: C , the minimal mean volume of no operational project costs; p , statistical frequency of the probability of timely project execution; S, the instant of project start; R 1 , the volume of leased B-resources of the first kind; R 2 , the volume of leased B-resources of the second kind; T(4; 6), the scheduled delivery time of the A-resources for the activity (4, 6); T(7; 10), the scheduled delivery time of the Aresources for the activity (7, 10).
The generalized experimental data are condensed in Table 3. We note that four iterations were required to obtain the final estimates. In doing so, the results of the fourth and third iterations are virtually identical, and in the course of solving the problem P1 the target function decreased more than two-fold already after the first iteration. In the course of four iterations, the target function decreased by a factor of seven to eight [8,9].

Conclusion
The developed model can be used for planning and supervising one or more Random PERT-like network projects. The model enables one to control several kinds of resources including rare and expensive ones transported from outside for a short time.
For any combination, the mean experimental estimate of the probability of timely project execution exceeds the predefined limit probability that was included as a parameter of the initial problem. Therefore, the constructed model performs probability supervision according to constraint (2).
The normal distribution results in an appreciable (more than by 20%) reduction in the no operational costs as compared with the uniform one. Therefore, the normal distribution makes project realization less costly. Therefore, development of more generalized resource models for network projects seems quite topical both from the scientific and applied standpoints.

Acknowledgment
Thanks to the Architecture Engineering Research Institute of General Logistics Department that give us a lot of help and has prepared and established three books within more than one year. It has taken efforts to bring it to the level of quality as it is today. In this respect, our greatest thank goes not only to all who have contributed and still participate but additionally to all students who give their confidence in our aim to transfer knowledge on a high level.