Project Scheduling Problem with Uncertain Variables

Project scheduling problem is mainly to determine the schedule of allocating resources in order to balance the total cost and the completion time. This paper chiefly uses chance theory to introduce project scheduling problem with uncertain variables. First, two types of single-objective programming models with uncertain variables as uncertain chance-constrained model and uncertain maximization chance-constrained model are established to meet different management requirements, then they are extended to multi-objective programming model with uncertain variables.


Introduction
Project scheduling problem is mainly to determine the schedule of allocating resources in order to balance the total cost and the completion time.A typical project scheduling problem can be described as follows: there are many activities in a project.There are tight-front relations among some projects because of the technical request.Activity can't be processed before its all tight-front works are finished.The structure of entire project can be described by a directed acyclic network graph.The pitch point represents transformation from an activity to another activity in the graph, and the arc represents tight-front relations among activities.A feasible plan can be defined as follows: the schedule of each activity has been determined; also each activity satisfies tight-front relation and resource restraint.
Researchers have studied project scheduling problem in certain or uncertain environments since 1960s.Kelley L. Lin et al.
[1] [2] initially presented function relationship between project cost and activity duration times, and established a mathematical model of deterministic project scheduling problem with objective of minimizing the total cost.In 1960, Freeman [3] [4] introduced probability theory into project scheduling problem.Charnes et al. [5] studied stochastic project scheduling problem via chance-constrained programming.Golenko-Ginzburg and Gonik [6] set up an expected cost minimization model of project scheduling problem under some deterministic resource constraint.Finally, Ke and Liu [7] built three stochastic models as expected cost model, α-cost model and probability maximization model via hybrid intelligent algorithm to relatively comprehensively solve stochastic project scheduling problem.But the uncertainty is assumed as randomness in the above work.Nevertheless, in real world, much uncertainty may not be replaced by randomness.For instance, fuzzy set theory, which was introduced by Zadeh [8], describes another uncertainty.Prade [9] first applied fuzzy set theory into the project scheduling problem in 1979.In 2004, Ke and Liu [10] built three fuzzy models via hybrid intelligent algorithm, and they quickly applied fuzzy set theory into the project scheduling problem successfully.Furthermore, in 2007, Ke and Liu [11] presented random fuzzy models, for example, the duration time of each activity is stochastic and stochastic parameters are fuzzy variables.
This paper chiefly introduces project scheduling problem with uncertain variables via chance theory.First, two types of single-objective programming models with uncertain variables as uncertain chance-constrained model and uncertain maximization chance-constrained model are established to meet different management requirements, and then they are extended to multi-objective programming model with uncertain variables.

Uncertain Variable
In many cases, randomness and fuzziness simultaneously appear in uncertain phenomena.In 1978, the concept of fuzzy random variable was introduced by Kwakernaak [12] [13] in order to describe these phenomena.Afterwards the concept of fuzzy random variable was developed by several researchers such as Puri and Ralescu [14], Kruse and Meyer [15], and Liu and Liu [16] according to different requirements of measure.Furthermore, Liu [17] first proposed the concept of random fuzzy variable.More generally, Liu [18] puts forward the concept of uncertain variables.As follows: Definition 1 (Liu [18]) an uncertain variable is a measurable function ξ from an uncertainty space ( ) , , L M Γ to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event.
It is very clear that uncertain variable is very different from random variable (Kolmogorov [19]) and fuzzy variable (Zadeh [8]).Roughly speaking, a random variable is a function from a probability space to the set of real numbers, and a fuzzy set is a function from a possibility space to the set of real numbers.

Chance Measure
In many cases, uncertainty and randomness simultaneously appear in a complex system.In order to describe this phenomenon, the concept of chance measure was proposed by Liu [20] in 2013: Definition 2 (Liu [20]) Let ξ be an uncertain random variable, and let B be a Borel set of real numbers.
Then the chance measure of uncertain random event Definition 3 (Liu [11]) Let ξ be a random fuzzy variable on the possibility space , and let B be a Borel set of real numbers.Then the chance of random fuzzy event

Problem Assumptions
In a real world application, a large engineering project is a big complex system.In order to establish the corresponding mathematical models, we must give some simplifications and assumptions to meet different management requirements.Assumption 1): All of the costs are obtained by loans with some given interest rate; Assumption 2): Each activity can start only if the loan needed is allocated and all the foregoing activities are finished; Assumption 3): Duration time of each activity is assumed to a continuous and an uncertain variable; Assumption 4): Each man-power needed for each activity is an uncertain variable; a part of duration time of each activity is inversely proportional to the number of workers; Assumption 5): The cost needed for each activity is only considered to workers' wages and loans with some given interest rate.

Models Establishment
Generally speaking, a project scheduling problem can be described by a directed acyclic graph like Figure 1.
Let ( ) is the set of nodes standing for the milestones and is the set of arcs representing the activities of the project.
Let us first introduce the following mathematical signs and symbols: , where ij ξ are uncertain variables and ij ξ are duration times of activities represented by ( ) , where ij η are uncertain variables and ij η are numbers of workers for activities represented by ( ) , , , n x x x x =  , where x is a decision vector and i x represents the allocating time of all the loans needed for activities represented by ( ) , , , n y y y y =  , where y is a decision vector and i y represents the allocating number of workers needed for activities represented by ( ) , , , i T x y ξ η : the starting times of activity ( ) ( ) M t : the number of workers needed for the project at the time point denoted as t ;

( ) ( )
, , 1 Obviously, ( ) According to the assumption (5), the costs needed for activity ( ) According to the assumption (2), ( ) , , , , , , max , , , , , , , , Then completion time of the total project can be calculated by The total cost of the project can be calculated by The maximal numbers of workers needed for the project can be denoted as , , , As these basic formulas have been given in the above section, we can establish different hybrid programming models to meet different management requirements.

C C x y C T x y T M x y M
where 0 C is the total cost of the project, 0 T is the total times of the project and 0 M is the total numbers of workers of the total project.

C x y C T x y T M x y M
x y where 0 C is the total cost of the project, 0 T is the total times of the project and 0 M is the total numbers of workers of the total project.

Multi-Objective Programming Models with Uncertain Variables
If we consider both minimal cost and minimal numbers of workers of the total project, we can build the corresponding multi-objective programming models with uncertain variable.
Model 1: , and are predetermined confidence levels.
where 0 C is the total cost of the project, 0 T is the total times of the project and 0 M is the total numbers of workers of the total project.
, where 0 C is the total cost of the project, 0 T is the total times of the project and 0 M is the total numbers of workers of the total project.

Conclusion
Considering the process of practical application, a large engineering project is a large complex system.This article embarks from the actual.On the basis of the time constraints, allocating the number of workers is also an important factor which cannot be ignored.That is also the innovation of this paper.We set up corresponding mathematical models, so as to adapt to different management needs.In the further study, we can put the above models to the practical problems to meet the needs of reality.
η : the duration time of activity ( ) , i j in A ; η : the cost needed for activity ( ) , i j in A ; ij c : the fixed cost needed for activity ( ) , i j in A ; r : the interest rate;