Times. Chunxiao Ding, Xingfang Zhang. School of Mathematical Sciences, Liaocheng University, Liaocheng , China

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1 Project Scheduling Problem with Uncertain Activity Duration Times Chunxiao Ding, Xingfang Zhang School of Mathematical Sciences, Liaocheng University, Liaocheng , China Abstract Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. This paper considers project scheduling problem with uncertain activity duration times and presents four new uncertain programming models for different management requirements. Moreover, taking advantage of some properties of uncertainty theory, these models can be transformed to their deterministic forms respectively. Finally, a numerical example is illustrated, and its solution is found by using hybrid intelligent algorithm. Keywords: Project scheduling; Uncertain programming; Genetic algorithm; Uncertain measure 1 Introduction Uncertainty theory, founded by Liu [6] in 2007 and refined by Liu [9] in 2010, has been developed to a fairly complete mathematical system based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Uncertain programming, a type of mathematical programming involving uncertain variables, was first proposed by Liu [7] as a branch of uncertainty theory. Through years of exploration, uncertain programming has been applied to system reliability design, facility location problem, vehicle routing problem, project scheduling problem and so on. This paper deals with the project scheduling problem primarily. Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. Due to the vagueness of project activity duration times, many researchers have studied project scheduling problem with uncertainty. Freeman [1, 2] firstly introduced probability theory into project scheduling problem in Then Ke and Liu [4] built three stochastic models and established a hybrid intelligent algorithm to solve project scheduling problem with stochastic activity duration times. In 1979, Prade [10] applied fuzzy set theory into the project scheduling problem. Then Ke and Liu [3, 5] presented three types of fuzzy models for project scheduling problem and its 1

2 hybrid intelligent algorithm. This paper considers project scheduling problem with uncertain activity duration times. what is more, four uncertain programming models for project scheduling problem are presented and a numerical example is given. This paper consists of 5 sections, and its frame is organized as follows. In Section 2, some basic concepts and properties on uncertainty theory used throughout this paper are introduced, then the 99 method and hybrid intelligent algorithm are discussed. In Section 3, four uncertain models for project scheduling problem are presented. An example is given in Section 4. At last a brief summary is presented. 2 Preliminary In this section, we introduce some foundational concepts and properties of uncertainty theory, which will be used throughout this paper. Definition 1. (Liu [6]) Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ L is called an event. The set function M is called an uncertain measure if it satisfies the following four axioms: Axiom 1. (Normality) M{Γ} = 1; Axiom 2. (Monotonicity) M{Λ 1 } M{Λ 2 } whenever Λ 1 Λ 2 ; Axiom 3. (Self-Duality) M{Λ} + M{Λ c } = 1 for any event Λ; Axiom 4. (Countable Subadditivity) For every countable sequence of events {Λ i }, we have { } M Λ i M{Λ i }. i=1 Liu [8] presented the product measure axiom of uncertainty theory in 2009 as follows: i=1 Axiom 5. (Product Measure Axiom) Let Γ k be nonempty sets on which M k are uncertain measures, k = 1, 2,..., n, respectively. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L 1 L 2... L n satisfying That is, for each event Λ L, we have M { n i=1 Λ k } = min 1 k n M k{λ k } M{Λ} = sup Λ 1 Λ 2 Λ n Λ if 1 sup Λ 1 Λ 2 Λ n Λ c if min M k{λ k }, 1 k n sup Λ 1 Λ 2 Λ n Λ 0.5, otherwise. min M k{λ k } > k n min M k{λ k }, 1 k n sup Λ 1 Λ 2 Λ n Λ c min M k{λ k } > k n 2

3 Definition 2. (Liu [6]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} = {γ Γ ξ(γ) B} is an event. Definition 3. (Liu [6]) The uncertainty distribution Φ : R [0, 1] of an uncertain variable ξ is defined by Φ(x) = M{γ Γ ξ(γ) x}. Definition 4. (Liu [6]) An uncertain variable ξ is called normal if it has a normal uncertainty distribution Φ(x) = ( ( )) 1 π(e x) 1 + exp, x R 3σ denoted by N (e, σ) where e and σ are real numbers with σ > 0. Definition 5. (Liu [6]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = + provided that at least one of the two integrals is finite. 0 0 M{ξ r}dr M{ξ r}dr Definition 6. (Liu [6]) The uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if { n } M {ξ i B i } = min M{ξ i B i } 1 i n i=1 for any Borel sets B 1, B 2,, B n of real numbers. Theorem 1. (Liu [9]) Let ξ and η be independent uncertain variables with finite expected values. Then for any real numbers a and b, we have E[aξ + bη] = ae[ξ] + be[η]. 99-Method and Hybrid Intelligent Algorithm 99-Method 1.2.(Liu [9]) The 99 method and genetic algorithm are powerful tools in numerical solution of the models for the project scheduling problems. In this section, we will use more words to introduce them. It is suggested that an uncertain variable ξ with uncertainty distribution Φ is represented on a computer by x 1 x 2 x 3 x 99 3

4 where 0.01, 0.02, 0.03,, 0.99 in the first row are the values of uncertainty distribution Φ, and x 1, x 2,, x 99 in the second row are the corresponding values of Φ 1 (0.01), Φ 1 (0.02), Φ 1 (0.03),, Φ 1 (0.99). The 99 method may be extended to the 999 method if a more precise result is needed. 99-Method 1.2.(Liu [9]) Assume ξ 1, ξ 2,, ξ n are uncertain variables, and each ξ i is represented on a computer by x i 1 x i 2 x i 3 x i 99 If the function f(x 1, x 2,, x n ) is strictly increasing with respect to x 1, x 2,, x m, and strictly decreasing with x m+1, x m+2,, x n, then the uncertain variable f(ξ 1, ξ 2,, ξ n ) is f(x 1 1,, x m 1, x m+1 99,, x n 99) f(x 1 99,, x m 99, x m+1 1,, x n 1 ) With the help of 99 method, some models can be approximately transformed to integer programming, which is a deterministic problem. In this paper we use genetic algorithm to find the numerical solution of the model. Hybrid Intelligent Algorithm (HIA).(Liu [9]) HIA is a numerical method for solving deterministic programming. From the mathematical viewpoint, there is no different between deterministic mathematical programming and uncertain programming.then,we may integrate the 99-method and the genetic algorithm to produce a hybrid intelligent algorithm for solving uncertain programming models. The HIA can be summarized as follows: Step 1: Initialize chromosomes whose feasibility may be checked by the 99- method.; Step 2: Update the chromosomes by the crossover operation in which the 99-method may be employed to check the feasibility of offsprings; Step 3: Update the chromosomes by the mutation operation in which the 99-method may be employed to check the feasibility of offsprings; Step 4: Calculate the objective values for all chromosomes by the 99- method; Step 5: Compute the fitness of each chromosome based on the objective values; Step 6: Select the chromosomes by spinning the roulette wheel; Step 7: Repeat the second to sixth steps a given number of cycles; Step 8: Report the best chromosome as the optimal solution. 4

5 3 The models for project scheduling problem with uncertain duration time Project scheduling is usually represented by a directed acyclic graph where nodes correspond to milestones, and arcs to activities which are basically characterized by the times and costs consumed. Let (ϑ, A) be a directed acyclic graph, where ϑ = {1, 2,, n, n + 1} is the set of nodes, A is the set of arcs, (i.j) A is the arc of the graph (ϑ, A) from nodes i to j. We can rearrange the indexes of the nodes in ϑ such that i < j for all (i, j) A. In order to model the project scheduling problem, we introduce the following indices and parameters: ξ ij : uncertain duration time of activity (i, j) in A; Φ ij : uncertainty distribution of ξ ij ; c ij : cost of activity (i, j) in A; r: interest rate; x i : integer decision variable representing the allocating time of all loans needed for all activities (i, j) in A. 3.1 Starting Times For simplicity, we write ξ = {ξ ij : (i, j) A} and x = (x 1, x 1,, x n ). Assume each uncertain duration time ξ ij is represented by a 99-table, t 1 ij t 2 ij t 3 ij t 99 ij (1) Let T i (x, ξ) denote the starting time of all activities (i, j) in A. According to the assumptions, the starting time of the total project should be whose inverse uncertainty distribution may be written as and has a 99-table, T 1 (x, ξ) = x 1 (2) Ψ 1 1 (α) = x 1 x 1 x 1 x 1 x 1 (3) From the starting time T 1 (x, ξ), we deduce that the starting time of activity (2, 5)is T 2 (x, ξ) = x 2 (x 1 + ξ 12 ) 5

6 Whose inverse uncertainty distribution may be written as and has a 99-table, Ψ 1 2 (α) = x 2 (x 1 + Φ 1 12 (α)) (4) x 2 (x 1 + t 1 12) x 2 (x 1 + t 2 12) x 2 (x 1 + t 99 12) x 1 Suppose that the starting time T k (x, ξ) of all activities (k, i) in A has an inverse uncertainty distribution Ψ 1 k (α) and has a 99-table, y 1 k y 2 k y 3 k y 99 k (5) Then the starting time T i (x, ξ) of all activities (i, j) in A should be whose inverse uncertainty distribution is T i (x, ξ) = x i max (k,i) A (T k(x, ξ) + ξ ki ) (6) and has a 99-table, Ψ 1 i (α) = x i max (k,i) A (Ψ 1 k (α) + Φ 1(α) (7) ki x i max (k,i) A (y1 k + t1 ki ) x i max (k,i) A (y99 k + t99 ki ) (8) where y 1 k, y2 k,, y99 k activities. are determined by (3). This recursive process may produce all starting times of 3.2 Completion Time The completion time T (x, ξ) of the total project is T (x, ξ) = whose inverse uncertainty distribution is max (T k(x, ξ) + ξ k,n+1 ) (9) (k,n+1) A and has a 99-table, Ψ 1 (α) = max (k,n+1) A (Ψ 1 k (α) + Φ 1 k,n+1 (α) (10) where y 1 k, y2 k,, y99 k max (k,n+1) A (y1 k + t1 k,n+1 ) max (k,n+1) A (y99 k + t99 k,n+1 ) (11) are determined by (3). 6

7 3.3 Total Cost Based on the completion time T (x, ξ),the total cost of the project can be written as C(x, ξ) = c ij (1 + r) T (x,ξ) xi (12) (i,j) A where a represents the minimal integer greater than or equal to a. It is well known that C(x, ξ) is a discrete uncertain variable whose inverse uncertainty distribution is Υ 1 (x; α) = c ij (1 + r) Ψ 1 (x;α) x i (i,j) A (13) for 0 < α < 1. Since T (x, ξ) is obtained by the recursive process and represented by a 99-table, s 1 s 2 s 3 s 99 (14) the total cost C(x, ξ) has a 99-table, c ij (1 + r) s 1 x i c ij (1 + r) s 99 x i (i,j) A (i,j) A (15) 3.4 models model(liu [6]) min E[C(x, ξ)] subject to M{T (x, ξ) T 0 } α x 0, integer vector. where T 0 is a due date of the project, α is a predetermined confidence level, T (x, ξ) is the completion time defined by (4), and C(x, ξ) is the total cost defined by (7). The purpose of this model is minimize the expected cost of the project under the completion time constraint. Based on the model of project scheduling problem proposed by Liu [6], some other models are given as follows: 7

8 model 1 If we want to minimize the expected time of the project under the uncertain measure of the cost is equal or greater than the given confidence level α, we may construct the following project scheduling model: min E[T (x, ξ)] subject to M{C(x, ξ) C 0 } α x 0, integer vector. model 2 If we want to maximize the uncertain measure of the cost of the project under the expected time constraint, we may construct the following project scheduling model: max M{C(x, ξ) C 0 } subject to E[T (x, ξ)] T 0 x 0, integer vector. model 3 If we want to maximize the uncertain measure of the time of the project under the uncertain measure of the cost is equal or greater than the given confidence level α, we may construct the following project scheduling model, max M{T (x, ξ) T 0 } subject to M{C(x, ξ) C 0 } α x 0, integer vector. model 4 If we want to maximize the uncertain measure of the time of the project under expected cost constraint,we may construct the following project scheduling model, max M{T (x, ξ) T 0 } subject to E[C(x, ξ)] C 0 x 0, integer vector. 8

9 4 Example An example of model 1 is given. Consider a project with 4 milestones and 4 activities, from milestone 1 to 4, there are two routes: milestone 2 and milestone 3. Assume that all duration times of activities are normal uncertain variables, ξ ij N (e ij, σ ij ), (i, j) A in which σ ij = 1, e ij = j + 2, and assume that the costs of activities are c ij = (i + j) 10. In addition, we also suppose that the interest rate is r = 0.03, the due cost is C 0 = 280, and the confidence level is α = In order to find an optimal project schedule, we take advantage of a hybrid intelligent algorithm integrated by the 99-method and genetic algorithm. A run of computer program shows that the optimal allocating time of all loads needed for all activities are Dete Node (16) Loan we can get the expected time is 10.27, and M{C(x, ξ) 280} = Conclusions In this paper, four uncertain programming models for project scheduling problem are introduced. Under uncertainty theory, these models can be transformed to their deterministic forms, and then we can use hybrid intelligent algorithm to find their solutions. At last, a typical example is given. References [1] R.J.Freeman, A generalized PERT, Operations Research, Vol.8, No.2, , [2] R.J.Freeman, A generalized network approach to project activity sequencing, IRE Transactions on Engineering Management, Vol.7, No.3, , [3] H.Ke and B.Liu, Fuzzy project scheduling problem and its hybrid intelligent algorithm, Technical Paper, [4] H.Ke and B.Liu, Project scheduling problem with stochastic activity duration times, Applied Mathematics and Computation, Vol.168, No.1, , [5] H.Ke and B.Liu, Fuzzy project scheduling problem and its hybrid intelligent algorithm, Applied Mathematical Modelling, Vol.34, , [6] B.Liu, Uncertainty theory, 2nd ed.,springer-verlag, Berlin, [7] B.Liu, Theory and practice of uncertain programming, 2nd ed., Springer-Verlag, Berlin,

10 [8] B.Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No.1, 3-10, [9] B.Liu, Uncertainty theory: A branch of mathematics for modeling human uncertainty, Springer-Verlag, Berlin, [10] H.Prade, Using fuzzy set theory in a scheduling problem: a case study, Fuzzy Sets and Systems Vol.2, No.2, ,