1
1*
1 Dongling School of Economics and Management, University of Science and Technology Beijing, Beijing, China
* Corresponding author
(hxiaoxia@manage.ustb.edu.cn)
Abstract - This study discusses the optimizing project selection and scheduling problems. In real life, cash inflows and cash outflows of each project are uncertain, we regard them as stochastic variables consequently. Considering time value of capital, Net present value is used as the standard to measure the projects and introduce chance-constraints to control the uncertainty and formulate the model. According to the logical relationship and the characters of projects, we introduce implicit enumeration algorithm to select appropriate projects and schedule them in a reasonable order. Finally, a numerical example is given to express the thought of the model.
Keywords - Chance-constrained programming, project selection, implicit enumeration algorithm
I. INTRODUCTION
Traditional project selection study is to select projects in order to maximize the whole profits, while in recent years scholars gradually recognize the importance of scheduling problem. Different projects have their own characters: different construction durations, lifetime durations, returns and the like. Considering time value of the capital, these characters may influence the selected results, for this reason appear especially important. S. Liu,
C. Wang [1] and J. Chen, R.G. Askin [2] systematically study the project selection and scheduling, while the returns of projects are definite value. X. Huang introduces chance-constrained programming to handle uncertainty in
[3] and employs net present value to measure capital in [4] .
A.F. Carazo, T. Gómez, J. Molina, A. G. Hernández-Díaz,
F. M. Guerreroa, R. Caballerob [5] propose a multiobjective binary programming model for portfolio selection. J.A. Sefair, A.L. Medalia [6] considers the covariance among different projects, and K. Ahsan, I.
Gunawan [7] studies the cost and schedule problems of international development projects. R.M. Henry, G.E.
McCray, R.L. Purvis, T.L. Roberts further assesses the subsequent impact of predictability on project success in
[8] . In solving period, P. Brucker, S. Knust, A. Schoo, O.
Thiele [9] introduces a branch and bound algorithm and R.
Dzeng, H. Lee employs genetic algorithm (GA) in [10] .
This paper will follow the thought of NPV, meanwhile, introduce chance-constraints to project selection and scheduling problem. We regard NPV as the index to evaluate the earnings-generating capacity. The rest of the paper is organized as follows. Section II formulates the mathematical model of chance-constrained
NPV programming model. The algorithm and a numerical example are provided in Section III. Section IV gives the concluding remarks.
II. MATHEMATICAL MODEL FORMULATION
We will select projects and schedule them in a reasonable order from n projects. Let x i
denote decision variables for project selection, which is zero-one variable: x i



1 if project is selected,
0 otherwise,

(1) where i

1, 2, , , respectively.
NPV is a vital index of dynamic evaluation for project investment and is to evaluate the earningsgenerating capacity during the whole counting period.
NPV is defined as the sum of net cash flow discounted to zero time reference using preset discount rate. For each project, there are three periods, before-construction period, construction period and lifetime. Assume that s i the construction start time of project i ,
τ i as
as the construction period duration of project i , and T i
as the lifetime duration of project i . Let IC it
denote the initial cost at time t in the construction period of project i , CI it and CO it
as the cash inflow and outflow at time t in the lifetime of project i , respectively. As a result, the NPV of project i at zero time reference can be formed as:
NPV i
 s i


 i
T i
CI it

CO it

1
 r
 t where i

1, 2, , , respectively.
 s

1 i 
 i
IC it

1
 r
 q
, (2)
Cash flow consists of initial cost, cash inflow and cash outflow, which are all uncertain. In order to come close reality, we assume these three are stochastic variables. The ultimate purpose of modeling is to maximize NPV value of selected projects. Because of the stochastic variables, the goal cannot be a crisp number, we can set the goal as maximizing NPV at a preset confidence level. The constraints can be changed to the requirement that credibility of investment not exceeding budgeting should be equal or greater than a preset level.
The object function and constraints of chance-constrained programming model are listed beneath.

max f .
(3)
Pr i n 

1 x NPV i
 f

  ， for i

1, 2, n .
(4)
Pr i n 

1 i

0
  ， for i

1, 2, n , (5) where (3) is the object function and (4) and (5) are chance-constraints. Inequation (4) is to make sure the sum
NPV of selected projects greater and better, then the investment would acquire more profits. Inequation (5) is to make sure the sum NPV of selected projects no less than zero, because if the NPV is below zero, returns cannot achieve the expected standard and the investment is failure.
The model also presents the logical relation among the alternative projects. There are three different kinds of relationship among the projects: interdependent, exclusive and mutual independence. If the two projects are interdependent, then one of the two is selected, the other one must be selected, and vice versa, one of the two is not selected, the other one should not be selected. One of the two projects is selected, the other one should not be selected if the two projects are exclusive. If the two projects are mutual independence, then whether to choose one project, has nothing to do with the other project. The mathematical form of logical relationship above is shown as follows: x i
 x j
, for ,

1, 2, n . and i
 j .
(6) x i
 x k
, for ,

1, 2, n . and i
 k .
(7)
In (6), project i and project j are interdependent, so they should be selected or not to be selected at the same time. In (7), project i and project k are exclusive, so their results of to be selected should be opposite.
We have already denoted s i
the construction start time of project i and τ i
the construction period duration of project i . Among the alternative projects there is one special relation which is succession relation. When one selected project is the successor of another project, its construction start time should not be earlier than complement time of the project which is succeeded. Let project l be the successor of project i , Equation (8) below shows their relation. s l s i
 i
, for ,

1, 2, n , and i
 l .
(8)
Assume that f i
as the construction finish time of project i . Each selected project should be finished before their own milestone time and also before the deadline of constructive period. s i
 i f i
, for i

1, 2, n .
(9) f i

M i
, for i

1, 2, n .
(10) f i

DL , for i

1, 2, n , (11) where M i is the milestone of project i and DL is deadline of constructive period.
The capital can be invested to accomplish the projects is limited. Let W t
denote the capital can be used to construct at time t , we assume that there is no cash inflow in the construction period, so the net cash flow derives from initial cost. Thus, the sum of the initial cost of all the projects on construction should be no greater than W t
. We formulate the constraint in chance-constrained form, then
(12) is acquired.
Pr i n 

1 x IC i it

W t
 
.
(12) where i

1, 2, n , and t

0,1, 2, , DL

1 .
From the above, the chance-constrained programming model for optimizing project selection and scheduling problem is obtained.




















 max f subject to
Pr
Pr x i x i
 x k s l
  f i

M i f i

DL
Pr

:
 i n 

1 i n 

1 x
 i n 

1 j x NPV i
 f

 
 i i

0 x IC it

W t 

 

,
(13) where , , ,

1, 2, n , and i

,

,
 l .
For the last constraint, t should be bounded as t

0,1, 2, , DL

1 .
III. ALGORITHM AND A NUMERICAL EXAMPLE
In this section, we introduce one kind of implicit enumeration algorithm. We can divide the constraints into three categories. One is constraint of logical relationship, one is constraint of time, another one is constraint of capital budget. According to constraint categories, we can acquire the procedure of the algorithm.

Step 1. Screen out the probable project groups on the basis of logical relationship of projects,
Step 2. Calculate the sum value of NPV of each probable project group on the condition of budget and time constraints.
Step 3. Compare the value of
Project
Duration
τ i
T i
A 3 b
8
B 2 7
C 2 8
D 3 6
E 4 6
F 3 5
G 3 7
0
0
0
0
0
0 f , select the maximal one, and the group is the final solution.
By means of calculation above, we can acquire the maximal NPV at a preset level, and homologous group to be constructed. To further illustrate the thought and algorithm of the model, a numerical example is presented below.
In the numerical example, we assume that the investor’s object is to maximize the total NPV with a predetermined confidence level 0.95, and to make sure that the probability of the total NPV less than zero is less than a predetermined confidence level 0.05. The logical relation among the alternative projects and the milestone time of each project are given in Table I. Also there exists the deadline of the construction period. W t
is assumed as
$80,000 and it is required the total cash outflow at time t should be less than W t
with a predetermined confidence level 0.95. The construction period and lifetime of project i with cash flow for each period are shown in Table II
( r =5.25%).
Project Successor
Inter-
T ABLE I
L OGICAL R ELATION AND T IME R ESTRICT dependency
Exclusive
Milestone
Time
A B G,D -
B G - E
C D,E - -
6 a
D - A F
E F - B
Deadline
12
F G
G -
-
A a time unit: year
D
-
8
N(8,2)
N(3,1)
N(5,1.5)
N(4,2)
N(6,1.5)
N(5,1) b time unit: year; c capital unit: $10,000
In Table II, N( a , b ) represents the variable follows a normal distribution, with a mean of a and variance of b . In the first two years of lifetime, the projects operate well, so the cash inflow and cash outflow every year are accordingly high. In the next two years, competitors appear, the returns decrease under influence. In the rest years of lifetime, the production and operation mode are stable, the cash flows would increase than the two years before.
We solve the example applying the implicit enumeration algorithm, the optimal solution is in Table
III.
A 0 d
D 4
T
Selected Project Start Time
ABLE III
SELECTED
P
ROJECT AND
S
TART
T
IME f
49.9459 e
G 8 d time unit: year; e capital unit: $10,000
From Table III, we can acquire the optimal solution is to select project A, D, and G to construct, and the start construction time of project A is zero, project D is the fourth year, project G is the eighth year. The sum value of
NPV is no less than $499,459 at 95% level.
IV. CONCLUSIONS
We proposes a model for optimizing project selection and scheduling problem in this paper. Chance-constrained programming is provided to manage the uncertainty of cash flows. In the solving period, an implicit enumeration algorithm is employed and the result of the numerical example shows the idea and application of the model.
While the iterations would add following the increase of project scale, so how to develop the algorithm is vital in further study, probably improve existing algorithm or try new algorithm.
T ABLE II
C ONSTRUCTION P ERIOD, L IFETIME AND C ASH FLOW t < s i s i
 t

s i+
τ i s i+
τ i
 t
 s i+
τ i
+ 2
Cash Flow IC it
CI it
CO it
0 c
N(4,1) N(8,1) N(2,0.1) s i+
τ i
+ 2
CI it

t
 s i+
τ i
+ 4
N(7,1.1)
CO it
N(2,0.5)
N(12,1) N(1,0.2)
N(11,1.5) N(3,1)
N(10,1.8) N(1,0.3)
N(8,1.2) N(3,0.8)
N(8,1) N(2,1)
N(10,1.1) N(2,0.2)
N(10,2)
N(9,1)
N(8,1)
N(5,1.1)
N(7,2)
N(6,1)
N(1,0.2)
N(2,0.5)
N(1,0.1)
N(2,0.6)
N(2,0.7)
N(1,0.1) s i+
τ i
+ 4
CI it
N(10,2)
N(14,3)
N(10,1)
N(9,2)
N(7,3)
N(8,3)
 t

s i+
τ i
+T i
N(9,1.5)
CO it
N(3,1)
N(2,0.9)
N(3,1)
N(1,0.1)
N(2,0.2)
N(2,0.3)
N(1,0.2)

ACKNOWLEDGMENT
This work was supported by National Natural Science
Foundation of China Grants Nos. 70871011 and
71171018, Program for New Century Excellent Talents in
University, and the Fundamental Research Funds for the
Central Universities.
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