Abstract-- A successful project is subjected to many different factors, which include timing and cash flow management as two determinative ones. Scheduling the projects is one of the most important fields of Operation Research (OR). Different objectives are considered in project scheduling and financial goals are the most challenging ones. Lack of enough liquidity is one of the problems that contractors encounter during the project, which causes a delay in the execution of project activities. To avoid these delays, project financing is essential. In most projects, a credit line (CL) is considered a source of cash. The contractors should remain below the credit limit, which is imposed by the lender bank. This limit may cause an extension in project duration because some activities cost too much, and the available cash cannot satisfy the expenses. In this paper, a new model for finance-based scheduling is proposed, in which, in addition to CL, other resources like long-term loans and short-term loans are taken to consideration as financial resources. The objective of this study is to minimize the project duration and financing costs. The results of this work show that financing costs and the extension of project duration can be decreased simultaneously by using different financing alternatives.
Keywords-- Project scheduling, Finance-based scheduling problem, Financing alternatives, Multi-objective programming.
I. INTRODUCTION AND LITERATURE REVIEW
In this paper, the Finance-Based Scheduling Problem (FBSP) is considered. FBSP is about scheduling the project activities subject to a Credit Limit (CL) (Elazouni and Gab-Allah, 2004), which is a kind of Resource-Constrained Project Scheduling (RCPSP) (Ozdamar and Ulusoy, 1995) and it is a useful tool to control required credit (Elazouni and Metwally, 2005). In general, project scheduling is sequencing the project activities so that some precedence relations and a set of resource constraints are fulfilled (Davis, 1973). A realistic schedule helps to minimize project failure (Elazouni and Gab-Allah, 2004). Financial factors are some of the most important reasons for a project’s failure (Arditi et al., 2000), and about 60% of failures are because of financial problems (Russell, 1991). Accordingly, project success is mainly dependent on cash flow management (Russell, 1970), which is not accessible due to long project duration and many other factors like cost estimation or cash retainage by the owners (Park et al., 2005). The contractors invoice the expenses of executed activities at the end of each reimbursement period, but the sponsors do not pay it completely. They withhold a percentage of it, as Retainage Percent (RP) paid at the end of the project (Hendrickson and Au, 2008). It leads to cash shortage at the end of each reimbursement period (Lu et al., 2016). Since the execution of project activities needs a considerable investment, the contractors’ savings cannot be reliable (Elazouni and Metwally, 2005). To pay the project expenditures, the contractors have a bank account. They can withdraw needed cash from it, but it has to be deposited back to remain under the Credit Limit (CL) imposed by the bank (Ahuja, 1976). The credit limit restricts the maximum negative cumulative balance during the project (Elazouni, 2009). Lack of enough liquidity can also happen when the payment by the sponsors is delayed (Lu et al., 2016). It makes the contractors delay the execution of some activities, and as a result, they cannot finish the project on schedule (Singh and Lokanathan, 1992); therefore, it is essential for contractors to have enough financial resources with minimum financing cost to execute the project activities on time (Elazouni and Gab-Allah, 2004). High financing costs can affect the project profitability and reduce it (Lu et al., 2016); consequently, cost control should be executed effectively by financing costs consideration (Turner, 2009).
Since the timely execution of the project activities is of particular importance to contractors and project owners and its delay imposes additional financing costs and overhead costs on the project, the issue considered in this paper is the impact of using another financial resource in addition to the credit line, on the delay imposed on project activities due to financial constraints. The financial resources used alongside with credit line are long-term loans and short-term loans, which in separate scenarios are combined with a credit line to create a new model.
According to the importance of cost management, which includes financing costs, this study also considers minimizing financing costs, and the single-objective model in classic FBSP is transformed to a multi-objective model. The particular importance of financing costs for project owners and contractors is since its reduction increases the profitability of the project.
Finance based scheduling problem was introduced for the first time in 2004 by Elazouni and Gab-Allah (2004) and developed by various researchers. This scheduling method is one of the branches of the Resource-Constrained Project Scheduling Problem in which financial resources are the primary constraint.
They used an integer-programming (IP) method for FBSP to balance the expenditures of activities and available financial resources in each period. The main goal of this paper is to minimize the total duration of the project considering financial constraints.
Elazouni and Metwally (2005) used a Genetic Algorithm (GA) method for finance-based scheduling problems to minimize the project duration and financing cost, which is related to project profitability. The financial resource considered in this paper is the credit line.
Elazouni and Metwally (2007) developed the FBSP by considering both cash shortage and cash surplus conditions by using time-cost trade-off (TCT) to balance increased direct cost and decreased overhead. A GA method is used for finance-based scheduling problems, and the results are compared to the results of the IP method. The main goal in this study is to minimize direct and overhead costs, as well as financing costs to maximize the profitability of the project under credit limit and resource constraints.
A heuristic method was presented by Elazouni (2009) to schedule multiple projects under cash constraints. In this method, the available liquidity in each period is determined, and all possible schedules for project activities are found. Then, the expenditures for each schedule are identified, and the schedules are ranked based on their efficiency in minimizing the extension of project duration. The chosen schedule is used for project scheduling, and the impact of activities is determined on project cash flow. Finally, the results of this method are compared to the results of the IP method.
Ali and Elazouni (2009) considered a Line of Balance technique (LOB) to schedule the projects with repetitive activities so that activities’ cash requirements and available cash are in balance. The objective of this study is to combine the Critical Path Method (CPM) with the LOB technique and create a CPM/LOB model to find schedules that are financially feasible. The GA technique is also used for maximizing project profitability, subject to cash constraints.
Afshar and Fathi (2009) considered uncertainty in the direct cost of each activity for multi-objective finance-based scheduling problems. They used a Non dominated sorting genetic algorithm II (NSGA-II) to find the optimal solutions. The objectives in this study are total project duration, financing cost, and the amount of required credit. To find the required credit and financing costs under direct cost uncertainty, a fuzzy sets theory is used. The results in this study show that the more extended project duration leads to less credit requirement.
Liu and Wang (2010) used cash flow for multi-project scheduling problems considering profit maximization. Since the contractors usually handle different projects simultaneously, financing management is complicated. In this study, cash flow and financial requirements are considered in a multi-project environment and a model for maximizing profitability is proposed. The financial resource used in this paper is the credit line.
Abido and Elazouni (2011) proposed a three-objective model for scheduling multiple projects simultaneously. The objectives of this study are to minimize the required credit, total duration, and financing cost, which is done using a Strength Pareto Evolutionary Algorithm (SPEA). The combinations of two-objective and three-objective models are considered in this paper to choose the optimal solution.
Elazouni and Abido (2011) considered a model for scheduling multi-projects within a portfolio to maximize each project’s profitability which results in conflicting objectives. The model is solved using an SPEA. In the end, the results of this method are compared to the results of the GA technique, which shows they both have the same results.
El-Abbasy et al. (2012) used NSGA II for scheduling multi-projects with multi-mode activities. In this model, two different resources are considered for each activity, and as a result, each activity has two different durations and two different direct costs. The credit limit is also ignored in this study. The objectives in this paper are to minimize multiple projects duration and financing costs. Minimizing maximum negative cumulative balance is also another objective in this study. The first step done in this paper is scheduling the projects using CPM and finding the cash flow. Then the NSGA II is used to solve the scheduling problem, which produces acceptable results.
Al-Shihabi and AlDurgam (2017) used a max-min ant system to solve Finance Based Scheduling Problem. In this study, a credit line is considered as a financial resource and the CL results in extension of project duration for J periods. To solve this model, in the first step, project scheduling is done using the CPM method and the cash flow is predicted. Then by imposing CL constraint and considering J=0, the created model is solved. This action continues by increasing J until a feasible solution can be found. In this model, the feasible solution is achieved in J=3, which means that three additional weeks are needed to fulfill the CL constraint. The proposed model is solved by using the max-min ant system, which has effectively good results.
Alavipour and Arditi (2018) calculated the financing cost based on project cash flow. This method is much more reliable and realistic than considering the financing cost as a percentage of total expenses because financing costs should be found according to an optimal combination of lenders. The objective of this article is to minimize the financing costs and bid price. The long-term loan and short-term loan are the financing alternatives that are used in addition to CL for project financing in this study.
Alavipour and Arditi (2019) suggested that an optimal project schedule with the minimum total cost and maximum profit can be found using the TCT method. According to this study, different alternatives for project finance like long-term loans, short-term loans, and credit lines lead to minimum financing costs.
Although various optimization models have been suggested by researchers, only a few of them considered finance-based scheduling problems with multiple financial resources and their focus is mainly on the credit line. It is evident that multiple financial alternatives can have a profound impact on financing costs and project duration. Thus, in this paper, two other resources are introduced to create a model, where two main objectives are followed.
The main objective of this study is to use other financial resources in addition to credit lines, such as long-term loans and short-term loans, to suggest an optimal financing model. Moreover, the problem in this study is a multi-objective model and is solved concerning two main objectives. These objectives are minimizing financing cost and minimizing project duration simultaneously. The innovation of the proposed model is different from the previous ones because the majority of them considered credit line as the only financial resource to solve FBSP, and there were no financial alternatives. Also, the studies that used multiple financial resources did not consider minimizing financing cost and project duration at the same time.
This paper is organized as following steps that the second part includes the cash flow model. The optimization model is presented in the third part, and solved in part four. The fifth part consists of the conclusion and discussion.
II. CASH FLOW MODEL
In IP formulation, we have three sets of parameters. Input parameters, decision variables, and auxiliary variables. The indices used in this model are:
, the
weekly periods
, the monthly periods
, the number of
weeks that the activity
, 
is delayed
from its early start time as calculated by the CPM
A. Input Parameters
: Credit Limit
: Duration of
activity i in weeks
: Indirect
costs
: Number of
weeks in each reimbursement period k
: Mark-up
percentage, which is the contractor’s profit
: Predecessor
activities of activity i
: Successor activities of
activity i
: Monthly
interest rate
: Weekly interest rate
: Yearly
interest rate
: Direct cost of
activity i in each week
: Retainage Percent
B. Auxiliary Parameters
The parameters that are used in mathematical calculations are auxiliary parameters.
: Actual finish
of each activity, which is found by the IP model.
: Actual start of each activity, which is found by the IP model.
: Sum of
expenditures, including interests, in each disbursement period k.
: Sum of
expenditures, excluding interests, in each disbursement period k.
Table 1. Predecessors, duration, and direct cost of activities
|
Direct cost per week ( |
Duration ( |
Predecessors ( |
Activity (i) |
|
2000 |
2 |
- |
A |
|
1500 |
1 |
A |
B |
|
1000 |
3 |
A |
C |
|
1250 |
4 |
C |
D |
|
2000 |
3 |
B, C |
E |
|
1500 |
5 |
- |
F |
Table 2 - ES, EF, and FF of each activity
|
FF |
EF |
ES |
Activity i |
|
0 |
2 |
0 |
A |
|
2 |
3 |
2 |
B |
|
0 |
5 |
2 |
C |
|
0 |
9 |
5 |
D |
|
1 |
8 |
5 |
E |
|
4 |
5 |
0 |
F |
: Early finish
of each activity, which is found by the CPM.
: Early start
of each activity, which is found by the CPM.
: Cumulative
net cash flow at the end of period k, before receiving 
.
: Free float of
each activity, which is found by the CPM.
: Interest value of the expenditures in week w at period k.
: Invoice value of work executed in period k.
: Maximum
extension of the project duration from the one found by CPM.
: Maximum extension that each activity i is allowed to have, 
: Net
cumulative cash flow at the end of period k, after receiving .
: Payment
received from the project sponsor at the end of period k.
: Time of
Project completion, which is found by the CPM.
: Integer variable, which is the number of weeks that activity i is delayed.
: Disbursements of each activity in week w.
: Total disbursements at week w for all activities.
C. Decision Variables
: Binary variable that is one if activity i is extended j
weeks; otherwise, it is equal to 0.
D. Illustrative Example
The finance-based scheduling problem in this study is the one used in Al-Shihabi and AlDurgam (2017). In this example, the following restatements are assumed:
CL=25000, IC=10%, m=4, MU=20%,
RP=10%, 
=10%
It is also supposed that this project comprises six activities A, B, C, D, E, F and the predecessors, duration, and direct cost of each activity is shown in Table 1.
The early start (ES), early finish (EF), and free float (FF) of each activity, calculated by the CPM method, are also available in Table 2.
Figure 1. The Gantt chart of the classic model.
The Gantt Chart of the project found by CPM, is also shown in Fig. 1.
If the expenditures of activities exceed the CL, the duration of the project, which is calculated by CPM must be increased for J weeks. The maximum possible extension of activity i is:
(1)
The expenditures of activity i are
between its actual start 
and its actual finish 
. The actual start of activity i is:
(2)
The equation for calculating is:
(3)
We also have:
(4)
The expenditures of week w due to activity i can be calculated by Eq. (5):
(5)
The total expenditures of activities in week w can be found using Eq. (6):
(6)
The interest value of the expenditures of week w in reimbursement period k is calculated using Eq. (7):
(7)
Each reimbursement period k consists of m weeks which
is considered 
in this
example. The total expenditures of reimbursement period k, excluding
interests, is:
(8)
After the contractor adds the mark-up percentage to using the equation (9), the invoiced value is:
(9)
The sponsors usually retain a certain percentage of the invoiced value as retainage percent (RP) and pay the rest of the invoice m weeks after receiving it. These retained amounts will be paid to the contractor after the project is finished. The Payment received from the project owner at the end of period k is:
(10)
The expenditures include the interest paid by the contractor and can be found by Eq. (11).
(11)
Net cumulative cash flow at the end of disbursement period k
is equal to the sum of the project balance 
and money
received from the sponsor ():
(12)
And before
receiving from the
sponsor is:
(13)
E. Objective Function and Constraints of the classic model
Finally, the optimization model can be illustrated as follows:
(14)
Subject to:
‚
(15)
‚ 
(16)
‚ 
(17)
(18)
III. OPTIMIZATION MODEL
A. Objective Function and Constraints added to the classic model
As explained before, the second objective added to the classic model
is minimizing financing cost (
).
There are also some new constraints associated with other financial resources, which are explained in two different scenarios.
Scenario 1
According to this scenario, the second resource for project
financing is the Long-Term Loan (
). In this
scenario, it is supposed that the loan is taken out at the beginning of the
project, and both principal and interest 
are paid back
monthly. The constraints related to LTL are:
(19)
According to this constraint, a limitation (L) for the amount
of is imposed on
the model, which doesn’t let the contractor take a long-term loan that exceeds L.
L may differ in other projects based on its financial requirements.
LTL (20)
(21)
The cumulative withdrawal from is the sum of
withdrawn money in each reimbursement period k which is shown in Eq.
(21), and it should be equal or less than the LTL according to constraint (20).
(22)
According to (22), the withdrawal from LTL in each period must be equal to or greater than 0.
In the following, by adding the long-term loan to the
constraint (18), we have (23), which says the cumulative net cash flow at the
end of period k must be equal or less than credit line (CL) plus the
withdrawn from long term loan in each reimbursement period k (
):
(23)
The financing cost in each period can be found according to Eq.
(26), and the total financing cost () can be
calculated by using Eq. (25):
(24)
(25)
(26)
In the end, the mathematical model for the first scenario is:
Min π
Min FC
Subject to:
‚
‚ 
‚
LTL
≤
Scenario 2
In this scenario, the second financial resource is a Short-Term Loan (STL). It is supposed that
this loan should be taken in each reimbursement period, that the CL cannot
fulfill the expenditures of activities. The principal must be repaid after r
periods from when the loan is taken out, and r must be at last 12 months
as it’s a short-term loan. The STL has a specific monthly interest rate 
which can be
paid back monthly or with the principal after r periods. In this study,
it is supposed that both the principal and the interest of STL must be repaid
together after three months, and based on this assumption, the interest rate is
calculated using Eq. (27):
(27)
(28)
The constraint (28) says that the short-term loan in each period must be equal to or greater than 0.
By adding short term loan to (18), we have:
(29)
Constraint (29) says that the cumulative net cash flow at the end of
period k must be equal or less than the credit line () plus the
short-term loan taken out in each reimbursement period k (STL(k)).
The objective added to the classic model in this scenario is minimizing FC, and it can be calculated by Eqs. (30) and (31).
(30)
(31)
Table 3. The results of solving classic model
|
π |
CL |
J |
|
Infeasible solution |
25000 |
0 |
|
Infeasible solution |
25000 |
1 |
|
Infeasible solution |
25000 |
2 |
|
9 |
25000 |
3 |
Figure 2. The Gantt chart of classic model in J=3
The final model for scenario 2 is:
Min π
Min FC
Subject to:
‚
‚
‚
IV. SOLVING THE MODEL
Firstly, the results of solving the classic model are shown in Table 3.
According to table 3, the feasible solution is found
in J=3, which means we need to extend the project duration for three
weeks to fulfill the . The Gantt
chart of the classic model can be seen in Fig. 2, considering J=3.
By comparing Fig. 1 and 2, we can understand that activities B, E, and F use their free float, but the project can be finished at the end of week 9.
To solve the proposed multi-objective in scenarios 1 and 2, we use the goal programming (GP) method. This method is used in numerous studies such as Torres-Ruiz and Ravindran (2019), Inan et al. (2018), and dos Santos Rubem et al. (2017). The proposed model in this study is solved by the GP method, which is used by Torres-Ruiz and Ravindran (2019) for solving a multi-objective problem for sustainable supplier management. This method is like (34):
(34)
The GP constraints that should be added to the model are:
Subject to
(35)
(36)
(37)
Table 4 - Results for solving the model for scenario 1
|
Parameters Scenario |
J |
LTL |
FC |
π |
|
Scenario 1 |
0 |
2759.396 |
356.8803 |
9 |
(38)
In constraints above, 
and 
are the goals
for our objectives, and 
are the
variables that show the negative and positive deviation from the goals. 
represents the
weight for each deviation.
The final optimization model for scenario 1, considering goal programming, is:
Subject to:
‚
‚
‚
LTL
≤
The final optimization model for scenario 2, considering goal programming, is:
Subject to:
‚
‚
‚
A. Scenario 1
The suppositions for solving the model are:
CL=25000, IC=10%, m=4, MU=20%,
RP=10%, =10%
, 
, 
,
These assumed values can be varied in different projects and are changeable. The model is coded and solved using LINGO, and the results are available in Table 4.
Table 5. The results of sensitivity analysis for scenario 1, considering CL and LTL
|
CL |
LTL |
FC |
π |
|
0 |
- |
- |
- |
|
1000 |
- |
- |
- |
|
2000 |
- |
- |
- |
|
3000 |
- |
- |
- |
|
4000 |
- |
- |
- |
|
5000 |
- |
- |
- |
|
6000 |
- |
- |
- |
|
7000 |
- |
- |
- |
|
8000 |
- |
- |
- |
|
9000 |
- |
- |
- |
|
10000 |
- |
- |
- |
|
11000 |
- |
- |
- |
|
12000 |
- |
- |
- |
|
13000 |
- |
- |
- |
|
14000 |
17619.15 |
1852.398 |
9 |
|
15000 |
13598.78 |
1450.361 |
9 |
|
16000 |
11262.25 |
1216.708 |
9 |
|
17000 |
10152.38 |
1102.536 |
9 |
|
18000 |
9059.957 |
993.2937 |
9 |
|
19000 |
7950.137 |
879.1274 |
9 |
|
20000 |
6840.356 |
764.9764 |
9 |
|
21000 |
4410.084 |
531.4861 |
9 |
|
22000 |
2839.155 |
374.3933 |
9 |
|
23000 |
2744.223 |
364.9001 |
9 |
|
24000 |
2759.396 |
356.8803 |
9 |
|
25000 |
2759.396 |
356.8803 |
9 |
|
26000 |
2759.396 |
356.8803 |
9 |
|
27000 |
2759.396 |
356.8803 |
9 |
|
28000 |
2759.396 |
356.8803 |
9 |
|
29000 |
2759.396 |
356.8803 |
9 |
|
30000 |
2759.396 |
356.8803 |
9 |
According to the table, the feasible solution is found in J=0, which means by using LTL as a second financial resource, we don’t need to extend the project duration to find a feasible solution for the problem.
In the classic model, minimizing financing cost is not considered as an objective, and the only objective function is the duration. The suggested model in this study includes a long-term loan as a second resource for financing the project, and a combination of resources with optimal financing cost is available.
This model also allows us to find the optimal mixture of financial resources with the least FC by having their values changed. For sensitivity analysis, the limitation of LTL is removed, and the model is solved with different amounts of CL to find the required quantity of LTL for a feasible solution at J=0 (see Table 5).
The sensitivity analysis results show that a minimum CL=14000
is required to find a feasible solution at J=0. If it is less than this
amount, there will be no feasible solution. In this situation, the needed
amount of LTL is equal to 17619.15. When the CL goes up, the
required LTL for a feasible solution goes down, and eventually, it
remains stable at 2759.396. It does not change with higher amounts of CL,
which shows the minimum amount of LTL for a feasible solution J=0.
By increasing CL and decreasing LTL, the part of FC goes down,
and when the LTL holds steady, FC stays constant, too. Also, the
amount of π remains steady at
nine during the sensi-
Figure 3. The changes of FC, LTL, and π according to the changes of CL.
Table 6. Results for solving the model for scenario 2
|
Param. Scenario |
J |
STL1 |
STL2 |
STL3 |
STL4 |
FC |
π |
|
Scenario 2 |
0 |
0 |
1261.662 |
317.0608 |
0 |
603.4980 |
9 |
tivity analysis. These trends are depicted in Fig. 3.
B. Scenario 2
There are also some assumptions for this scenario like scenario 1, which are:
CL=25000, IC=10%, m=4, MU=20%,
RP=10%, =10%
-CL
, , ,
The model is coded in LINGO like the first scenario, and the results are in Table 6.
According to the results in Table 6, the feasible solution is found in J=0, and the amount of required STL in each period is also available in this table.
For sensitivity analysis, the problem is solved with different values of CL, and the results are depicted in Table 7. The minimum CL required for a feasible solution at J=0 is 15000. In this state, the total STL equals 11309.46, which includes all short-term loan amounts in each period. When the CL rises, the quantity of STL goes down, and at CL=24000, it reaches its minimum amount equal to 1570.042, and in this situation, FC is similar to 610.1617. When CL grows to 25000, STL rises to 1578.723, and in higher amounts of CL, it stays constant at the same amount. At STL=1578.723, the minimum FC is found, equal to 603.498.
The trend of Table 7 is also depicted in Fig. 4.
For solving both models with different weights, the
value of W is assumed equal to 0.5. Since this
value is hypothetic and, in reality, it is probably different, the model is
solved with different values for . The results
are represented in Table 8, which shows that by raising , the value of
FC lowers at the first step, then it remains constant. The value of π is
also stable during the changes of . This trend
can be seen in Fig. 5 for scenario 1, and for the second scenario, the same
trend occurs.
Figure 4. The changes of FC, LTL, and π according to the changes of CL
Table 7. The results of sensitivity analysis for scenario 2, considering CL
|
CL |
STL1 |
STL2 |
STL3 |
STL4 |
STL |
FC |
π |
|
0 |
- |
- |
- |
- |
- |
- |
- |
|
1000 |
- |
- |
- |
- |
- |
- |
- |
|
2000 |
- |
- |
- |
- |
- |
- |
- |
|
3000 |
- |
- |
- |
- |
- |
- |
- |
|
4000 |
- |
- |
- |
- |
- |
- |
- |
|
5000 |
- |
- |
- |
- |
- |
- |
- |
|
6000 |
- |
- |
- |
- |
- |
- |
- |
|
7000 |
- |
- |
- |
- |
- |
- |
- |
|
8000 |
- |
- |
- |
- |
- |
- |
- |
|
9000 |
- |
- |
- |
- |
- |
- |
- |
|
10000 |
- |
- |
- |
- |
- |
- |
- |
|
11000 |
- |
- |
- |
- |
- |
- |
- |
|
12000 |
- |
- |
- |
- |
- |
- |
- |
|
13000 |
- |
- |
- |
- |
- |
- |
- |
|
14000 |
- |
- |
- |
- |
- |
- |
- |
|
15000 |
0 |
11309.46 |
0 |
0 |
11309.46 |
3833.915 |
9 |
|
16000 |
0 |
10293.49 |
0 |
0 |
10293.49 |
3494.441 |
9 |
|
17000 |
0 |
9293.485 |
0 |
0 |
9293.485 |
3163.441 |
9 |
|
18000 |
0 |
8277.555 |
0 |
0 |
8277.555 |
2823.984 |
9 |
|
19000 |
0 |
7277.555 |
0 |
0 |
7277.555 |
2492.984 |
9 |
|
20000 |
0 |
4598.962 |
1064.935 |
0 |
5663.897 |
1965.228 |
9 |
|
21000 |
0 |
3598.962 |
64.935 |
0 |
3663.897 |
1303.228 |
9 |
|
22000 |
0 |
2598.962 |
0 |
0 |
2598.962 |
950.7343 |
9 |
|
23000 |
0 |
1598.962 |
0 |
0 |
1598.962 |
619.7343 |
9 |
|
24000 |
0 |
1570.042 |
0 |
0 |
1570.042 |
610.1617 |
9 |
|
25000 |
0 |
1261.662 |
317.0608 |
0 |
1578.723 |
603.498 |
9 |
|
26000 |
0 |
261.662 |
1317.061 |
0 |
1578.723 |
603.498 |
9 |
|
27000 |
0 |
1578.723 |
0 |
0 |
1578.723 |
603.498 |
9 |
|
28000 |
0 |
1578.723 |
0 |
0 |
1578.723 |
603.498 |
9 |
|
29000 |
0 |
1578.723 |
0 |
0 |
1578.723 |
603.498 |
9 |
|
30000 |
0 |
1578.723 |
0 |
0 |
1578.723 |
603.498 |
9 |
This figure shows the results based on different W amounts. When W=0, it means that only project duration is important, and this is why the financing cost is higher. For other quantities of W, identical results are achieved.
C. Comparison
In this part, the results of solving two scenarios are compared in table 9. The Gantt chart for each scenario is shown in Figs. 6 and 7.
Figure 5. The trend of FC and π during the changes of weights in scenarios 1and 2
Table 8. The results of solving the problem for scenarios 1 and 2, using different weights
|
|
FC (scenario 1) |
π (scenario 1) |
FC (scenario 2) |
π (scenario 2) |
|
0 |
372.9420 |
9 |
6833.997 |
9 |
|
0.1 |
356.8803 |
9 |
603.498 |
9 |
|
0.2 |
356.8803 |
9 |
603.498 |
9 |
|
0.3 |
356.8803 |
9 |
603.498 |
9 |
|
0.4 |
356.8803 |
9 |
603.498 |
9 |
|
0.5 |
356.8803 |
9 |
603.498 |
9 |
|
0.6 |
356.8803 |
9 |
603.498 |
9 |
|
0.7 |
356.8803 |
9 |
603.498 |
9 |
|
0.8 |
356.8803 |
9 |
603.498 |
9 |
|
0.9 |
356.8803 |
9 |
603.498 |
9 |
|
1 |
356.8803 |
9 |
603.498 |
9 |
Table 9. Comparing the results of two scenarios
|
Parameters Scenarios |
J |
FC |
π |
|
Scenario 1 |
0 |
356.8803 |
9 |
|
Scenario 2 |
0 |
603.498 |
9 |
By comparing the two charts, we can understand that in both scenarios, the feasible solution is found in J=0. We can also know that activities B, E, and F, use their FF but the total duration of the project is still nine weeks.
Finally, by comparing the results we can say that in this particular example, the LTL is a better choice than STL due to its lower financing cost. However, the result can be completely different in other projects concerning various factors.
V. DISCUSSION
As discussed before, correct timing and cash flow management are two
fundamental factors of the success or failure of a project. Different
objectives should be considered in project scheduling, and financial goals are
of high importance. Lack of enough cash is one of the drawbacks that the
contractors face, which leads to delay in the execution of project activities.
To tackle this issue, project financing is essential. In most projects, a
credit line (CL) is used as a resource for project financing and cash
flow management. The expenses of each period shouldn’t exceed the CL
imposed by the lender bank. This constraint can lead to the extension of
project dura-
Figure 6. The Gantt chart after solving the model in scenario 1.
Figure 7. The Gantt chart after solving the model in scenario 2.
Table 10. Predecessors, duration and direct cost of activity G
|
Direct Cost per week ( |
Duration ( |
Predecessors ( |
Activity (i) |
|
1000 |
1 |
- |
G |
tion since the expenditures of some activities in a particular period might be high, and CL cannot satisfy them. In this study, a multi-objective model is developed in which other resources like long-term loans and short-term loans are considered financial resources in addition to CL. The objective of this model is to minimize the project duration and the financing cost (FC). This model helps the contractors to find an optimal mixture of resources for project financing with the least FC. The model is divided into two different scenarios. In scenario1, LTL is the second financing resource, and the amounts of financing cost and project duration, which are found in this scenario, are FC=356.8803 and π=9. STL is another financial resource that is considered in scenario 2. In this scenario, the value of FC is 603.498, and π is equal to 9, and the feasible solution is found in J=0 in both scenarios, which means that there is no need to extend the project duration to find a feasible solution. The scenarios suggest that it is possible to execute project activities on time and control the financing costs at the same time by choosing an optimum combination of financial resources.
Both scenarios have been solved concerning a new project, where activity G is added to the previous activities. The predecessors, duration, and direct cost of activity G can be seen in Table 10.
