Modified Finance-Based Scheduling with Activity Splitting

Abstract

Construction contractors often rely on external funding to manage financial deficits caused by irregular cash inflows and outflows. To address these cash flow challenges, contractors typically adjust the start times of project activities to prevent shortages while minimizing the overall project duration. However, in severe cases, operations may need to halt if cash flow issues cannot be adequately resolved. This study introduces an alternative strategy to prevent cash shortages by allowing for the temporary suspension of activities, known as activity splitting. In this approach, operations are paused to conserve cash and then resumed when sufficient funds become available. The potential benefit of this method is illustrated through a simple, four-activity project. Extending the application to more complex projects with a set of splittable activities—each having different cash requirements, durations, and associated splitting costs—the challenge lies in identifying the optimal activities to split and determining the precise suspension and resumption times, all while minimizing or avoiding project delays and additional costs. To address this, we present a novel mixed-integer linear programming (MILP) model that optimizes the scheduling and splitting of activities to minimize project duration without breaching financial constraints. The MILP model effectively identifies the best trade-off between activity splitting and project extension, taking into account the financial and cost implications of the contractor’s constraints. This is demonstrated through a case study that considers different scenarios related to splitting costs and financial limitations.

1. Introduction

From the perspective of construction contractors, project cash flow consists of both outflows and inflows. Cash outflows refer to the contractor’s expenditures, while inflows represent payments received from project owners. Whenever cash outflows exceed inflows, the contractor experiences a cash deficit. Typically, project owners hold greater bargaining power than contractors when negotiating payment terms [1], and throughout much of a project’s duration, contractors frequently encounter persistent cash deficits, necessitating external financing to bridge the gap. The magnitude and timing of these deficits are largely dictated by specific payment-related clauses in the contract between the contractor and owner, such as payment delays [2]. In practice, cash flow issues stemming from these deficits are among the leading causes of contractor failures [3].

Effective cash flow management is crucial for the financial health of construction contractors. Numerous studies have addressed finance-based scheduling (FBS) models to optimize cash flow in construction projects [4,5,6]. FBS has generated significant attention from both practitioners and researchers since its introduction [7]. The fundamental approach behind FBS ensures that the contractor’s cash deficit remains within the credit limit (CL) set by lines of credit (LOC), which are widely used as the standard method for financing projects [8]. According to the Surety Information Office (SIO), which monitors surety bond data in the United States, there are six key warning signs indicating that a construction company may be in financial distress. One critical indicator is when the LOC is continuously borrowed to its limit [9].

When adequate cash is not available, contractors can reschedule the start times of certain activities. As long as these adjustments do not consume the activities’ total float, the project can still be completed on schedule. Otherwise, delays may occur, potentially resulting in penalties for late completion. Another approach to addressing cash deficits involves employing different execution modes for project activities [10]. By utilizing advanced execution technologies and deploying additional resources, contractors can shorten the duration of activity execution. This, in turn, impacts both the timing and amount of contractor expenditures, as well as the corresponding payments received, thereby influencing the cash flow situation. However, while this approach can help alleviate cash flow issues, it typically incurs higher costs and is justifiable only when sufficient incentives, such as early completion bonuses, are available.

In addition to the two primary FBS approaches identified in the literature for meeting financing constraints—rescheduling activities and employing different execution modes—contractors can also temporarily halt activities to conserve cash and resume them once sufficient funds become available. This strategy, known as activity splitting (AS), is typically employed as a reactive measure during the construction phase to address cash flow shortages

• It integrates AS into FBS, introducing a new problem, FBS-AS.
• It develops an MILP model to solve the FBS-AS problem.
• It demonstrates the significance of integrating AS and FBS for contractors, showcasing its practical benefits.
• It shows that the optimal solution, which involves selecting the activity to split and the timing of the split, is case-sensitive and requires solving the developed MILP model.

Despite its potential to mitigate contractors’ cash deficit challenges and its established use in resource leveling (e.g., [11]), AS has not been considered in FBS research. In this work, we address this gap by introducing AS as a proactive strategy during the planning phase to prevent potential cash shortages. To this end, we propose a mixed-integer linear programming (MILP) model that optimizes project scheduling to minimize duration while adhering to financial constraints through activity splitting. This novel approach is termed finance-based scheduling with activity splitting (FBS-AS).

In summary, this study makes four key contributions:

• It integrates AS into FBS, introducing a new problem, FBS-AS.
• It develops an MILP model to solve the FBS-AS problem.
• It demonstrates the significance of integrating AS and FBS for contractors, showcasing its practical benefits.
• It shows that the optimal solution, which involves selecting the activity to split and the timing of the split, is case-sensitive and requires solving the developed MILP model.

The structure of this paper is as follows. Section 2 reviews previous research related to FBS and AS. Section 3 introduces the MILP developed to solve the FBS-AS problem. To demonstrate the concept, Section 4 presents a simple example that highlights the effectiveness of AS in managing cash deficits, while in Section 5, a more detailed example, adapted from the literature, illustrates the practical application of the MILP model. Finally, Section 6 concludes the paper and offers directions for future research.

2. Literature Review

The application of AS within the context of FBS has not been explored in existing research. Nevertheless, a review of the literature on FBS and AS is conducted to establish a comprehensive foundation for this study.

2.1. Finance-Based Scheduling Problem

FBS was first introduced by Elazouni and Gab-Allah [7] as an extension to the traditional project scheduling problem under capital constraints. FBS aims to schedule construction project activities based on cash availability during project billing periods. During any given billing period, funds are typically sourced from two primary channels: (1) payments received for activities completed and billed in previous periods and (2) external funds acquired through CLs. This approach allows contractors to effectively manage their CLs. Elazouni and Gab-Allah [7] formulated an FBS model and solved it using an integer programming technique to generate schedules that adhere to constraints on negative overdraft amounts by the end of each project billing period. Their model incorporated extended total floats for activities, enabling it to handle stricter financial constraints, which may result in extended project durations. Building on this work, Liu and Wang [12] developed a more comprehensive scheduling model that integrates both resource usage and cash flow constraints to satisfy resource and CL limitations. They utilized constraint programming to solve the model. However, both integer programming and constraint programming approaches are primarily limited to solving small-scale FBS problems due to their computational complexity. Subsequently, meta-heuristic techniques such as genetic algorithms (GA), simulated annealing, and shuffled frog-leaping algorithms were employed to solve large-scale FBS problems [13]. Additionally, rule-based heuristic procedures were developed for FBS solutions [14].

Recent studies have continued to explore and enhance FBS models. For instance, Alavipour and Arditi [6] introduced a model that minimizes financing costs by incorporating various financing alternatives, expanding beyond the traditional reliance on CLs. Lucko [15] utilized a singularity function to model cash flows more accurately, capturing the non-uniform distribution of activity costs during execution. Other researchers have integrated subcontractor payment time delays into FBS models, accounting for varying contractor-to-subcontractor payment arrangements [16]. Multi-objective optimization techniques have also been applied to FBS problems, balancing factors such as time, cost, resources, and cash flow using methods like genetic algorithms and particle swarm optimization [4,5].

Unlike previous FBS models that considered CLs as the sole external financing method, Alavipour and Arditi [6] proposed a model that minimizes financing costs by incorporating various financing alternatives. A different approach to modeling cash flow in construction projects, initially introduced by Au and Hendrickson [17], was further developed by Lucko [15] using a singularity function to capture the non-uniform distribution of activity costs during execution. In contrast to the typical FBS strategy of adjusting activity start times to address cash flow limitations, Tabyang and Benjaoran [16] proposed a model that considers subcontractor payment time delays. This model accounts for varying contractor-to-subcontractor payment arrangements, including billing dates and payment time delays for individual work packages.

Several studies have extended FBS problems to multi-objective optimization using various techniques. GA, for example, was applied to solve several multi-objective FBS optimization problems [4,18,19]. Jiang et al. [20] proposed a cash flow planning model considering financial constraints and banking instruments, and [5] used particle swarm optimization for project scheduling considering time, cost, resources, and cash flow. Moreover, the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) technique for optimizing multiple projects with multi-mode activities and resources was also used [21].

2.2. Activity Splitting

The critical path method (CPM) has a notable limitation in its inability to accommodate activity splitting [22]. This limitation can be problematic because, in some cases, it may be more cost-effective to temporarily halt an activity and reallocate its resources to another task, as it is often argued that “it is often more cost-effective to stop a specific activity for a certain period, thus allowing its resources to be used in another activity” [22]. Despite the potential cost implications of stopping and resuming activities [23], activity splitting remains a common practice in construction projects. To support this approach, commercial project management software such as MS Project and Primavera includes features that allow activity splitting to facilitate resource leveling.

Activity splitting has been employed to optimize resource utilization in construction projects [24,25,26]. The approach has been particularly useful in linear projects, such as highways and tunnels, which involve repetitive activities. In these contexts, a linear integer program was proposed to allow for activity interruptions, thereby optimizing the use of crews and machinery [25]. This concept was later expanded to non-linear projects through the development of a binary resource leveling model that incorporates activity splitting, demonstrating that this method significantly enhances resource utilization [27]. An activity’s duration can be affected by splitting and executing mode, where the combination of both was used to minimize project duration [11] or level resources [28]. Researchers have also assumed that activities can be split once [29] or multiple times [30].

2.3. New Contribution

There is a gap in the literature as no previous research has integrated AS into FBS models, making this study a novel approach to managing financial constraints in construction projects. By developing an MILP model that incorporates activity splitting, we provide a tool that helps contractors proactively plan for cash flow shortages. The model considers the additional costs associated with splitting, such as financial penalties incurred when work is halted, but machinery remains on-site, resulting in opportunity costs due to idle equipment. These managerial considerations, combined with the proposed mathematical model and the demonstration of how splitting can ease financial constraints, distinguish this study from previous research. The model offers optimal scheduling solutions that balance splitting costs against potential project delays, providing a practical framework for addressing financial limitations.

3. Mathematical Model

This section begins with a list of core notations used in developing the MILP model for a project having M activities. Additional notations will be introduced as needed throughout the model description. Any time unit, such as a week or a month, can be used in the model, where time is indexed by k.

• $C_i$: Direct cost of activity $i\in M$.
• ${\mathrm{SC}}_i$: Stopping cost of activity $i\in M$.
• ${\mathrm{RC}}_i$: Resuming cost of activity $i\in M$.
• $T_i$: Duration of activity $i\in M$.
• W: Project duration.
• $P^i$: Set of predecessor activities for activity $i\in M$.

Decision variables of the MILP model include:

• $x_{ik}$: A binary decision variable equal to 1 if activity $i\in M$ is executed in time unit k, and 0 otherwise.
• $y_{ik}$: A binary variable equal to 1 if activity $i\in M$ is stopped due to a split decision in time unit k, and 0 otherwise.
• $z_{ik}$: A binary variable equal to 1 if activity $i\in M$ is resumed after being stopped in time unit k, and 0 otherwise.

The contractor’s objective is to minimize the overall project duration, denoted as W, to ensure project completion as efficiently as possible. This objective is formulated as the primary goal of the optimization model, as shown in Equation (1). Equation (2) shows that for any activity $i\in M$, it needs to be executed for $T_i$ unit times. In addition to minimizing the project duration, the schedule must also satisfy the project’s precedence relations, which ensure that activities are executed in the correct order. These precedence constraints are represented in Equation (3), where $P^i$ denotes the set of predecessor activities for activity $i\in M$. The system of equations suggested in [24] is used in the proposed model to determine the start times ($S{T}_i$), finish times ($F{T}_i$), and the values of the decision variables $y_{ik}$ and $z_{ik}$.

$$\mathrm{minimize}z=W$$

(1)

$$T_i=\sum _{k=1}^W{x}_{ik}$$

(2)

$$S{T}_i>F{T}_j,\forall i,j\in M,\forall j\in P^i$$

(3)

Three sets of parameters influence the project’s cash inflows and outflows, as outlined in the following list. These parameters will introduce additional constraints to the MILP model, which will be discussed in detail later.

• Contractor’s bidding parameters, including:

  (a)

  Expected contract bidding time (CBT), denoted as W.

  (b)

  Contract bidding price (CBP).

• Contracting parameters that define the payment conditions imposed by the contractor.
• Financing parameters provided by financial institutions to the contractor.

3.1. Calculating CBP

The calculation of the contract bidding price (CBP) is detailed in Equation (4).

$$CBP=D{C}^{\mathrm{sum}}+FO{C}^{\mathrm{sum}}+VO{C}^{\mathrm{sum}}+MOB+MPV+BND$$

(4)

where:

• $D{C}^{\mathrm{sum}}={\sum }_{i\in M}{C}_i$: Total direct costs.
• $FO{C}^{\mathrm{sum}}=FOC\times CBT$: Total fixed overhead costs.
• $VO{C}^{\mathrm{sum}}=O_V\times D{C}^{\mathrm{sum}}$: Total variable overhead costs.
• $MPV$: Markup value representing the contractor’s targeted profit.
• $O_V$: overhead multiplier, project-dependent.
• $O_M$: Profit multiplier, chosen by the contractor.
• $O_B$: Bonding multiplier, chosen by the owner.
• $FOC$: Fixed overhead cost per time unit.
• $MOB$: Mobilization cost.

The expressions for the markup value (MPV) and bonding (BND) are given by Equations (5) and (6), respectively:

$$MPV=O_M\times (D{C}^{\mathrm{sum}}+FO{C}^{\mathrm{sum}}+VO{C}^{\mathrm{sum}}+MOB)$$

(5)

$$BND=O_B\times (D{C}^{\mathrm{sum}}+FO{C}^{\mathrm{sum}}+VO{C}^{\mathrm{sum}}+MOB+MPV)$$

(6)

3.2. Contracting Clauses

Contracting decisions primarily involve payment details between the contractor and the owner. The calculation of project expenditures, denoted as $E_k^{\mathrm{project}}$, in time unit k is detailed as shown in Equation (7). The owner’s payments, $P_k$, are calculated as shown in Equation (8).

$$E_k^{\mathrm{project}}=FOC+\sum _{i\in M}\left((1+O_V)\times {\displaystyle \frac{C_i}{T_i}}\times x_{ik}+R{C}_i\times y_{ik}+S{C}_i\times z_{ik}\right),1\le k\le W$$

(7)

$$P_k=(1-RP)\times MU\times \sum _{t=k-L_P-R}^{t=k-L_P}{\displaystyle \frac{C_i}{T_i}}\times x_{it}-{\displaystyle \frac{AP}{IP}},k=R+L_P,2R+L_P,\dots ,\lceil W/R\rceil R+L_P$$

(8)

The new notations used in formulating these equations are as follows:

• $MU$: Mark-up factor equal to ${\displaystyle \frac{CBP}{D{C}^{sum}}}$.
• $R{C}_i$: Resuming cost for activity i.
• $S{C}_i$: Stopping cost for activity i.
• $RP$: Retainage percentage deducted from each payment.
• $L_P$: Invoice payment delay in time units.
• $AP$: Advanced payment provided at the beginning of the project.
• $IP$: Number of invoice payments throughout the project.
• R: Reimbursement period.

3.3. Financing Obligations

The suggested MILP model allows the contractor to use three financing sources: long-term loans (LTL), short-term loans (STL), and lines of credit (LOC). The LOC interest payment value, $I{P}_k^{LOC}$, for a monthly compounding period is defined as shown in Equation (9). The total financing expenditures in time unit k, denoted as $E_k^{\mathrm{finance}}$, are calculated as shown in Equation (10).

$$I{P}_k^{LOC}=\sum _{t=k-V+1}^{t=k}{r}_M\times O{D}_{t-1},k=V,2V,\dots ,\lceil W/V\rceil V$$

(9)

$$E_k^{\mathrm{finance}}=I{P}_k^{LTL}+I{P}_k^{STL}+I{P}_k^{LOC},k\ge 0$$

(10)

Variables and parameters used to derive Equations (9) and (10) include:

• $I{P}_k^{LOC}$: Interest payment for the line of credit in time unit k.
• V: Compounding period for interest calculation.
• $r_M$: Monthly interest rate for the line of credit.
• $O{D}_{t-1}$: Overdraft amount in the previous time unit $t-1$.
• $I{P}_k^{LTL}$: Interest payment for long-term loans in time unit k.
• $I{P}_k^{STL}$: Interest payment for short-term loans in time unit k.
• $E_k^{\mathrm{finance}}$: Total financing expenditures in time unit k.

3.4. Cash Flow Balance

To determine the contractor’s cash needs, we calculate the available cash in time unit k, as shown in Equation (11). The overdraft value $O{D}_k$ must satisfy the CL limit condition, as shown in Equation (12).

$$B_k=B_{k-1}+P_k-E_k^{\mathrm{total}}+(LT{L}_k^{+}-LT{L}_k^{-})+(ST{L}_k^{+}-ST{L}_k^{-})+(LO{C}_k^{+}-LO{C}_k^{-}),k\ge 1$$

(11)

$$O{D}_k\le CL,k\ge 0$$

(12)

The following is the new notation used to derive Equations (11) and (12):

• $B_k$: Available cash balance in time unit k.
• $B_{k-1}$: Available cash balance in the previous time unit.
• $P_k$: Payments received from the owner in time unit k.
• $E_k^{\mathrm{total}}$: Total expenditures in time unit k.
• $LT{L}_k^{+}$: Amount of long-term loan received in time unit k.
• $LT{L}_k^{-}$: Amount of long-term loan repaid in time unit k.
• $ST{L}_k^{+}$: Amount of short-term loan received in time unit k.
• $ST{L}_k^{-}$: Amount of short-term loan repaid in time unit k.
• $LO{C}_k^{+}$: Amount of line of credit utilized in time unit k.
• $LO{C}_k^{-}$: Amount of line of credit repaid in time unit k.
• $O{D}_k$: Overdraft value in time unit k.
• $CL$: Credit line limit.

3.5. Iterative Search

To guarantee the linearity of the model without adding extra complexity, we adopt the solution methodology suggested by Elazouni and Gab-Allah (2004). We add Equation (13) to our model and search for a feasible solution such that the project duration is $W^{\max }$. We iteratively increment $W^{\max }$ by one time unit if the MILP solution is infeasible until we find a feasible solution or reach an upper limit for $W^{\max }$:

$$W\le W^{\max }$$

(13)

4. Proof of Concept

A four-activity project, detailed in

Table 1. Project parameters of the illustrative example.

Activity	Predecessors	Time (Month)	Direct Cost ($)	Stopping Cost ($)	Resuming Cost ($)
A	-	5	250,000	N/A	N/A
B	A	5	1,000,000	N/A	N/A
C	B	2	80,000	N/A	N/A
D	A	2	600,000	30,000	90,000

, is used to show how splitting allows the project to remain on schedule despite financial constraints and incurring extra costs for splitting. Among the four activities, only activity D is eligible for splitting, which introduces an additional cost of USD 120,000—30,000 to pause the activity and USD 90,000 to resume it. The original project schedule spans 12 months and is shown with solid bars in Figure 1. The modified schedule, where Activity D is split, is illustrated with hatched bars: the first half of Activity D is completed in month six, and the remaining half in month ten.

Table 2. Bidding parameters of the illustrative example.

Parameter	Value
Fixed overhead cost per month (FOC)	$50,000
Mobilization cost (MOB)	$200,000
Variable overhead multiplier ($O_V$)	10%
Bonding multiplier ($O_B$)	2%
Markup multiplier ($O_M$)	20%

lists the bidding parameters, which, in conjunction with Equations (4)–(6), are used to compute the project’s CBP. The CBP components are summarized in

Table 3. Bidding cost calculations of the illustrative example.

Parameter	Value ($)
Sum of direct costs (DCsum)	1,930,000
Sum of variable overhead cost (VOCsum)	193,000
Sum of fixed overhead cost (FOCsum)	600,000
Markup value (MPV)	584,600
Bonding cost (BND)	70,152
Contract bidding price (CBP)	3,577,752

. Since $CBP=USD3,577,752$ and $D{C}^{sum}=USD1,930,000$, the MU used by the contractor and owner is 1.853757, leading to contractual activity prices shown in

Table 4. Activity prices.

Activity	Price ($)
A	463,439
B	1,853,757
C	148,300
D	1,112,254
Contract bidding price	3,577,752

. Note the sum of these prices is equivalent to the contract price.

Table 5. Contract parameters of the project financial life cycle example.

Parameter	Value
Advanced payment (AP)	$500,000
Reimbursement period (R)	2 months
Invoice payment lag (LP)	2 months
Retainage percentage (RP)	25%
Extension penalty per month (EP)	$500,000

provides the contract payment parameters and shows that at the start of the project, the contractor receives an AP of USD 500,000. The contractor submits invoices to the owner every two months, with payments made two months later. Since the project is expected to last for 12 months, the contractor is expected to receive six payments, from which USD 83,333 are deducted to pay AP. The owner will also retain 25% of each payment, releasing the retained amount at the end of the project. A delay penalty of USD 500,000 per month is imposed if the project exceeds the 12-month duration.

To finance the project, the contractor obtained an LTL of USD 500,000 at the start of the project, with an annual percentage rate (APR) of 6%. The LOC has a CL of USD 800,000 and an APR of 20%. LOC interest is compounded monthly and paid every two months.

Table 6. Cash flow calculations without splitting activity D.

Month	Project Expenditure	Invoice	Payment	Cash Balance *	LTL Changes	Cash Balance	Overdraft	Interest Payment
	${\mathit{E}}_{\mathit{k}}^{\mathbf{project}}$	${\mathit{I}}_{\mathit{k}}$	${\mathit{P}}_{\mathit{k}}$	${\mathit{B}}_{\mathit{k}}$	${\mathit{LTL}}_{\mathit{k}}$	${\mathit{B}}_{\mathit{k}}$	${\mathit{OD}}_{\mathit{k}}$	${\mathit{IP}}_{\mathit{k}}^{\mathit{LOC}}$
0	270,152	0	500,000	229,848	500,000	729,848	0	0
1	105,000	0	0	124,848	0	624,848	0	0
2	105,000	185,376	0	19,848	0	519,848	0	0
3	105,000	0	0	−85,152	0	414,848	0	0
4	105,000	185,376	55,698	−134,454	0	365,546	0	0
5	105,000	0	0	−239,454	0	260,547	0	0
6	600,000	1,019,567	55,698	−783,755	0	0	283,755	0
7	600,000	0	0	−1,383,755	0	0	883,755	0
8	270,000	1,297,630	681,342	−972,413	0	0	490,287	17,874
9	270,000	0	0	−1,242,413	0	0	760,287	0
10	270,000	741,503	889,889	−622,524	0	0	159,543	19,146
11	94,000	0	0	−716,524	0	0	253,543	0
12	94,000	148,301	472,794	−337,730	−530,000	0	411,074	6324
13	0	0	0	−337,730	0	0	411,074	0
14	0	0	922,330	584,600	0	498,670	0	12,587

* Cash balance without external financing (LTL and LOC).

and

Table 7. Project cash flow calculations of the project schedule with splitting activity D.

Month	Project Expenditure	Invoice	Payment	Cash Balance *	LTL Changes	Cash Balance	Overdraft	Interest Payment
	${\mathit{E}}_{\mathit{k}}^{\mathbf{project}}$	${\mathit{I}}_{\mathit{k}}$	${\mathit{P}}_{\mathit{k}}$	${\mathit{B}}_{\mathit{k}}$	${\mathit{LTL}}_{\mathit{k}}$	${\mathit{B}}_{\mathit{k}}$	${\mathit{OD}}_{\mathit{k}}$	${\mathit{IP}}_{\mathit{k}}^{\mathit{LOC}}$
0	270,152	0	500,000	229,848	500,000	729,848	0	0
1	105,000	0	0	124,848	0	624,848	0	0
2	105,000	185,376	0	19,848	0	519,848	0	0
3	105,000	0	0	−85,152	0	414,848	0	0
4	105,000	185,376	55,698	−134,454	0	365,546	0	0
5	105,000	0	0	−239,454	0	260,547	0	0
6	630,000	1,019,567	55,698	−813,755	0	0	313,755	0
7	270,000	0	0	−1,083,755	0	0	583,755	0
8	270,000	741,503	681,342	−672,413	0	0	186,154	13,740
9	270,000	0	0	−942,413	0	0	456,154	0
10	690,000	1,297,630	472,794	−1,159,619	0	0	683,193	9833
11	94,000	0	0	−1,253,619	0	0	777,193	0
12	94,000	148,300	889,889	−457,730	−530,000	0	533,662	22,358
13	0	0	0	−457,730	0	0	533,662	0
14	0	0	464,600	464,600	0	372,328	0	16,340

* Cash balance without external financing (LTL and LOC).

present the project’s cash flow calculations without and with the splitting of activity D, respectively. Columns 2, 3, and 4 detail the monthly contractor expenditures, invoice values, and owner’s payments, respectively. Column 5 displays the contractor’s cash balance (B) without external financing, highlighting negative values at certain points. At the start of the project, the contractor receives an advanced payment (AP) of USD 500,000 from the owner, of which USD 270,152 is allocated to MOB and BND expenses. Project expenditures include direct, variable, and fixed costs, while payments are determined using activity prices as per Equation (8). Column 6 in Table 6 and Table 7 lists the LTL transactions, including principal and interest payments due after 12 months.

In the no-splitting scenario, the contractor’s maximum cash deficit reaches USD 1,383,755 by the end of month 7 (Table 6, column 5). In contrast, applying splitting reduces the peak deficit to USD 1,253,619 by the end of month 11 (Table 7, column 5). Splitting activity D not only lowers the magnitude of the cash deficit but also delays its occurrence, altering LOC usage to meet financing constraints. When external financing is used, cash balance values remain either positive or zero, as shown in column 7 of Table 6 and Table 7. Additionally, columns 8 and 9 of these tables detail the overdraft amounts and interest payments, respectively.

In the no-splitting scenario, the maximum overdraft reaches USD 883,755 in month 7, exceeding the credit limit (CL) of USD 800,000, rendering the project infeasible without splitting. By splitting activity D, the maximum overdraft decreases to USD 777,193 in month 11, allowing the contractor to complete the project within 12 months while adhering to financial constraints. However, this reduction comes at the cost of a USD 120,000 decrease in profit, as reflected in the ending balances shown in column 5 of Table 6 and Table 7. In summary, splitting proved to be a viable strategy to complete the project on time while meeting financial constraints; however, the contractor had to incur an additional cost associated with splitting activity D.

5. Sensitivity Analysis

As a test scenario, the project described in Hariga and El-Sayegh [23] is extended.

Table 8. Project parameters of the illustrative example.

Activity	Predecessors	Duration (Month)	Cost ($1000)	Stopping Cost ($1000)	Resuming Cost ($1000)	ES (Month)	EF (Month)
A	-	5	500	15	35	1	5
B	A	1	100	NA	NA	6	6
C	A	4	400	12	28	6	9
D	A	5	500	15	35	6	10
E	B	1	250	NA	NA	7	7
F	B	4	800	24	56	7	10
G	C	5	250	7.5	17.5	10	14
H	C	1	100	NA	NA	10	10
I	D	1	100	NA	NA	11	11
J	E	4	250	7.5	17.5	8	11
K	F	4	400	12	28	11	14
L	G	3	300	9	21	15	17
M	H	3	900	27	63	11	13
N	J, K, L, M, I	1	600	NA	NA	18	18

outlines the project activities, including their names, predecessors, durations, and associated costs. In this example, three scenarios are analyzed to understand how the optimal solution, derived by solving the MILP models, responds to changes in splitting costs. The scenarios considered are as follows:

• No splitting allowed.
• Splitting with an additional cost of 10%.
• Splitting with an additional cost of 20%.

To finance the projects, the contractor relies solely on an LOC with an APR of 20%. The interest is compounded monthly and paid every two months. Six different CL values are imposed on the project to examine how the optimal solution changes across the three scenarios listed above. The developed MILP models were solved using the CPLEX 12.10 solver with its default branch-and-bound (B&B) settings. The scenarios were analyzed using the following CL values:

• USD 2,000,000
• USD 1,850,000
• USD 1,800,000
• USD 1,750,000
• USD 1,700,000
• USD 1,650,000

Splitting costs are allocated such that 30% of the cost is incurred when stopping the activity and 70% is incurred when resuming it, as illustrated in Table 8 for the scenario with a 10% splitting cost. Table 8 also provides the early start (ES) and early finish (EF) times for each activity. Based on these early times, the minimum project duration is determined to be 18 months. However, the model allows for an extension of up to 22 months to meet the CL limit.

In this example, we use the same bidding and contracting parameters as in the previous example, as specified in Table 2 and Table 5, with no extension penalties applied. The bidding cost calculations for this scenario are detailed in

Table 9. Bidding cost calculations of the illustrative example.

Parameter	Value ($)
Sum of direct costs (DCsum)	5,450,000
Sum of variable overhead cost (VOCsum)	545,000
Sum of fixed overhead cost (FOCsum)	900,000
Markup value (MPV)	1,419,000
Bonding cost (BND)	170,280
Contract bidding price (CBP)	8,684,280

.

For CL= USD 2,000,000, the MILP solution shows that the contractor can finish the project based on the CPM solution without the need for splitting, extension, or changing activities’ start times, as shown in Figure 2a. When decreasing CL to USD 1,850,000, some activities utilized their float time, allowing the project to be completed on time without the need for an extension or splitting, as illustrated in Figure 2b. Comparing Figure 2a with Figure 2b, it can be observed that only the first four activities started at their ES times.

Figure 2c shows the case where splitting is allowed for CL = USD 1,800,000, considering both 10% and 20% additional splitting costs. The MILP solution indicates that splitting activity J is more economical than extending the project duration by one month when splitting is not allowed, as illustrated in Figure 2d. These solutions highlight the cost-efficiency of activity splitting under tighter financial constraints.

When further reducing the CL to USD 1,750,000 or below, the same solutions are obtained for the scenarios of no splitting and splitting with an additional 20% cost. However, splitting with an additional cost of 10% presents different solutions. Figure 2e illustrates that with a 10% extra splitting cost, it is still preferable to split activity J, although it requires extending the project by one month. In the no-split case, the project must be extended by two months, and for the 20% split cost scenario, it is more economical not to split any activity, following the no-split schedule, as shown in Figure 2f.

For CL = USD 1,700,000 and CL = USD 1,650,000, the no-split and 20% split cost schedules necessitate extending the project by three and four months, respectively, as depicted in Figure 2h,j. In contrast, for these two CL values, the 10% split cost scenario allowed the project delay to be limited to just two months, as shown in Figure 2g,i. It is important to note that the selection of activities to split changes as the CL value is adjusted.

Table 10. Comparison between project scheduling with and without allowing splitting.

Credit Limit ($1000)	Extension & Splitting (10%)			Extension & Splitting (20%)			Extension & No Splitting
	Total Cost ($)	Split Activities	Extension (Month)	Total Cost ($)	Split Activities	Extension (Month)	Total Cost ($)	Extension (Month)
1800	7,290,270	J	0	7,315,270	J	0	7,315,280	1
1750	7,340,170	J	1	7,365,280	None	2	7,365,280	2
1700	7,405,180	C	2	7,415,270	None	3	7,415,270	3
1650	7,415,280	D	2	7,465,270	None	4	7,465,270	4

presents the results for the 12 studied cases that showed different solutions. Columns 2–4 display the total costs, selected activities for splitting, and duration extensions for the 10% splitting cost scenario. Columns 5–7 provide the same information for the 20% splitting cost scenario, while columns 8 and 9 show the results when no splitting is allowed.

The case solutions reveal a clear trade-off between splitting costs and delay costs, as demonstrated by the different solutions obtained for the various scenarios. For the 20% splitting cost, it was advantageous to split activity J when CL = USD 1,800,000; however, as the CL was further reduced, extending the project duration proved to be more cost-effective. In contrast, the 10% splitting cost scenario consistently showed that splitting was preferable across the different tested CL values, although different activities were selected for splitting depending on the CL value. The decision on which activity to split was influenced by both the amount and the timing of the CL limit surpassed.

6. Conclusions

FBS has been extensively studied as a method to adjust the start times of project activities, with the goal of maintaining the contractor’s cash deficit within the credit limits imposed by lines of credit. If these adjustments do not consume the activities’ total float, the project can stay on schedule; otherwise, delays become unavoidable. AS presents a valuable approach to further reduce cash deficits, allowing contractors to meet credit limits with minimal impact on project timelines. However, suspending and resuming activities can incur additional costs. Therefore, identifying which activities to split and the optimal timing for such splits, when allowed, is crucial for contractors to complete projects within the assigned deadlines or, at the very least, to minimize costs related to splitting and delay penalties.

This paper presents an MILP model designed to tackle FBS challenges by incorporating AS. The model combines rescheduling and activity splitting strategies to effectively manage the contractor’s cash deficit while minimizing project completion delays. The results demonstrate that FBS can be significantly improved by leveraging AS, leading to reduced delays and more optimized project timelines.

The concept of activity splitting is illustrated using a simplified example that includes four activities and various financing options, with one activity eligible for splitting at an additional cost. The analysis reveals that splitting can be a crucial strategy for keeping the project on schedule when financial constraints are tight. The proposed MILP model is then applied to a case study exploring different scenarios with varying splitting costs and credit levels. The MILP results demonstrate that the optimal solution effectively balances splitting costs against delay penalties, providing a robust trade-off under different financial conditions. The findings also indicate that the decision on which activity to split and when to do so can vary depending on the contractor’s cash flow status. Furthermore, using activity splitting as a strategy to address financial challenges may sometimes be complemented by extending the project timeline to ensure feasibility.

To strengthen the conclusions about optimal activity splitting, it is crucial to analyze and solve additional scenarios that closely mimic real-world conditions using the proposed MILP model. Given that the computational time required to solve the MILP model significantly increases with the number of activities and the complexity of the project network, developing a heuristic or meta-heuristic algorithm is recommended as a practical approach to generate faster solutions to this problem.

Advancing the practical applications of activity splitting could significantly enhance the utility of widely used project management tools like MS Project^® and Primavera^™. Although these platforms already support activity splitting for resource leveling, future developments could focus on incorporating features designed specifically to address cash flow constraints. By enabling construction firms to simulate activity splitting scenarios under financial limitations and integrate third-party financing tools, such advancements would provide a robust framework for predicting cash requirements and optimizing project schedules. These innovations would bridge the gap between theoretical approaches and real-world practices, delivering actionable solutions for better financial and schedule management.