A New Bi-Objective Model for Resource-Constrained Project Scheduling and Cash Flow Problems with Financial Constraints under Uncertainty: A Case Study

Mohammad Ghasemi; Reza Shahabi-Shahmiri; Seyed-Ali Mirnezami

arxiv: 2509.00002 · v1 · pith:QECV3S5Mnew · submitted 2025-08-06 · 🧮 math.GM

A New Bi-Objective Model for Resource-Constrained Project Scheduling and Cash Flow Problems with Financial Constraints under Uncertainty: A Case Study

Seyed-Ali Mirnezami , Mohammad Ghasemi , Reza Shahabi-Shahmiri This is my paper

Pith reviewed 2026-05-21 23:25 UTC · model grok-4.3

classification 🧮 math.GM

keywords resource-constrained project schedulingcash flow optimizationfinancial constraintsuncertainty modelingmulti-objective optimizationinterval-valued fuzzy numbersmixed integer linear programmingoil and gas construction project

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The pith

A new bi-objective model maximizes final cash flow while shortening project duration in uncertain environments with financial constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a multi-mode multi-objective linear programming model for resource-constrained project scheduling that treats cash flow and project length as conflicting goals. It incorporates payment delays, initial capital, interest rates, credit limits, and credit line usage while representing uncertain parameters through interval-valued fuzzy numbers. An extended IVF-TH solution method converts the bi-objective problem into a single-objective one that is solved with standard MILP solvers. The approach is demonstrated on a real oil-and-gas construction project, followed by sensitivity analysis and comparison against earlier multi-objective techniques.

Core claim

The authors present a new comprehensive multi-mode multi-objective linear programming model with two conflicting objectives, which are maximizing final cash flow for profit optimization and shortening the duration of project execution, considering payments delays, project finance constraints, initial capital, different types of interest rates, credit limit to assuage financial distress, and credit line usage, in an uncertain environment. A new extended interval valued fuzzy - Torabi and Hassini (IVF-TH) approach is proposed to tackle the problem, the mixed integer linear programming model is solved with CPLEX, and the model is illustrated with a real construction project in the oil and gas行业

What carries the argument

The bi-objective mixed-integer linear program that tracks net cash position over time and is solved by the extended IVF-TH method to generate trade-off schedules under interval-valued fuzzy uncertainty.

If this is right

Project managers can simultaneously improve liquidity and meet tighter deadlines by choosing points on the computed trade-off curve.
Credit limits and interest-rate assumptions directly alter feasible start times and resource allocations in the generated schedules.
Sensitivity analysis on the case study shows how changes in initial capital or payment delays shift the optimal cash-flow versus duration frontier.
The extended IVF-TH method produces solutions that outperform standard multi-objective techniques on both the real project and larger test instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same structure could be applied to infrastructure or software projects that face similar cash-flow timing risks.
Embedding the model in a rolling-horizon framework would allow updates as actual payments and durations become known.
Adding stochastic programming or robust optimization variants could test whether the fuzzy representation is the most robust choice for this class of problems.

Load-bearing premise

Uncertainties in activity durations, costs, and payments can be represented accurately enough by interval-valued fuzzy numbers so that the extended IVF-TH procedure yields stable scheduling decisions.

What would settle it

If re-running the model on the oil-and-gas case study with different uncertainty representations or with actual realized cash flows produces schedules whose final cash position or duration deviates substantially from the reported trade-off front.

Figures

Figures reproduced from arXiv: 2509.00002 by Mohammad Ghasemi, Reza Shahabi-Shahmiri, Seyed-Ali Mirnezami.

**Figure 1.** Figure 1: Periodic and daily costs representation There are circumstances in which start and end dates of activities are not in the same period. For instance, the start date of activity B lies in the first period and its end date is in the second period. Meaning that costs of activity B are divided into the first and second periods. Even though costs of activity B are in two periods, considering these costs periodic… view at source ↗

**Figure 2.** Figure 2: Project cash flow network of the proposed model Two types of interest rates for these two loans are exerted using: 𝐿𝑇𝐿 (1 + 𝛾) 30 (1) 𝑆𝑇𝐿𝑦−1 (1 + 𝛿) 30 (2) Where 𝐿𝑇𝐿 is the long-term loan, 𝑆𝑇𝐿𝑦−1 indicates the short-term loan of the previous period, and 𝛾 as well as 𝛿 are interest rates on long-term loan and short-term loan respectively. Furthermore, two different upper bounds are considered for them, mean… view at source ↗

**Figure 3.** Figure 3: A normalized interval-valued triangular fuzzy (NIVTF) number 𝐴̃ ̃ (Stanujkic, 2015) 4.1. A New Extended IVF-TH Approach Since the proposed mixed-integer linear programming model is regarded in a fuzzy environment, an efficient approach is exerted in accordance with Jiménez et al. (2007) and Parra et al. (2005) to convert the model into the equivalent auxiliary crisp model. According to Jiménez et al. (2007… view at source ↗

**Figure 4.** Figure 4: An overview of proposed solution approach With respect to the inherent ambiguity in the boundaries of generalized fuzzy numbers, activity durations (𝑑̃ ̃ 𝑖𝑚), required renewable resources (𝑅̃̃ 𝑖𝑘𝑚), and required nonrenewable resources (𝑊̃̃ 𝑖𝑙𝑚) are considered as NIVTF numbers in this study. NIVTF numbers enable decision-makers to deal with lack of information and uncertainties by relying on intuition and … view at source ↗

**Figure 5.** Figure 5: An illustration of the project 5.2. Implementation As it was mentioned before, although several projects have been implemented by the company, one of them with twenty-two activities is taken into account in this paper. This project, that is steam generation, consists of different activities, including construction, precommissioning, start-up, and performance test, for boilers B, C, D, and E. However, only … view at source ↗

**Figure 6.** Figure 6: WBS of project [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: The project Gantt chart Similarly, the project Gantt chart is illustrated in Fig. 8A and Fig.8B. However, the difference between this Gantt chart and the last one in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 9.** Figure 9: The considered interest rates on excess cash, delayed payments, long [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

read the original abstract

Owing to the importance of project cash flow, which comprises an entire history of all cash inflows and cash outflows, to economic survival of firms, it is vital to coping with project scheduling issues considering resource constraints in circumstances involving cash flow. Furthermore, since appropriate project management is subject to the innate uncertainties involved in most projects, they are required to be appraised respecting their profound impact. In this paper, a new comprehensive multi-mode multi-objective linear programming model with two conflicting objectives, which are maximizing final cash flow for profit optimization and shortening the duration of project execution, considering improving assumptions, that is, payments delays, project finance constraints, initial capital, different types of interest rates, credit limit to assuage financial distress, credit line usage, is presented in an uncertain environment. Since the model is considered as multi-objective with uncertain parameters, a new extended interval valued fuzzy - Torabi and Hassini (IVF-TH) approach is proposed to tackle the problem. The presented mixed integer linear programming (MILP) model is solved applying CPLEX solver. In addition, a real construction project in oil and gas industry is presented as a case study to illustrate the model applications. Ultimately, for the purpose of assessing the outcomes, a sensitivity analysis is implemented, and the performance of the proposed solution approach is compared to the previous multi-objective optimization methods using both case study and large problem instances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper folds payment delays, credit limits, and interest rates into a multi-mode RCPSP model under interval-valued fuzzy uncertainty, solves it with an extended Torabi-Hassini method, and tests the whole thing on a real oil-and-gas project.

read the letter

The main takeaway is a bi-objective MILP that adds concrete financial constraints—payment delays, initial capital, different interest rates, credit limits, and credit line usage—to multi-mode resource-constrained project scheduling, all under uncertainty, then solves it with an extended interval-valued fuzzy Torabi-Hassini approach and shows results on an actual construction case. The model aims to maximize final cash flow while shortening duration. They use CPLEX and run sensitivity checks plus comparisons to earlier multi-objective methods. That combination of financial detail and a real-world example is the practical contribution. It gives operations-research readers working on construction projects a usable template for cash-flow aware scheduling. The case study and sensitivity work add some grounding that pure theoretical papers often lack. The citation pattern follows standard RCPSP and fuzzy multi-objective lines without obvious holes in what is shown. The full text appears to deliver the promised MILP formulation and solver steps, which moves it past pure abstract claims. The soft spot sits in the extended IVF-TH method itself. The stress-test note is on target: if the extension keeps adjustable compensation coefficients or satisfaction weights from the original Torabi-Hassini approach, then the reported mode selections and start times could shift with those settings rather than being fixed by the financial constraints alone. The paper claims unbiased trade-offs, but without explicit checks that the scheduling variables stay stable across reasonable parameter values, the case-study outcomes risk looking tuned. That issue is not fatal for an applied paper, but it is the part a referee would press hardest. This is the sort of work that project-scheduling researchers and construction-management analysts will want to read, especially if they already handle fuzzy or multi-objective tools. A reader focused on cash-flow integration in RCPSP gets the clearest value. It is coherent and applied enough to deserve a serious referee, even though the fuzzy extension will need tighter justification on invariance. I would send it for peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a new multi-mode, multi-objective mixed-integer linear programming (MILP) model for resource-constrained project scheduling problems that integrates cash flow optimization and financial constraints under uncertainty. The two objectives are maximizing the final cash flow and minimizing project duration. The model accounts for payment delays, initial capital, various interest rates, credit limits, and credit line usage. Uncertainties are handled via interval-valued fuzzy numbers, and an extended interval-valued fuzzy Torabi-Hassini (IVF-TH) method is introduced to generate trade-off solutions. The approach is demonstrated on a real oil-and-gas construction project case study, solved with CPLEX, and evaluated through sensitivity analysis and comparisons with prior multi-objective methods.

Significance. If the extended IVF-TH method produces scheduling decisions that are robust to its internal parameters and the case-study results hold under the stated financial constraints, the work could provide a useful integrated framework for handling cash-flow and uncertainty in RCPSP. The inclusion of multiple financial elements (credit limits, interest rates, delays) and the real-world validation are positive features. However, the contribution rests on an extension of an existing aggregation technique rather than a parameter-free derivation, so the overall significance is moderate pending verification that the reported trade-offs are not sensitive to method-specific choices.

major comments (2)

The central claim that the extended IVF-TH approach yields unbiased trade-off solutions whose mode selections and start-time decisions are independent of post-hoc parameter choices is load-bearing for the paper's contribution. The manuscript must explicitly state the full formulation of the extended IVF-TH (including any compensation coefficient, satisfaction-level weights, or additional parameters) and demonstrate, either analytically or via additional experiments, that varying these parameters does not alter the optimal project schedule in the case study.
In the case-study results and sensitivity analysis, the reported solutions for the oil-and-gas project should be accompanied by a table or figure showing how the selected modes and activity start times change (or remain stable) when the IVF-TH parameters are varied within reasonable ranges; absence of such a check leaves open the possibility that the claimed financial-constraint effects are confounded by tuning.

minor comments (3)

The abstract uses the phrase 'improving assumptions'; replace with a clearer description of the specific financial features added to the model.
Ensure consistent notation for interval-valued fuzzy numbers and define all acronyms (MILP, IVF-TH, etc.) on first use in the main text.
The comparison with 'previous multi-objective optimization methods' should specify exactly which methods and metrics are used, and include the same large problem instances for all methods to allow direct performance assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. The points raised concerning the transparency and robustness of the extended IVF-TH method are well taken, and we will revise the manuscript to address them directly. Our point-by-point responses follow.

read point-by-point responses

Referee: The central claim that the extended IVF-TH approach yields unbiased trade-off solutions whose mode selections and start-time decisions are independent of post-hoc parameter choices is load-bearing for the paper's contribution. The manuscript must explicitly state the full formulation of the extended IVF-TH (including any compensation coefficient, satisfaction-level weights, or additional parameters) and demonstrate, either analytically or via additional experiments, that varying these parameters does not alter the optimal project schedule in the case study.

Authors: We agree that the full formulation must be stated explicitly to support the contribution. In the revised manuscript we will provide the complete mathematical formulation of the extended IVF-TH method, including the compensation coefficient, satisfaction-level weights, and all other parameters. To verify independence of the scheduling decisions, we have performed additional experiments on the oil-and-gas case study in which the key parameters were varied over reasonable ranges. These experiments show that the optimal mode selections and activity start times remain unchanged, although objective values exhibit minor adjustments. The formulation and the new experimental results will be added to the methods and sensitivity-analysis sections. revision: yes
Referee: In the case-study results and sensitivity analysis, the reported solutions for the oil-and-gas project should be accompanied by a table or figure showing how the selected modes and activity start times change (or remain stable) when the IVF-TH parameters are varied within reasonable ranges; absence of such a check leaves open the possibility that the claimed financial-constraint effects are confounded by tuning.

Authors: We accept the referee's recommendation. The revised manuscript will include a new table (and, if space permits, a supplementary figure) that reports the selected modes and start times obtained for multiple values of the IVF-TH parameters on the oil-and-gas instance. This table will confirm the stability already observed in our additional experiments and will make clear that the reported financial-constraint effects are not an artifact of parameter tuning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper explicitly formulates a multi-mode multi-objective MILP model incorporating payments delays, financial constraints, interest rates, credit limits, and interval-valued fuzzy uncertainties for the two objectives (max final cash flow, min duration). It proposes an extension of the existing Torabi-Hassini method to convert the fuzzy bi-objective problem into crisp equivalents, then solves the resulting MILP via CPLEX on a real oil-and-gas case study with sensitivity analysis and comparisons to prior methods. No load-bearing step reduces by construction to fitted inputs, self-definitional loops, or self-citation chains; the model equations, objective functions, and solver outputs stand independently of any internal parameter fitting that would force the reported scheduling decisions. The approach rests on standard MILP assumptions and an external fuzzy aggregation technique rather than tautological redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; therefore free parameters, axioms and invented entities cannot be audited from the text. The model presumably inherits standard MILP axioms and fuzzy-set assumptions from the Torabi-Hassini literature.

pith-pipeline@v0.9.0 · 5800 in / 1249 out tokens · 35895 ms · 2026-05-21T23:25:29.458598+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

new extended interval valued fuzzy - Torabi and Hassini (IVF-TH) approach ... normalized interval-valued triangular fuzzy (NIVTF) numbers
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

maximizing final cash flow ... minimizing project duration ... credit limit, interest rates

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

https://doi.org/10.1080/0951192X.2014.880948 He, Y., Jia, T., & Zheng, W. (2024). Simulated annealing for centralised resource - constrained multiproject scheduling to minimise the maximal cash flow gap under different payment patterns. Annals of Operations Research , 338(1), 115 -149. https://doi.org/10.1007/s10479-023-05580-3 He, Y., Jia, T., & Zheng, W...

work page doi:10.1080/0951192x.2014.880948 2014
[2]

alphorn of uncertainty

https://doi.org/10.1016/j.fss.2007.08.010 30 Van Peteghem, V., Vanhoucke, M., 2014. An experimental investigation of metaheuristics for the multi-mode resource-constrained project scheduling problem on new dataset instances. European J. Oper. Res. 235 (1), 62 –72. https://doi.org/10.1016/j.ejor.2013.10.012 Wang, C. W., Lee, C. C., & Wu, L. T. (2023). The ...

work page doi:10.1016/j.fss.2007.08.010 2007

[1] [1]

https://doi.org/10.1080/0951192X.2014.880948 He, Y., Jia, T., & Zheng, W. (2024). Simulated annealing for centralised resource - constrained multiproject scheduling to minimise the maximal cash flow gap under different payment patterns. Annals of Operations Research , 338(1), 115 -149. https://doi.org/10.1007/s10479-023-05580-3 He, Y., Jia, T., & Zheng, W...

work page doi:10.1080/0951192x.2014.880948 2014

[2] [2]

alphorn of uncertainty

https://doi.org/10.1016/j.fss.2007.08.010 30 Van Peteghem, V., Vanhoucke, M., 2014. An experimental investigation of metaheuristics for the multi-mode resource-constrained project scheduling problem on new dataset instances. European J. Oper. Res. 235 (1), 62 –72. https://doi.org/10.1016/j.ejor.2013.10.012 Wang, C. W., Lee, C. C., & Wu, L. T. (2023). The ...

work page doi:10.1016/j.fss.2007.08.010 2007