# A Practical and Robust Execution Time-Frame Procedure for the Multi-Mode Resource-Constrained Project Scheduling Problem with Minimal and Maximal Time Lags

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. The RCPSP and Its Variations

#### 2.1.1. RCPSP

#### 2.1.2. MRCPSP

#### 2.1.3. MRCPSP/Max

#### 2.2. Objectives Solved and Solution Procedures

_{j}, and the deviation of an activity’s completion time (C

_{j}) from a given due date (d

_{j}). Tardiness (T

_{j}), the positive value of lateness (C

_{j}≥ d

_{j}) and earliness (E

_{j}), the negative value of lateness (C

_{j}≤ d

_{j}), have been researched within the last 15 years by [12].

#### 2.3. Measuring Uncertainty

_{i}, represents the set of unfavorable events for activity i. The value δt is a relevant time interval and is dependent on the nature of the project in the sense that its value is negatively correlated to the level of project risk. From a practitioner’s point of view, we can think of δt as checkpoints or meetings in which the project manager receives updated project status information. The higher the risk of a schedule disruption, the more control needs to be exerted, requiring more frequent updates. For this study, we fixed the value of δt = 1 based on a sensitivity analysis performed in our previous research. The slack, s

_{i}, is determined by the standard forward (backward) recursion procedure. The values d

_{iu}, d

_{i}and d

_{il}respectively stand for the longest, most probable and shortest possible durations, while EFT and LFT respectively represent the earliest and latest finish times with the corresponding durations indexed. E

_{i}results as the time difference between $LF{T}_{{d}_{i}}$ and $EF{T}_{{d}_{iu}}$, as shown in Equation (3).

_{iu}, d

_{i}and d

_{il}, the more uncertain the scheduler considers the activity to be, and thus, the higher the value of its entropy. Figure 1 gives a graphical representation of E

_{i}and δt. Here, the solid grey bar represents an activity scheduled with a most probable duration d

_{i}, and slack, s

_{i}, is the time interval between $EF{T}_{{d}_{i}}$ and $LF{T}_{{d}_{i}}$. The white bar shows the $EF{T}_{{d}_{il}}$ and $EF{T}_{{d}_{iu}}$; E

_{i}represents the unfavorable events; and the relevant time interval parameter defined by the user is shown as δt.

#### 2.4. Schedule Robustness

_{i}, frac × d

_{i})). The user can define the value of frac; however, based on a sensitivity analysis performed in the authors’ previous study [36], the value of frac is fixed to 0.25 for the experiments performed in this study. Section 3 explains the use of this robustness measure in the proposed method.

## 3. Model and Methodology

#### 3.1. Mathematical Formulation of MRCPSP/Max

_{i}$\in $ M out of the total available for every activity M

_{i}= {1, …, |M

_{i}|}. The duration of every activity, ${d}_{i{\mu}_{i}}$, depends on the mode µ selected, and the beginning and end of every activity is denoted by S

_{i}and C

_{i}correspondingly. Therefore, if activity i begins at time S

_{i}, it will continue to be executed in mode µ

_{i}throughout time t = [S

_{i}, S

_{i}+ ${d}_{i{\mu}_{i}}$], so if S

_{0}:= 0, then the duration of the project is given by S

_{n}

_{+1}. Between the start times of activities i and j, a time lag p can be included. This time lag is dependent on the mode µ

_{i}of activity i and µ

_{j}of activity j (i ≠ j) and can be maximal ${p}_{i{\mu}_{i}j{\mu}_{j}}^{max}$ or minimal ${p}_{i{\mu}_{i}j{\mu}_{j}}^{min}$.

_{i}consumes ${r}_{i{\mu}_{i}k}^{\rho}$ type k renewable per period of time out of a total ${R}_{k}^{\rho}$ renewable resources available in the project and ${r}_{i{\mu}_{i}k}^{v}$ nonrenewable resource out of a total ${R}_{k}^{v}$. Dummy activities consume no resources and have duration = 0.

_{i}) assigns only one mode µ to every activity i for it to be executed. Each mode vector M has an associated project, N(M), obtained by choosing the durations, resources and time lags corresponding to the modes of M. A start time vector S = (S

_{i}) assigns to each activity i exactly one point in time t ≥ 0 as start time S

_{i}with S

_{0}:= 0.

_{i}≤ t < S

_{i}+ d

_{iµ}} denoting the set of activities being executed at a time t for a schedule (M, S), the MRCPSP/max can be stated as follows:

_{il}, d

_{i}and d

_{iµ}, respectively. Column ${R}_{k}^{v}$. shows the amount of renewable resources consumed by the activities in each of their modes. Finally, the column “arc weight” shows the time lags when going from mode $\mu $

_{i}of activity i to mode $\mu $

_{j}of activity j. If the activity i is succeeded by the final dummy activity, only one arc weight is needed, i.e., Activities 3, 4 and 5. Finally, notice that Activity 4 is also a precedent for Activity 2, which, as mentioned previously, could not occur in the case of the MRCPSP. This indicates that Activity 4 is a successor and predecessor of Activity 2 and exemplifies the loops that make the MRCPSP/max both more realistic and complicated to solve.

#### 3.2. Solution Procedure

_{1}, MS

_{2}and MS

_{R}respectively represent the makespan for Stage 1, Stage 2 and Stage 3. The rest of Section 3 presents the details for each of the stages. First, three tools used in two of the stages are presented: a discrete version created from a powerful optimization algorithm, followed by rules that help select execution modes (mode selection rules) and rules that help prioritize the activities (activity priority rules). Finally, how these tools are integrated into the 3-stage procedure is introduced.

#### 3.2.1. Discrete Artificial Bee Colony

- Initialization: n-dimensional solutions are generated randomly throughout the search space. After the initialization phase, the algorithm is repeated a Maximum Number of Cycles (MNC), executing three improvement phases in each cycle:
- Employed bees phase: Each solution (food source) is assigned a bee, which thus becomes an employed bee. This bee seeks to improve the solution by applying modifications (local search operators), and the quality (nectar) of the obtained solution is later compared to that of the original solution. If the modified solution is better, the old solution is forgotten, and the new solution is memorized. The employed bee will keep modifying the assigned food source until either a better solution is found or the abandonment limit is reached.
- Onlooker bees phase: After all employed bees have finished their local search cycle, they share the nectar information of their food sources with the onlookers, each of which then selects a food source for further exploration based on the following probability:$${p}_{i}=\frac{{f}_{i}}{{\sum}_{i=1}^{SN}{f}_{i}}$$The onlookers tend to choose a food source i with higher probability p
_{i}among SN total food sources, each with a fitness f_{i}. - Scout bees phase: If the employed and onlooker bees cannot improve a solution after a number of trials (i.e., they reach the abandonment limit), a scout bee searches for a new food source (i.e., a new solution is generated randomly), and the previous food source is abandoned.

#### 3.2.2. Mode Selection Rules

#### 3.2.3. Activity Priority Rules

#### 3.2.4. Three-Stage Procedure Execution Time-Frame

Algorithm 1: Repeat until all instances are solved |

Stage 1: Minimize Makespan (Upper Bound Makespan, MS_{1}) |

Initialization Phase |

While i < population |

Evaluate Mode Selection Rules (MSR) |

Evaluate Activity Priority Rules (APR) |

End |

Repeat until MNC |

Employed Bees Phase |

Onlooker Bees Phase |

Scout Bees Phase |

End |

Stage 2: Compute schedule’s entropy (Upper Bound Makespan) |

Stage 3: Maximize Robustness |

Initialization Phase |

While i < population |

Evaluate Mode Selection Rules (MSR) |

Evaluate Activity Priority Rules (APR) |

End |

Repeat until MNC |

Employed Bees Phase |

Onlooker Bees Phase |

Scout Bees Phase |

End |

End |

#### Stage 1: Lower Bound Makespan

_{1}in Figure 3) and used as the input for Stage 2.

#### Stage 2: Upper Bound Makespan

_{il}, d

_{i}, and d

_{iu}, respectively) estimated for every activity. Here, Equation (3) determines the set of unfavorable events, and finally, Equation (2) is used for the schedule’s total entropy. As a short example, assume there is an activity whose selected execution mode requires 7 time units. Furthermore, the scheduler determined that the activity’s most probable duration (d

_{i}) was 9 time units and that the longest possible duration (d

_{iu}) was 10 time units. This activity happens to be part of the critical path and, therefore, has slack s = 0. Furthermore, suppose the relevant time interval δt is set to 1. Therefore, after using Equation (2), the entropy for this activity is 1 time unit. To determine the complete schedule’s entropy, this same procedure is applied to all project activities. Nonetheless, the total time added to the current makespan is not necessarily equal to the sum of the entropy for all activities. If activities have slack greater than their entropy values, these activities will simply be shifted without affecting the overall makespan. The new entropy-containing makespan will serve as the upper bound for the execution time-frame and will also serve as an input for Stage 3 (MS

_{2}in Figure 3).

#### Stage 3: Robust Makespan

_{1}≤ MS

_{R}≤ MS

_{2}for all of the initial solutions i). Second, the objective function in Stage 3 is not to minimize the makespan, but rather to maximize the makespan’s robustness. Equation (9) shows the objective function solved in Stage 3:

_{i}denotes the number of immediate successors of activity i and s

_{im}represents the slack of activity i if executed in mode μ. Equation (9) establishes that the objective is to maximize the robustness measure, which is based on the slack of each activity in each available mode.

## 4. Computational Results

#### 4.1. Parameters

_{1}); this will serve as the lower bound of the execution time-frame. In Stage 2, the entropy Equation (1) determines the upper bound (MS

_{2}). Finally, in Stage 3, the third makespan is computed also using the discrete ABC algorithm, but in this stage, seeking to maximize robustness (Equation (9)), while keeping the makespan lower than the upper bound, though not necessarily higher than the lower bound.

#### 4.2. Results

_{s}− M*) M*, where M* represents the reference makespan when comparing the results of Stage 1 and Stage 3 (M* can be either a confirmed optimal or the Best-Known Makespan), and M

_{s}denotes the makespan of the current stage. Furthermore, S1, S2 and BKO represent the makespan of Stage 1, Stage 2 and the Best-Known Optima (BKO), respectively.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ABC | Artificial Bee Colony |

ACOSS | Ant Colony Optimization and Scatter Search |

ACTIM | Activity Time |

ANGEL | Ant Colony and Genetic Algorithm with Local Search |

APR | Activity Priority Rules |

BKO | Best-Known Optima |

BMAP | Best Mode Assignment Problem |

CPM | Differential Evolution |

DE | Differential Evolution |

EDA | Estimation of Distribution Algorithm |

EFT_{i} | Earliest Finish Time of activity i |

GCUMRD | Greatest Cumulative Resource Demand |

LFT_{i} | Latest Finish Time of activity i |

LNRJ | Least Non-Related Jobs |

LPSRD | Least Product Sum of Resource and Duration |

LRP | Least Resource Proportion |

LRS | Least sum of Non-renewable Resource |

LTRU | Least Total Resource Usage |

MIS | Most Immediate Successors |

MNC | Maximum Number of Cycles |

moEDA | Multi-Objective Estimation Distribution Algorithm |

MRCPSP | Multi-mode Resource Constrained Project Scheduling Problem |

MRCPSP/max | Multimode RCPSP with minimal and maximal time lags |

MSi | Makespan of Stage i |

MSR | Mode Selection Rules |

MTS | Most Total Successors |

NP-Hard | Non-deterministic polynomial time hard |

NPV | Net Present Value |

PERT | Program Evaluation and Review Technique |

PSO | Particle Swarm Optimization |

RCPCP | Resource Constrained Project Scheduling Problem |

ROT | Resource Over Time |

RSM | Resource Scheduling Method |

SA | Simulated Annealing |

SFLA | Shuffled Frog-Leaping Algorithm |

SFM | Shortest Feasible Mode |

SGS | Serial Generation Scheme |

SLK | Minimum Slack First |

TS | Tabu Search |

WRUP | Weighted Resource Utilization Ratio and Precedence |

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Activity (i) | Modes ($\mathit{\mu}$_{i}) | Duration (d_{il};d_{i};d_{iu}) | ${\mathit{R}}_{\mathit{k}}^{\mathit{v}}$ | Arc Weights (δ) |
---|---|---|---|---|

1 | 1 | - | - | (5,3,1); (4,5,5) |

2 | 1 | 3;3;4 | 2 | (4,1,4); (1,1,7) |

2 | 2;3;4 | 3 | (2,1,4); (1,6,3) | |

3 | 1;1;2 | 4 | (−1,2,0); (2,3,7) | |

3 | 1 | 2;3;5 | 1 | (5) |

2 | 2;2;2 | 3 | (1) | |

3 | 1;3;5 | 5 | (4) | |

4 | 1 | 3;3;4 | 3 | (1,4,9); (3) |

2 | 2;3;4 | 3 | (7,3,2); (1) | |

3 | 1;2;3 | 4 | (1,2,4); (2) | |

5 | 1 | 2;3;3 | 4 | (1) |

2 | 3;3;4 | 5 | (2) | |

3 | 1;2;2 | 7 | (3) | |

6 | 1 | - | - | - |

Priority Rule | Description |
---|---|

SFM (Shortest Feasible Mode) | Find the feasible mode combination for which the makespan is minimal |

LRP (Least Resource Proportion) | Choose the mode that leads to the smallest value of the criterion, max($\frac{{r}_{i\mu k}^{\rho}}{{K}_{\rho}}$)$\forall \mu $ |

LPSRD (Least Product Sum of Resource and Duration) | For each activity, choose the execution mode that has the minimum product sum of non-renewable resource usage and its corresponding mode duration, $\mathrm{min}{\sum}_{k=1}^{{K}^{v}}\left({r}_{i\mu k}^{v}*{d}_{i\mu}\right)\forall \mu $ |

LTRU (Least Total Resource Usage) | Choose the execution mode that requires the least total non-renewable usage, $\mathrm{min}{\sum}_{k=1}^{{K}^{v}}{r}_{i\mu k}^{v}\forall \mu $ |

LRS (Least sum of Non-renewable Resource) | Choose the execution mode that requires the least sum of the ratio of the non-renewable consumption to its corresponding resource limitation, $\mathrm{min}{\sum}_{k=1}^{{K}^{v}}\frac{{r}_{i\mu k}^{v}}{{R}_{k}^{v}}\forall \mu $ |

Priority Rule | References | Description |
---|---|---|

max ACTIM | [51] | CPM − LST_{i} |

max GCUMRD | [52] | The sum of the renewable resource demand of the activity considered and the renewable resource demands of all its immediate successors |

max MTS | [53] | $\left|\overline{{S}_{i}}\right|$ the total number of successors for activity i |

max MIS | [53] | $\left|{S}_{i}\right|$ the number of immediate successors for activity i |

max ROT | [54] | $\frac{{\sum}_{k=1}^{{K}^{\rho}}\frac{{r}_{ik}^{\rho}}{{R}_{k}^{\rho}}}{{d}_{i}}$ sum of the ratio of the renewable resource requirement over the resource availability divided by the activity duration for activity i |

max WRUP | [55] | $0.7*\left|{S}_{i}\right|+0.3*\frac{{\sum}_{k=1}^{{K}^{\rho}}\frac{{r}_{ik}^{\rho}}{{R}_{k}^{\rho}}}{{d}_{i}}$, a weighted sum of the number of immediate successors and an average resource use over all renewable resource types |

min EFT | [56] | EFT_{i} |

min LFT | [56] | LFT_{i} |

min SLK | [56] | LFT_{i} − EFT_{i} |

min LNRJ | [52] | $\left|\overline{N{S}_{i}}\right|$ the total number of activities that are not precedence related with activity i |

min RSM | [57] | d_{ij} = max[0, (EFT_{i} − LST_{j})] |

Parameter | Value |
---|---|

Population size | 30 |

Abandonment limit | 5 |

MNC | 20 |

δt (relevant time interval) | 1 |

frac | 0.25 |

Benchmark Set | MM30 | MM50 | MM100 | Overall Average | |
---|---|---|---|---|---|

Optima Found | 260 | 123 | 84 | ||

Stage 1 | Avg. Dev. vs. BKO | 0.18% | 4.57% | 4.42% | 3.06% |

Stage 2 | Avg. Dev. vs. BKO | 9.69% | 10.13% | 8.50% | 9.44% |

Avg. Dev. vs. S1 | 6.02% | 5.84% | 4.27% | 5.38% | |

Stage 3 | Avg. Dev. vs. BKO | 5.04% | 5.39% | 4.37% | 4.93% |

Avg. Dev. vs. S1 | 1.10% | 0.79% | −0.12% | 0.59% | |

Avg. Dev. vs. S2 | −5.36% | −5.48% | −4.62% | −5.16% |

H µ_{0}:_{BKO} = µ_{S1} |

H µ_{1}:_{BKO} < µ_{S1} |

H µ_{0}:_{BKO} = µ_{S2} |

H µ_{1}:_{BKO} < µ_{S2} |

H µ_{0}:_{BKO} = µ_{S3} |

H µ_{1}:_{BKO} < µ_{S3} |

Benchmark Set | p-Value | ||
---|---|---|---|

Stage 1 | Stage 2 | Stage 3 | |

MM30 | 0.067 | 0.000 | 0.024 |

MM50 | 0.057 | 0.000 | 0.029 |

MM100 | 0.049 | 0.000 | 0.047 |

H µ_{0}:_{S1} = µ_{S2} |

H µ_{1}:_{S1} < µ_{S2} |

H µ_{0}:_{S1} = µ_{S3} |

H µ_{1}:_{S1} < µ_{S3} |

Benchmark Set | p-Value | |
---|---|---|

Stage 2 | Stage 3 | |

MM30 | 0.008 | 0.318 |

MM50 | 0.015 | 0.379 |

MM100 | 0.047 | 0.488 |

H_{0}: Avg. Dev._{S2} − Avg. Dev._{S1} = 5% |

H_{1}: Avg. Dev._{S2} − Avg. Dev._{S1} > 5% |

Benchmark Set | p-Value |
---|---|

MM30 | 1 |

MM50 | 1 |

MM100 | 1 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, A.H.-L.; Liang, Y.-C.; Padilla, J.D.
A Practical and Robust Execution Time-Frame Procedure for the Multi-Mode Resource-Constrained Project Scheduling Problem with Minimal and Maximal Time Lags. *Algorithms* **2016**, *9*, 63.
https://doi.org/10.3390/a9040063

**AMA Style**

Chen AH-L, Liang Y-C, Padilla JD.
A Practical and Robust Execution Time-Frame Procedure for the Multi-Mode Resource-Constrained Project Scheduling Problem with Minimal and Maximal Time Lags. *Algorithms*. 2016; 9(4):63.
https://doi.org/10.3390/a9040063

**Chicago/Turabian Style**

Chen, Angela Hsiang-Ling, Yun-Chia Liang, and Jose David Padilla.
2016. "A Practical and Robust Execution Time-Frame Procedure for the Multi-Mode Resource-Constrained Project Scheduling Problem with Minimal and Maximal Time Lags" *Algorithms* 9, no. 4: 63.
https://doi.org/10.3390/a9040063