{MFPP(R). An R package for matrix-based flexible project planning

Databases related to project management

Two project databases are directly included in this package. The first is the²⁷ database, which contains 240 simulated projects, all of which include four completion modes. This database can be used to test algorithms for the multimode resource-constrained project scheduling problem (MM-RC-PSP). The second database is provided by²⁸ and contains 125 real projects. These projects each include only one completion mode, but they also include the cost demands of the tasks.

Simulated project structures

Build fixed matrix-based projects and calculate their demands to support TPMa

We start with the tpt function of the mfpp package to build fixed matrix-based projects and calculate their demands to support the TPMa. The LD specifies the structure of the project. The LD is an N-by-N (sub)matrix, where N is the number of tasks. The diagonal values represent the task priorities for task completion, where a value of 1 indicates a mandatory task, and a lower value indicates a supplementary task. The off-diagonal values represent the dependencies between tasks, where a value of 1 indicates a fixed dependency, and lower values indicate flexible dependencies between tasks. In the case of an acyclic graph, the LD can be reordered as an upper triangular matrix, which is assumed in this package. In the following example, a binary logic plan with one completion mode is specified. The tpt function determines the duration of the project (total project time, TPT) by calculating the task schedule, including time specifications such as the early start time (EST), early finish time (EFT), late start time (LST), late finish time (LFT), scheduled start time (SST), and scheduled finish time (SFT) for each task. The plot function draws a Gantt chart for ESTs, LSTs, or SSTs. The LD and TD must be specified as necessary and sufficient arguments for this function. The output of the function is as follows:

LD<-rbind(c(1,0,1), c(0,1,0), c(0,0,1))
colnames(LD)<-rownames(LD)<-paste("a",1:3,sep = "")
TD<-c(3,4,5)
TPT<-tpt(LD,TD)
summary(TPT)

##
##
##  Table of schedule
##    Dur EST EFT LST LFT TF SST SFT SF Is.Crit
## a1   3   0   3   0   3  0   0  3   0    TRUE
## a2   4   0   4   4   8  4   0  4   4   FALSE
## a3   5   3   8   3   8  0   3  8   0    TRUE

SST <- c(1,1,0)
TPT<-tpt(LD,TD,SST)
summary(TPT)

##
##
##  Table of schedule
##     Dur EST EFT LST LFT TF SST SFT  SF Is.Crit
##  a1   3   0   3   0   3  0   1   4  –1    TRUE
##  a2   4   0   4   4   8  4   1   5   3   FALSE
##  a3   5   3   8   3   8  0   4   9  –1    TRUE

The plot of the schedule can be drawn as follows (Figure 1(a)):

Figure 1. The scheduled Gantt chart and resource diagram for a binary logic plan with one completion mode.

The total project resources based on two resources domains (R₁ and R₂) extracted through mfpp package.

Next, we illustrate the use of the tpc, tpq and tpr functions to calculate the total project cost, the total project quality, and the total project resources. The tpr function calculates the maximal resource demands and can specify a resource graph. In these functions, the CD and QD are N-by-w matrices, while the RD is an N-by-$\left(w^{\ast }r\right)$ matrix, where r is the number of resources. The CD is mandatory, while the QD and RD are optional matrices. The necessary arguments of these functions are the LD, CD, quality parameters (q), QD with completion modes, and RD. The output of this function for the given example is as follows:

set.seed(6)
CD<-c(10,20,24)
cat("\nTotal Project Cost (TPC): ",tpc(LD,CD))

##
## Total Project Cost (TPC): 54

q<-runif(3)
cat("\nTotal Project Quality (TPC): ",tpq(LD,LD,q))

##
## Total Project Quality (TPC):  0.5316529

QD2<-cbind(q,runif(3)) # Generate two completion modes
cat("\nRelative TPQ: ",tpq(LD,LD,q,QD2))

##
## Relative TPQ:   0.7169994

RD<-round(cbind(runif(3,min=0,max=5),runif(3,min=0,max=5)))
cat("\n\nTotal Project Resources (TPR)\n\n")

##
##
## Total Project Resources (TPR)

tpr(TPT$SST,LD,TD,RD)

##     R_1 R_2
## TPR   9   8

A plot of the total project resources can be called with the additional argument “res.graph=TRUe”, and the output is seen in Figure 1(b).

tpr(TPT$SST, LD, TD, res.graph = TRUE)

##     R_1 R_2
## TPR   9   8

Optimizing resource allocation is a combinatorial (NP-hard) problem; therefore, a metaheuristic method is used to minimize the maximum resource demands. In this case, the nondominated sorting genetic algorithm II (NSGA-II) method is used to minimize the maximum resource demands. In the case of multiple resources, a Pareto-optimal solution is found. The paretores function can help a project engineer determine the minimal resource demands while maintaining a specified project duration. The output of this function is as shown in Figure 1(c) and in Figure 1(d).

The set of domains, namely, the set of the LD, TD and CD and optionally the QD and RD specifies the PDM. A logic network can be constructed using this set of domains as follows:

PDM<-cbind(LD,TD,CD,q,RD)
class(PDM)<-"PDM_matrix"
summary(PDM,w=1,Rs=2)

##
## summary PDM matrix:
##    a1 a2 a3 TD  CD         q  
## a1  1  0  1  3  10 0.6062683 5 0
## a2  0  1  0  4  20 0.9376420 4 3
## a3  0  0  1  5  24 0.2643521 3 5
## attr(,"class")
## [1] "PDM_matrix"
##
## Minimal constraints:
##
## Summary of the PDM constraints structure:
##
## Time constraint (Ct): 8
## Const constraint (Cc): 54
## Score/scope constraint (Cs):  1
## Quality constraint (Cq):  0.5316529
##
## Maximal constraints:
##
## Summary of the PDM constraints structure:
##
## Time constraint (Ct): 8
## Const constraint (Cc): 54
## Score/scope constraint (Cs): 1
## Quality constraint (Cq): 0.5316529
## Resource constraint(s) (CR):
##     R_1 R_2
## TPR   9   8

Generation of a flexible project network to support the HPMa

The HPMa combines the features of the TPMa (multiple completion modes), APMa (a flexible project structure), and XPMa (new, unplanned tasks). Project plans to support the HPMa can also be designed using the proposed mfpp package. For this purpose, the first function that is needed is generatepdm. The input arguments of this function are the number of tasks (N); the flexibility factor (ff); the connectivity factor (cf); the maximum values of the TD (mTD), CD (mCD), RD (mRD); the number of modes (w); the number of resources (nR); the number of possible extra tasks (nW); and the scale and QD of the project scenario. The function returns either the PDM only or a PDM list that also contains the number of completion modes (w) and the number of resources (Rs). Consider a flexible plan with 4 planned tasks, 2 completion modes and 2 resources. The project schedule generated using the HPMa is as follows:

# Generation of PDM matrix for flexible project planning MFPP package.

# Define number of modes, flexibility factor and connectivity factor of a project scenario.
N=5;ff=0.30;cf=0

# Define maximum value of time domain, cost domain and resource domain of a project scenario.
mTD=3;mCD=4;mRD=3

# Define number of modes, number of resources, number of possible extra tasks (nW), scale and# quality domain of a project scenario.

w=2;nR=2;nW=1
scale=1.6
PDM<-generatepdm(N,ff,cf,mTD,mCD,mRD,w,nR,nW,scale,QD=TRUE,lst=TRUE)
rownames(PDM$PDM)<-colnames(PDM$PDM)[1:(N+nW)]<-paste("a",1:(N+nW),sep="")

summary(PDM) # Summary of PDM list

##
## summary PDM list:
##
## Number of completion modes (w): 2
## Number of resources (Rs): 2
## summary PDM matrix:
##   a1        a2        a3        a4 a5 a6       t_1      t_2       c_1
## a1 1 0.6348251 0.0000000 1.0000000  0  0 1.8922066 2.206381 2.9384623
## a2 0 1.0000000 1.0000000 0.0000000  0  0 0.1397774 0.160757 0.5917513
## a3 0 0.0000000 0.9458257 0.0000000  0  0 2.7119461 2.784497 3.1012793
## a4 0 0.0000000 0.0000000 0.7420972  0  0 1.8202474 1.896929 0.5478731
## a5 0 0.0000000 0.0000000 0.0000000  1  0 2.6490032 2.922064 2.3434704
## a6 0 0.0000000 0.0000000 0.0000000  0  0 0.0000000 0.000000 0.0000000
##          c_2       q_1       q_2     r_1.1    r_2.1     r_1.2     r_2.2
## a1 2.9456676 0.4591163 0.4597432 1.7602166 1.767857 0.7749634 0.8299197  
## a2 0.7051308 0.2476199 0.2898660 0.9645262 1.109172 1.9847076 2.0104963  
## a3 3.8735180 0.3979600 0.4327975 2.3338386 2.723688 2.3193125 2.3997095  
## a4 0.6072272 0.2989701 0.3343502 1.8691629 2.192587 2.5611961 2.9716221  
## a5 2.5560046 0.4891549 0.5480301 2.2484773 2.280209 2.2384662 2.4961772  
## a6 0.0000000 0.0000000 0.0000000 0.0000000 0.000000 0.0000000 0.0000000  
## attr(,"class")
## [1] "PDM_matrix"
##
## Minimal constraints:
##
## Summary of the PDM constraints structure:
##
## Time constraint (Ct): 2.711946
## Const constraint (Cc): 5.873684
## Score/scope constraint (Cs): 0.4907664
## Quality constraint (Cq): 0.1039251
##
## Maximal constraints:
##
## Summary of the PDM constraints structure:
##
## Time constraint (Ct): 2.945254
## Const constraint (Cc): 10.68755
## Score/scope constraint (Cs): 0.9427112
## Quality constraint (Cq): 0.3830448
## Resource constraint(s) (CR):

##          R_1      R_2
## TPR 7.196484 7.867509

The main components of flexible project structures can be drawn using the plot function of the mfpp package, where these structures are defined as follows.

Minimal structure: This structure contains only mandatory tasks and fixed dependencies. It provides the lowest quality, cost, and time demands when the lowest quality, lowest cost, and shortest completion modes are specified for each task.

Maximal structure: This structure contains both mandatory and supplementary tasks as well as both fixed and flexible dependencies. It provides the highest quality, greatest cost, and greatest time demands when the highest quality, greatest cost, and longest completion modes are specified for each task.

Minimax structure: This structure contains only mandatory tasks but both fixed and flexible dependencies. It can provide the lowest resource demands.

Maximin structure: This structure contains both mandatory and supplementary tasks, but only fixed dependencies. It can provide the greatest resource demands.

Most-likely structure: Rounding the values of the LD yields the most-likely structure. The plot of the PDM structure is as follows in Figure 2(a):

Figure 2. Logic structure (a), minimal and maximal constraints (b) of the hybrid-project; Gantt chart (c) and Resource graph (d) of optimal schedule for completion mode 1.

The plot of the constraints is as follows in Figure 2(b). The most likely/most desired structures of the project for multiple modes can be determined using mfpp. The input and output are as follows:

PSM_list<-get.structures(PDM,type = "most")$moststruct
summary(PSM_list)

##
## summary PDM list:
##
## Number of completion modes (w):  2
## Number of resources (Rs):  2
## summary PDM matrix:
##   a1 a2 a3 a4 a5 a6       t_1      t_2       c_1       c_2       q_1       q_2
## a1 1  1  0  1  0  0 1.8922066 2.206381 2.9384623 2.9456676 0.4591163 0.4597432
## a2 0  1  1  0  0  0 0.1397774 0.160757 0.5917513 0.7051308 0.2476199 0.2898660
## a3 0  0  1  0  0  0 2.7119461 2.784497 3.1012793 3.8735180 0.3979600 0.4327975
## a4 0  0  0  1  0  0 1.8202474 1.896929 0.5478731 0.6072272 0.2989701 0.3343502
## a5 0  0  0  0  1  0 2.6490032 2.922064 2.3434704 2.5560046 0.4891549 0.5480301
## a6 0  0  0  0  0  0 0.0000000 0.000000 0.0000000 0.0000000 0.0000000 0.0000000
##        r_1.1    r_2.1     r_1.2     r_2.2 
## a1 1.7602166 1.767857 0.7749634 0.8299197
## a2 0.9645262 1.109172 1.9847076 2.0104963
## a3 2.3338386 2.723688 2.3193125 2.3997095
## a4 1.8691629 2.192587 2.5611961 2.9716221
## a5 2.2484773 2.280209 2.2384662 2.4961772
## a6 0.0000000 0.000000 0.0000000 0.0000000
## attr(,"class")
## [1] "PDM_matrix"
##
## Minimal constraints:
##
## Summary of the PDM constraints structure:
##
## Time constraint (Ct): 2.851723
## Const constraint (Cc): 9.522836
## Score/scope constraint (Cs): 1
## Quality constraint (Cq): 0.3665425
##
## Maximal constraints:
##
## Summary of the PDM constraints structure:
##
## Time constraint (Ct): 2.945254
## Const constraint (Cc): 10.68755
## Score/scope constraint (Cs): 1
## Quality constraint (Cq): 0.4025323
## Resource constraint(s) (CR):

##          R_1      R_2
## TPR 7.196484 7.867509

The optimal schedules for different completion modes can be calculated using the paretores and truncpdm functions by dropping excluded tasks and their demands. The optimal resource allocation can be calculated once the completion modes are selected for each task as follows:

# Get PSM matrix

# Drop excluded tasks, and their demands

PSM<-round(truncpdm(PSM_list$PDM))

w<-PSM_list$w # Get number of completion modes
Rs<-PSM_list$Rs # Get number of resurces

DSM<-PSM[,1:nrow(PSM)] # Get PSM matrix

# Get time demands (time domain, TD)
TD<-PSM[,paste("t",1:w,sep = "_")]

# Get resource demands (resource domain, RD)
RD<-PSM[,tail(colnames(PSM),w*Rs)]

# Calculate optimal schedules for both completion modes

RES1<-paretores(DSM,TD[,1],RD[,1:Rs])
RES2<-paretores(DSM,TD[,2],RD[,(Rs+1):(2*Rs)])

Figure 2(b) shows the Gantt chart and Figure 2(c) shows the resource graph of the optimal schedule for the completion mode 1.

Obtaining a project structure from the collection

Figure 3(a) shows the original project structure and Figure 3(b) shows the minimal/maximal constraints of a project 2 from the Ref. 27’s project collection.

Figure 3. Logic network and minimal/maximal demands of project based on 4 completion mode using the Boctor database.

The Gantt charts of the selected project of completion mode 1 is constructed by extracting the PDM and TD as follows (see Figure 4(a)):

Figure 4. Gantt chart of the selected project for the 1st completion mode scheduled by early starting time (a); after varying project demands (b-c) (see phase 1 and phase 2).

Sensitivity analysis

For project risk analysis, one of the most frequently used methods is sensitivity analysis. The sensitivity analysis procedure in mfpp consists of three phases. Phase 1: Effects of uncertainty Phase 2: Effects of shocks Phase 3: Structural changes.

Notably, phase 1 and phase 2 involve varying the demands, while phase 3 varies the project structure. Phase 1 affects the entire project, while phase 2 and phase 3 affect only selected demands or structural elements. These phases can be performed in a sequential or parallel manner. The phases follow the logic of.¹⁰ The mfpp package also analyzes the effects of uncertainty on the scheduled project using the phase1 and phase2 functions. Furthermore, the existing project plan can be modified to a flexible structure using the phase3 function of the developed package. These functions are also demonstrated on the selected project from the database.

Effects of uncertainty

Phase 1 serves to analyze the uncertainty of the estimation. Modified demands are generated in the interval of $\left(\right[o+a,o+b\left]\right)$, where $o$ is the original value. The random generator can be specified to follow either a uniform (=default) or beta distribution. These modifications are applied to all kinds of demands for each task (compare Figures 4(a), 4(b), and 4(c)).

Effects of shocks

Phase 2 simulates shock effects, by increasing $p$ percent of the task demands by a factor of up to $s$. Phase 2 investigates the effects of shocks, in which not all task demands are changed but the changes are significant (see Figure 4(b)).

Structural changes

In phase 3, $P$ percent of the nodes (i.e., tasks) or arcs (i.e., dependencies)are selected to change the corresponding scores by up to the maximal change effect $S$. Phase 3 is simulates changes in customer priorities and technological changes. In the case of $S<0$, the plan will be more flexible, whereas $S>0$ increases priorities and can produce new dependencies (compare Figures 5(a), 5(b), and 5(c)).

Figure 5. Logic network of the original structure (a), with more dependencies (b), and with fewer restrictions (c) (see phase 3 function).