# A method for integrating personnel into project schedule

Michel Gagnon, PhD, PMP

### Introduction

Project scheduling is traditionally resolved by means of the “Resource Constrained Project Scheduling” (RCPS) model (Ozdamar & Ulusoy, 1995). This model describes project activities related by precedence constraints. Typically, each activity requires one or several resources from various categories, also known as resource types. The amount of resources needed for the execution of each activity remains constant, while resource availability may be constant or variable throughout the project duration. All these quantities are inserted into constraints that must be respected for the duration of the project for the obtained schedule to be considered feasible. The solution of the model mainly supplies an activity schedule while optimizing a management objective. By analyzing this schedule, one notices the use rate of every resource type.

In practice, the RCPS model is the most widely used, and the design of common commercial software programs having a wide distribution are derived from this model. The most frequent management objective is to minimize the project duration of the schedule. The RCPS model, which belongs to the class of NP-hard problems (Blazewicz, Lentra, & Rinnooy, 1983), is best resolved by heuristic methods. However, these methods supply the first found schedule having the minimal project duration. To estimate the project cost, many scheduling tools derived from the RCPS model determine the project cost by considering only the use time of resources allocated and the mean use cost (e.g., Microsoft Project^{®}). These scheduling tools ignore the fact that, in many situations, making resources available is dependent on time, and it has an effect on personnel resource management and, actually, on the project cost.

In project management, one generally recognizes that personnel resource management is more complex and drives particular requirements in scheduling. Personnel resources are, for the most part, made up of specialized resources that are often rare and expensive and, incidentally, also represent the major part of the overall project cost. Software projects and technological development projects are examples in which the personnel resource occupies a dominant place. Having obtained a project schedule, the project manager must resolve a new scheduling problem in order to assign specific resources to tasks. This schedule may present important gaps in personnel management as personnel may be allocated to the project but not necessarily scheduled to work at every period. Note that the difficulty is greater when it is necessary to *build* a project schedule while respecting a project deadline—a scheduling problem routinely faced by many consultant firms.

Project managers faced with this problem have to look for practical resource allocation strategies to minimize the project cost. The proposed strategy is, first, to consider the cost of an allocated resource to a project not as a function of its use time, but rather as a measure of its real assignment duration to the project. For now, we introduce the concept of the assignment duration of a resource as the time interval of its participation to the project. So, the assignment of resources is generally made up of a continuous period. It can include periods of inactivity not considered by traditional scheduling methods. Therefore, the personnel management objective is to minimize these gaps, also minimizing the project cost.

The proposed method advances that, because of the interrelation of the numerous factors influencing the allocation of resources, the process of resource assignment must be associated with the process of activities scheduling to optimize the selection criteria of a project schedule based on cost. So, we introduce the scheduling of a project based on a resource allocation management policy denoted as (personnel) resource assignment. The resource assignment duration can be better optimized by taking into account the *assignment cost* of every resource, instead of a mean work cost by resource category. In the context of the assignment cost of a resource as a function of its *assignment duration,* an economic strategy then becomes possible by assigning the more expensive resources needed over the shortest possible periods. Consequently, the corresponding project cost will be based on scheduling decisions more accurately reflecting the real assignment cost of the allocated resources. Such a scheduling approach allows the determination of which resources must be assigned to each activity so as to minimize the project cost. The study of this problem, which we denote by *RASC (Resource ASsignment Cost)* has received limited attention in the literature.

The purpose of this paper is to present a mathematical model which integrates the activity scheduling of a project with resource scheduling based on an affectation management rule; that is, minimization of assignment costs. The management objective is to minimize the total assignment cost of resources allocated to the project (referred to as project assignment cost) while respecting a project deadline. This last constraint has a direct effect on the selection of the resources as well as the project cost. It is a combinatorial optimization problem NP-hard more difficult to resolve than the RCPS problem because it requires the assignment of resources in a way that minimizes the project cost in addition to producing a schedule that respects the project deadline. To resolve exactly such optimization problems is very difficult because the computation time of exact methods grows exponentially with the size of the problem. That is why the use of a heuristic procedure to obtain a good solution in an acceptable time is privileged.

The paper is organized as follows: Section 2 gives a brief literature review of scheduling related to this research. Section 3 presents the mathematical model formulation, integrating the activity scheduling and the resource scheduling according to the resource management policy proposed and a project deadline. Section 4 explains the method developed to find the best combination of resources minimizing project cost. Section 5 discusses results of the computational experiment made from a set of modified test problems. The conclusion follows with the advantages of such an approach in project personnel management.

### Literature Review

In the context of project management, personnel scheduling is mainly concerned with the determination of appropriate manpower needs, manpower allocation, and project activity allocations to personnel in order to meet specific management objectives. This involves the affectation of personnel to work package and timeslots. The terms personnel (or labor, or manpower staff workforce) scheduling (or timetabling, or rostering, or tour scheduling) are often used interchangeably to refer to the general problem (Soubeiga, 2003). The personnel scheduling problem is typically encountered in service organizations (e.g., hospital, aircraft maintenance, airport ground personnel, etc.). Bedworth and Bailey (1987) provided a detailed literature review on the work scheduling. Dean, Denzler, and Watkins (1992) also combined the scheduling of projects with the allocation of the resources according to the workload. Personnel scheduling has traditionally been associated with cyclical scheduling problems—e.g., shift scheduling, days-off scheduling, and tour scheduling. Personnel scheduling has traditionally been more closely associated with cyclical scheduling problems, shift scheduling, days-off scheduling, and tour scheduling. These problems relate more closely to scheduling of operations than to scheduling of projects tackled in project management.

One of the first preoccupations found in order to optimize models which integrate project activities (tasks) and manpower scheduling is by Bailey, Alfares, and Lin (1995). The problem is formulated using an integer programming (IP) program that relates the staffing level requirements to the start times and duration of the project activities as constraints. A dynamic-programming heuristic is then used to solve the problem. This results in personnel (labor) cost and total cost savings over the traditional two-step heuristic procedure, which first determines the start time and duration of the project activities before calculating the level of resources (labor) per period. On the other hand, Alfares and Bailey (1997) studied the problem of the scheduling of projects by incorporating the scheduling of the staff working on schedule in the context of production scheduling, where an activity (operation) is executed by one worker (resource). The authors proposed an integrated approach for solving project activities (operations) and personnel scheduling simultaneously. Project scheduling determines a schedule of the different activities to be performed within a certain period of time, subject to temporal and precedence constraints on each activity. The personnel scheduling problem is set from the different activities that must be performed by workers. Both problems were usually solved using a two-phase approach that first solves the project scheduling problem and then the personnel scheduling problem based on the results of the first problem. The authors modeled both problems as a unique IP problem and solved the resulting model using an IP solver. Computation results showed that this approach was superior to the two-phase approach in terms of total project cost, workforce cost, and workforce utilization. Limitations of the approach are the exact method used, which limits its applicability to small problems, and the fact that resource costs are identical instead of being specific to each resource.

On the other hand, Möhring (1984) and Demeulemeester (1995)—and, more recently, Drexl and Kimms (2001)—approached the problem of availability cost minimization. In these papers, the availability cost can be seen as a resource investment problem or a renting cost. Möhring (1984) first elaborated an optimal solution method determining from various combinations of the available resources the increase of the corresponding cost. The proposed solution procedure finds its application from the concepts and the algorithms of the comparability graphs theory and from the graphs of intervals. However, in the models the authors described, they considered an identical cost for every resource belonging to a resource type. And the project cost obtained was not a function of the resulting project duration and did not consider the periods when resources were not required.

Note that Demeulemeester (1995) proposed an iterative procedure which appeals to an exact procedure developed before to resolve the RCPS problem (Demeulemeester & Herroelen, 1992). This last procedure is used iteratively but with various resource combinations. At each iteration, the procedure looks for an answer to the problem of the following decision: “For a project given with available funds of resources and a maximal duration to realize the project, is there a solution which does not exceed the maximal duration and which respects all the precedence relations and all the constraints of resources?” (Demeulemeester, 1995). This procedure is based on an exact method, which limits its application to problems of small size. The efficiency of the iterative procedure thus developed depends on the search strategy adopted to find the adequate number of resources to realize the project according to the time constraint specified and at a minimal cost. We adopt a similar approach wherein the developed heuristic procedure is applied to try to find the best resource combination. To the best of our knowledge, there is no procedure proposed to this day that resolves the assignment cost minimization problem of a project with a project deadline.

### Mathematical Modeling

#### Project Scheduling

We can represent a project by a network of activities on nodes, by a directed acyclic graph denoted by *G* = (A, P); that is, which has to be without circuit: *A* denotes all the nodes representing the activities of the project and *P* denotes the set of precedence relations between the activities. The precedence relations define a partial order relation. The set *P _{j}* denotes all the immediate predecessors of the activity

*j*so that

*j*’∈

*P*if and only if

_{j}*j*’ precede

*j*. We number the activities of 1 in

*J*where the activities 1 and

*J*, often of null duration, represent respectively the start node and the end node of the network. To execute the project, there are different resources grouped by type (category) given by the set

*R*= {1,…,

*k,*…,

*K*}. For every activity

*j*, the project manager defines its fixed duration, noted

*d*

_{j}, and the quantity of resources of type

*k,*noted

*q*

_{jk}, required by period for a duration

*d*

_{j}. The rate of use of every resource type is constant during the execution of the activity and the activities cannot be interrupted during their execution. Without loss of generality,

*d*

_{j}and

*q*

*jk*are integer values.

#### Scheduling of Resources

In project scheduling, we use the term “resource allocation” to indicate in a general way the process of allocating the resources required for the execution of the project activities. We consider a resource as being allocated if it is required for at least one time period during the project execution. Most of the project scheduling procedures estimate the project cost from the quantity of resources used by the activities and according to their execution duration. To ease the resource allocation process, the resources are grouped together by resource type and allocated accordingly. So, a resource type represents a set of resources having similar properties corresponding to a given skill set required to accomplish different activities. The resources of a given type are interchangeable and the cost-in-use to execute a set of activities remains the same. Consequently, the cost-in-use and the availability cost of these resources are average costs identical to all the resources of the same type. By calculating the cost of the project on this basis, we only obtain an estimation of the cost of the project calculated according to the mean cost by resource type. It also implies that many different schedules have the same cost.

By allocating resources to a project, we can distinguish three cost structures to calculate the resources working on the project: the use duration, the availability duration, and the assignment duration of every resource. The allocation cost of the resources to the project can be made according either to their use duration, their availability duration, or their assignment duration. In the first case, the resources are paid only during the periods when they execute activities. In the second case, the resources are rented for the duration of the project, including the periods when they are inactive. In the last case, the resources are paid (rent) for a continuous period; that is, the project duration.

To evaluate the cost of the resources, we identify the unit cost for every resource. We number each of the resources of the same type by a number *r*, where *r* = 1, 2,…, *R _{k},* and

*R*is the maximal quantity of resources of type

_{k}*k*available to the project. To simplify the notation, a resource

*r*of type

*k*is identified by juxtaposed numbers

*kr.*For example, if the resources of type

*k*are the senior analysts used in a software project, the proposed numbering identifies every senior analyst in the following way: k1: the senior analyst 1, k2: the senior analyst 2, and finally

*kR*the senior analyst

_{k}:*R*We denote by

_{k}.*u*the total use duration of the resource

_{kr}*r*of type

*k*contributing to the execution of activities to which it is allocated. It is the sum of the execution durations of the activities to which this resource is allocated.

#### Definitions of Assignment Duration and Cost of the Assigned Resources

##### Definition 1: Assignment duration of a resource

The assignment duration of a resource *r* of type *k,* noted *t*_{kr}, corresponds to the time interval between the beginning of execution of the first activity and the end of execution of the last activity to which this resource is allocated, the first one and the last activity being determined according to the chronological order of the project schedule.

In fact, this duration corresponds to the total use duration of the resource by the activities plus the inactivity periods of this resource. Denoting by T* the duration of the schedule (project), one obtains three durations respecting the following relations:

The concept of nominative assignment—that is, specific to a given resource—implies a unit assignment cost and a unit cost-in-use specific to every resource.

##### Definition 2: Unit assignment cost of a resource

The unit assignment cost of a resource *r* of type k, noted *c _{kr},* is the assignment cost of this resource allocated to the project for one period.

In this paper, we make the assumption that the unit assignment cost is the same as the cost-in-use per period. In practice, these two costs may be different and include other cost elements.

##### Definition 3: Cost-in-use of a resource

The cost-in-use of the resource *r* of type k, noted by *C ^{U}_{kr},* is equal to the unit assignment cost of this resource working on this activity multiplied by its total use duration

*u*:

_{kr}To estimate the project cost according to the assignment duration of all resources, it is necessary to determine the assignment cost according to the assignment duration of each of the resources allocated to the project.

##### Definition 4: Assignment cost of a resource

The assignment cost of a resource *r* of type *k* allocated to the project, noted C^{A}_{kr}, is obtained by the product of the unit assignment cost of this resource by its assignment duration:

#### Formulation of the Mathematical Model

The formulation of the mathematical model to schedule both the project and the resources requires three sets of decision variables:

*The end of an activity*

*X*_{jt} Zero-one decision variable equal to 1 if the activity *j* finishes in the period *t* and 0 otherwise.

*The resource assignment start time and the assignment finish time*

These variables are used to determine the assignment duration of every allocated resource. For every resource *r* of type k, the following variables of decision specify:

*s _{krt}* Zero-one decision variable equal to 1 if the assignment of the resource

*r*of type

*k*starts in the period

*t*and 0 otherwise.

*f _{krt}* Zero-one decision variable equal to 1 if the assignment of the resource

*r*of type

*k*finishes in the period

*t*and 0 otherwise.

Without loss of generality, the unit cost of every resource type is specified in non-decreasing order of the resource number *r.*

Figure 1 gives the formulation of the mathematical model whereas Table 1 gives the notation used to describe it.

E_{j}, L_{j} | Earliest start and latest finish times of the activity j |

x_{jt} | Zero-one decision variable equal to 1 if the activity j ends in the period t and equal to 0 otherwise |

s_{krt} | Zero-one decision variable equal to 1 if the assignment of the resource r of type k starts in the period t and 0 otherwise |

f_{krt} | Zero-one decision variable equal to 1 if the assignment of the resource r of type k ends in the period t and 0 otherwise |

C_{kr} | Unit assignment cost of the resource r of type k when allocated to the project |

d_{min} | Minimal duration of the project activities, d = min _{min}{d_{j} | d_{j} > 0, j = 1, 2,…, J} |

T | Project deadline |

P_{j} | Set of immediate predecessors of the activity j |

d_{j} | Duration of the activity j |

R_{k} | Maximal quantity of resource type k, k = 1, 2,…,K |

q_{jk} | Quantity of resources of type k required by period to execute the activity j |

**Table 1 - Notation of the Model for Scheduling the Project and Resources**

**Figure 1 - Assignment Cost Minimization Model with Project Time Constraint**

The objective function (4) minimizes the assignment cost of all the resources, thus minimizing the project assignment cost. The constraint set (5) verifies that every activity *J ∈ A* has an ending period. The constraint set (6) verifies that the relations of precedence between the activities of the activity network of the project are respected. The constraint sets (7 and 8) verify that the number of assigned resources is sufficient to meet the needs of necessary resources to execute the activities in the foreseen periods. The quantity of resources of every type required at every period thus has to be lower or equal to the assigned resources. The constraint set (9) assures that every resource cannot be assigned more than once during the project. These constraints are respected when there is not more than one assignment start period for every resource *r* of type *k.* The constraint set (10) assures that if a resource possesses an assignment start period, then it also has to have an assignment end period to the project. To maintain the completeness of the time interval measuring the assignment duration of the resources during the project, the constraint set (11) verifies that the assignment start period of a resource *r* of type *k* is not superior to its assignment end period. The constraint (12) assures that the end of the terminal activity of the project does not exceed the project deadline *T*. The constraint sets (13, 14 and 15) specify the admissible values for every decision variable.

### Cost Minimization Procedure under Time Constraint

#### Description of the Minimum Cost Procedure

We propose an iterative procedure which relies, at every iteration, on a Taboo search method developed to resolve the RCPS problem ( Gagnon, Boctor, & d‘Avignon, 2002). This procedure, denoted by MC procedure, implements a management mechanism of resource allocations generated. At every iteration, the MC procedure generates, at first randomly, a non-dominated resource combination, noted Q. This resource combination specifies the maximal quantity of each resource type that can be allocated to the project. It then calls a Taboo search procedure, denoted here by TS procedure, which tries to obtain a project schedule with the resource allocation given by *Q,* and which respects the time constraint *T* at a minimal cost for the resolute RCPS problem.

Figure 2 describes the steps of the developed procedure whereas Table 2 gives the notation used.

x | A project activity schedule |

x* | The schedule presenting the lowest project assignment cost |

f_{x} | Project finish time calculated for the schedule x |

f_{x}* | Finish time of the schedule x* |

C_{x} | Project assignment cost according to the resource allocation of schedule x |

n | Counter of the number of iterations made by the MC procedure |

nMax | Stopping criterion controlling the maximal number of resource allocations estimated by the MC procedure |

L_{k}, U_{k} | Lower and upper bounds of the quantities of resources of type k |

Q | Allocation of resources to execute the project, Q = [q_{1},…,q_{k},…,q_{K}] |

T | Project deadline |

**Table 2 - Notation for the MC Procedure**

**Figure 2 - Minimum Cost Procedure (MC Procedure)**

The MC procedure determines the lower and upper bounds for resources of every type, quantities required in order to be able to execute the project according to the specified project deadline (line 1). The iterative process of the MC procedure then begins (line 2). First, it randomly generates a *non-dominated* resource allocation *Q* (line 3, see definition 5). Second, it uses a construction method as described in Boctor (1993) to generate a starting feasible solution according to the quantities of allocated resources *Q* (line 4). Next, the TS procedure is applied to try to find a solution respecting the project deadline *T,* while offering a minimal assignment cost for the project (line 5). If the best solution *x* obtained by the TS procedure respects the project deadline *T*, then one keeps the resource allocation of this schedule in the dominance tree (line 7). Now, if the calculated project assignment cost *C _{x}* is lower than the project assignment cost of the best schedule found up to this point, then the new schedule

*x*replaces the best schedule found up to this point

*x**(line 8). The procedure ends when the stopping criterion is verified (line 11).

#### Determination of Bounds of Assigned Resources Quantities

The MC procedure determines, at first, for every type of resource, the minimal and maximal quantity of resources *L*_{k} and *U*_{k}, which can be assigned as follows:

where *E _{j}* is the earliest start time of activity

*j*and

*L*is the latest finish time of activity

_{j}*j*determined by the CPM using the time constraint

*T*as the latest finish time of the last activity of the project, and

*L*is the latest finish time of the project determined by the Critical Path Method without resource constraints.

_{H}#### Constituting a Dominance Tree for the Non Dominated Resources Allocations

The use of a dominance tree (“Quad-Tree” in Sun & Steuer, 1996) allows us to direct the search effort of the MC procedure. This data structure is used to manage the non-dominated resource allocations (i.e., resource combinations) by keeping in nodes the arborescence of (approximate) non-dominated allocations previously generated.

##### Definition 5: Dominated Allocation of Resources

A resource allocation *Q* = [*q _{1}*,…,

*q*dominates an allocation of resources

_{k},…,q_{K}]*Q*’ = [

*q’*,…,

_{1},…q’_{k}*q’*] ⇔

_{K}*Q ≠ Q’*and

*q*

_{k}≤ q’_{k}∀ k.The MC procedure only generates non-dominated allocations of resources by comparing them to those already recorded in the dominance tree. If this were the case, a new allocation would be generated. This verification thus limits the search efforts to the values representing resource allocations said to be admissible; that is, which are not dominated by those recorded in the dominance tree. Every time a resource allocation *Q* results in a schedule which respects the time constraint *T* to execute the project, it is inserted into the dominance tree. The actual resource allocations in the dominance tree that are dominated by the new resource allocation are then removed. This thus restricts the search for an optimal schedule to the resource allocations which are potentially not dominated.

#### Determination of the Project Assignment Cost

The objective of the exact and approximate procedures proposed up to day to resolve the RCPS problem is only to produce a schedule of minimal duration. To do so, these procedures thus propose most of the time the first schedule of minimal duration found. However, there can be numerous other schedules also of minimal duration but that require fewer resources and thus a lesser assignment cost. In fact, a procedure seeking a schedule simultaneously minimizing the project duration and resource requirement would necessitate much more computation time. The proposed procedure for calculating the project assignment cost, noted CA procedure, makes this search but only for selected schedules. It is presented in Figure 3. The notation used is given in Table 3.

The AC procedure verifies every period of the schedule *x* (line 2 to line 14). For every resource type (line 3), it calculates the amount of resources needed for all the activities (line 5) that are executed during a considered period *t* (line 6), and it accumulates the quantity of resources required (line 6). It then determines the maximal quantity of resources required for the project (line 8). For every resource (line 9), it verifies whether or not this resource is already assigned (line 10). If not, the assignment is made and the assignment start time and finish time are maintained (line 10). Otherwise, only the assignment finish time is updated (line 11). At the end of the iteration (line 14), one obtains the duration of assignment of every resource *r* of type *k.* The maximal quantity of every resource type *q _{k}* is then adjusted accordingly (line 16). For every resource assigned to the project, the assignment cost is calculated according to its assignment duration and its unit cost (line 18). The project assignment cost

*C*and the maximal quantities of resources assigned, specified by having updated

*Q*, are returned to the MC procedure (line 21).

C | Project assignment cost calculated from the assignment cost of the resources |

A_{kr} | Zero-one decision variable equals to 1 if the resource r of type k is assigned |

s_{j}, f_{j} | Scheduled start and finish times of the activity j |

s_{kr} | Assignment start time of the resource r of type k |

f_{kr} | Assignment finish time of the resource r of type k |

f_{x} | Project duration corresponding to the schedule x |

U_{k} | Maximal quantity of resources of type k required by the schedule x |

Q | Resource allocation, Q = [,…, q,…, q_{1}_{k}q]_{K} |

**Table 3 - Notation for the AC Procedure**

**Figure 3 - Procedure Calculating the Project Assignment Cost (AC Procedure)**

### Computational Experiment

The computational experiment aims at estimating the efficiency of the procedure by measuring the monetary gains in resource management by resolving the RASC problem. The obtained results are compared with the assignment costs which we obtain by means of a parallel construction method with the rule of priority MIN LFT (MINimum Late Finish Time)—( Boctor, 1993). This method at first orders the eligible activities which have the smallest late finish times.

#### Description of the Test Problems

The experiment is performed by means of 110 test problems assembled by Patterson (1984). In these problems, the project activities can require up to three types of resources. The number of activities of each test problem varies between 7 and 51 activities. Three test problems use a single type of resources and four test problems use two types of resources. Thus, more than 90 % of the test problems use three resource types. The optimal duration of every original RCPS test problem is known, and it is used as compulsory constraint of time *T*. The unit cost of every resource is generated randomly within the interval [1, 10].

#### Context of the Experiment

The results presented in this section were obtained by using a Dell Latitude D600 computer equipped with an Intel Pentium^{®} M processor running at 1,6 GHz and one (1) GB memory. The procedures were developed with the Microsoft© Visual Basic 6.0 programming language. These procedures are run in the Windows XP© environment. Two parameters control the intensity of the search made by the MC procedure and the TS procedure: *nMax* and *kMax.* For purposes of experiment, the number of iterations of the MC procedure, *nMax,* varies between 30 and 80 whereas the maximal number of schedules estimated by the TS procedure, *kMax,* remains constant and fixed at 1000. Every individual experiment uses the same series of random numbers.

#### Performance Criteria

The performance criteria include, for every experiment, the sum of the cost-in-use of the problems original tests, the sum of the project assignment costs according to the construction method and the MC procedure, the total amount saved and the average CPU time by test problem. The total amount saved is the difference between the sum of the assignment costs of the schedules generated by the MC procedure and the sum of the project assignment costs of the schedules generated by construction method. We also give the average increase percentage. This average increase percentage measures the difference, in percentage, between the project assignment cost and the project cost-in-use of the test problems considered. The project cost-in-use offers a useful comparison base because it does not vary according to the schedule obtained.

#### Computational Results

Table 4 gives the cost compilation made on the Patterson's problems tests according to the variation of the values of the *nMax* parameter. The highest saving obtained is $17,612. Table 5 gives the average increase percentage and the difference between the approaches. On average, one saves about 13.5 % on the assignment costs of the starting schedules. A reminder: the increase percentage is established from the cost-in-use to have a common comparison base.

We notice a resulting slight improvement in spite of an increase of the nMax parameter value. This is attributable to the fact that MC procedure manages non-dominated resource allocations. The amount saved by the MC procedure thus remains relatively stable even if one increases the amount of resource allocations used 12 by the construction method. It remains that the proposed procedure also provides shorter project durations than the ones obtained by the construction method.

nMax Parameter | ||||||

30 | 40 | 50 | 60 | 70 | 80 | |

Cost-in-use of the test problems ($) | 112,186 | 109,353 | 106,543 | 108,811 | 107,874 | 112,309 |

Assignment costs of construction method ($) | 245,588 | 235,983 | 235,742 | 232,478 | 227,493 | 235,681 |

Assignment costs of the MC procedure ($) | 230,718 | 221.323 | 218,130 | 218,150 | 212,014 | 219,089 |

Total amount saved ($) | 14,870 | 14,660 | 17,612 | 14,328 | 15,479 | 16,592 |

Average CPU Time (s) | 0.625 | 0.832 | 1.049 | 1.256 | 1.411 | 1.674 |

**Table 4 - Experimental Results**

Parameter nMax | ||||||||

30 | 40 | 50 | 60 | 70 | 80 | |||

Construction method | 111.5 | 107.7 | 111.7 | 105.4 | 102.4 | 102.7 | ||

MC procedure | 98.2 | 94.8 | 96.5 | 93.1 | 88.6 | 88.5 | ||

Difference | 13.3 | 12.9 | 15.2 | 12.3 | 13.8 | 13.2 |

**Table 5 - Increase Percentage from Cost-in-Use**

### Conclusions

This research proposes a minimum cost procedure to resolve the RASC problem. In order to appreciate its applicability in project personnel management, a computational experiment is performed with the modified RCPS test problems originally assembled by Patterson (1984). The proposed procedure offers, on the basis of the parameters used, a reduction of the project assignment costs of about 13.5 % on all the test problems processed with regard to the schedules generated by a construction method, this obtained in a modest average computation time.

The integration of personnel into project scheduling based on the resource assignment concept establishes an important advantage of the approach developed with regard to the other scheduling methods employed in two stages. Indeed, the resulting schedule also specifies the resource assignment durations to the project largely facilitating the personnel management.

The concept of resource assignment incorporated into the integrated scheduling model also offers the possibility of inserting many other (personnel) resource management rules, as the choice of resources to assign to the project according to specific criteria, the execution of activities by a given resource, the maximal assignment duration of a resource, and a maximal assignment cost of a resource to the project.

An important characteristic of the proposed procedure is to consider the resource allocations that could admit a feasible schedule as (approximated) non-dominated points (i.e., resource allocations and approximated project duration) along the effective frontier. The management of these (approximated) non-dominated points is made by means of a dominance tree. The use of this data structure decreases significantly the computation time which would consequently be employed on the evaluation of allocations of resources dominated by those previously estimated. Consequently, this increases the quality of the obtained results by restricting the search space to the generation of potentially effective resource combinations.

Lastly, the procedure developed can be generalized to resolve a multi-objective scheduling problem by the mean of a heuristic evaluation of multiple objective functions. A data structure similar to the one used would be of use to the management of (approximated) non-dominated solutions. In the context of an interactive approach, the project manager would pass from a set of (approximated) non-dominated solutions to another set by estimating changes in the project duration, the project cost, and the quantities of resources required in order to execute the project.

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© Her Majesty the Queen, as represented by the Minister of National Defence, 2006. Published with permission.

©2006 Project Management Institute

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