- 1012 - 978-1-4799-5376-9/14/$31.00 ©2014 IEEE

Multi-project Scheduling Problem with Human Resources Based on Dynamic Programming and Staff Time Coefficient

CHEN Jun-jie，ZHU Jiang-li，ZHANG Ding-ning

School of Management, Northwestern Polytechnical University, Xi’an 710072, P.R. China

Abstract: The effect of personnel qualified degree to tasks has an important role for solving multi-project human resource constrained scheduling problem. This paper conducts research on multi-mode multi-project human resource constrained scheduling problem based on dynamic programming method. First, this paper put forward the concept and estimated formula of the staff time coefficient from the perspective of how to match staff and task based on the improved cube model and establish the mixed integer programming model. There are some constrains which should be considered in the model, such as the task workload that should be completed, limited supplies, and others. And the target of the model is completing the project in the shortest period and with the lowest total cost under all of these constrained. Then use the enumeration method to give the problem feasible solution that satisfied the constraints and find the optimal solution based on the dynamic programming method. Finally demonstrate the feasibility and effectiveness of the model and algorithm using the example analysis method, the results show that this method can effectively shorten the total duration of the multi-project Compared to the initial schedule, and is helpful for decision-makers to solve the multi-mode human resource scheduling problem in enterprise.

Keywords: dynamic programming, multi-project management, multi-mode, human resource scheduling, time coefficient 1 Introduction

With the development of global network economy and the increasingly fierce competition in the world market, some large companies often meet a situation that multiple projects should be executed simultaneously. While project managers usually adopt a new approach of project management, that is multi-project management. Resource-constrained project scheduling problem is the current focal point of international research on the field of project management. However, the Institute has been considered mostly general resources, seldom considering

Supported by the National Natural Science Foundation of China(70702026)

the particularity of human resources in project scheduling problem. Resource-constrained project scheduling problems is actually Multi Skill Project Scheduling Problem (MSPSP)[1-11]: Judging feasibility from the human resource capacity and possible time, every staff mastered one or more skills, every activity requires personnel with a certain number of specific skills, and preemption is forbidden.

Vairaktarakis[12] has studied Resource-Constrained Job Assignment Problem. He constructed a mixed integer programming model, proposed a method to solve the problem’s lower bound, and analyzes the impact of resource flexibility on project completion time. Bellenguez[13] has conducted in-depth research in Multi Skill Project Scheduling Problem. He also constructed a mixed integer programming model, designed the branch and bound method, heuristic algorithm, Tabu search and genetic algorithms. Researches of the two scholars belong to single-mode resource constrained project scheduling problem. Then Kadrou and Najid[14-15] has studied multi-skilled labor multi-mode resource constrained project scheduling problem. They proposed Stereotype heuristic algorithm and constructed a simulated annealing algorithm for the minimum target of chemical in the shortest period as the target. Drezet and Billaut[16] established integer programming model, taking into account constraints such as personnel with a variety of abilities and some of the factors required by law, and tried to be solve the problem by the greedy algorithm and tabu search method. Wu and Sun[17] used genetic algorithms to solve multi-project scheduling problem of human resources. They utilized the learning curve theory that the longer the time engaged in the same work, the higher average efficiency is. Heimerl and Kolisch[18] considering that the efficiency of human resources may affect the work time, established a mathematical model in a multi-project environment and tried to solve it using optimization software. The goal to minimize labor costs, but the exact model required to give each employee's work volume, it is difficult to precisely estimate the actual work. The target of the model is to minimize the cost of human resources. The exact amount of work of each employee required to be given, that is difficult to achieve in practice. These studies have considered only the ability of human resources, but did not quantify the

2014 International Conference on Management Science & Engineering (21th) August 17-19, 2014 Helsinki, Finland

- 1013 -

ability and not explain how the human resource ability affect work time and work efficiency. In addition the above algorithms regard the project scheduling problem as a single project to solve the problem, and limit the scale of the problem in certain extent. Fu Fang[19] studied the column generation method for solving multi-mode multi-project schedule management problems with human resource constraints. Construct a mathematical model, according to the column generation process, the use of heuristic algorithms and immune genetic algorithm, its multi-execution mode corresponding demand for human resources in a variety of activities, targets only minimize labor costs.

To solve these problems, considering that both of the ability and the play conditions of the ability of human resources can affect Efficiency, mission time and cost of the work, we introduce the concept and estimated formula of the staff time coefficient in multi-project human resource constrained scheduling problem, to reflect the differences of work ability and efficiency, and as the same time to estimate the workload. In this paper, we definite that multi-mode means that each of the activities correspond various personnel combinations. To complete the project with the shortest time and Minimal cost, we give every activity the personnel combinations (execution mode) with the most appropriate staff time coefficient, also considered the overall coordination of the match in the allocation of staff in order to achieve multi-mode multi-objective optimization problem. This will make the model more practical, because of the complex of the problems, this paper uses dynamic programming approach to decompose and optimize the problem by phase, numerical results show the effectiveness of this scheduling method.

2 Multi-project human resource scheduling model

2.1 Problem description

This paper mainly aims at the overall duration and cost optimization problem of multi-project to conduct a study in the context of resource-constrained and limited duration of each project. We definite that different execution modes mean the different staff combinations for each activity. Characteristics of the specific problems are defined as follows: the enterprise's internal is working on concurrent projects, the predecessor and successor relationships of each task in the project is known, limited duration of each project is known, resource constraints and personnel constraints(the number of requirements to perform each task) of each task, work duration estimation and restrictions are all known, personnel material resource consumption is known, and there are task can be assigned to people in human resource library.

2.2 Assumed conditions

In order to make complex issues reasonable simplification, and as far as possible in line with the

actual background and research purposes of this article, therefore make the following four assumptions:

Assume (1): In the combination solutions of human allocation in every activity, personnel cooperation starts meanwhile, and finishes meanwhile, simultaneously personnel can evacuate until the task is completed.

Assume (2): Types of resources all tasks consumed (Except human resources) are the same, and assume that each independent resource used in the project activities can be met.

Assume (3): The personnel to consumption rate for the same of resources are the same in different tasks.

Assume (4): The total cost of multi-project does not consider the human cost, considering only the direct costs of materials.

2.3 Staff time coefficient

In order to reflect the differences of the ability and efficiency of the staff, but also reflect the competence of the staff to tasks, the concept and estimation formula of staff time coefficient is introduced in this paper.

This paper improves the cube mode[20], and uses the ratio of the value of a person to perform a task to the estimated value of the task, which reflects the degree of a person qualified to perform certain tasks, and helps decision makers to make staff assignments.

Fig.1 Estimated value of task k of project j

Estimation formula of staff time coefficient can be derived using these following procedures: First of all, establish a coordinate system with time (t), quality (q) and cost (c). The cuboid A（Fig.1） established using table 1 is on behalf of project j task k. jk

T , jkQ , jk

C in table 1 respectively means estimated time, estimated quality and estimated cost. In figure 1, the value of point t in axis T, point c in axis C and point q in axis Q can be calculated from formula 1 to 3. The value of T from formula 1 reflects that the time of executing task k of project j is same as the estimated value which means that this task is performed by standard man. So the value of point T, point Q and point C will be one. The integration of time, quality and cost is the volume of the cuboid jk,

which reflects the estimated value of task k of project j calculated from formula 4. When the project is executed strictly by plan, the integration of volume jk is one.

Then derive the estimation formula with coefficients t, q and c, establish cuboid B with table 2 estimated by expert evaluation method. jk

Ti , jkQi , jk

Ci in table 2 respectively means predictive time, quality and cost of personnel i performing task k of project j. Value of

jk

T t C c

q Q

- 1014 -

point Ti in axis t, point Ci in axis c can be calculated by formula 5~6. Notably, time and cost are cost index, the smaller the better, and the quality are efficiency index, the bigger the better. So the value of point Qi in axis q can be calculated by formula 7. The integration of time, quality and cost is the volume of the cuboid jk

i which reflects the estimated value of personnel i performing task k of project j calculated from formula 8.

Finally, calculate the staff time coefficient of personnel i performing task k of project j by formula 9.

Tab.1 Task plan

Task Time(T) Quality(Q) Cost(C) Task 1 of Project j jk

T jkQ jk

C …… …… …… ……

Task k of Project j jkT jk

Q jkC

Tab.2 Predictive values

Personnel Time(T) Quality(Q) Cost(C) Personnel a jk

Ta jkQa jk

Ca …… …… …… ……

Personnel i jkTi jk

Qi jkCi

=kT

kT

jTj

(1) =kCkC

jCj

(2) =kQkQ

jQ

j (3)

0 0 0

=QT C

kj dt dq dc∫ ∫ ∫ (4)

=i

kT

i kT

jTj

(5) =i

kC

i kC

jCj

(6)

= i

kQ

i kQ

jQ

j (7)

0 0 0

=i i iT Q C

kij dt dq dc∫ ∫ ∫ (8)

ki

ijk k

jKj

= (9)

2.4 Mixed integer programming model

This section will establish a mixed integer programming model for this problem. Assumption of manpower, material resources constraint; Each activity provides the demand for employees, time coefficients are different , so the working efficiency of the staff are also different; Only consider the direct costs of material resources. Symbols are as follows:

(1) Problem parameters ajk represents task k of project j (1≤j≤NP，1≤k≤Nj，

NP is the total number of projects, Nj is the total number of tasks of project j) ;

Pjk represents precedence activities set for task k of project j;

z represents material resources，1≤z≤Z，Z is the types of material resources；

Ljkz represents the limited material resources z for implementing task k of project j；

Pz represents unit price of material resource z; NR represents the total number of available staff i; Njk represents specified assigned number of

employees for task k of project j； Ni represents specified the most number of

assignable employees during the same period；

m represents a feasible task execution mode for task k of project j，which corresponds to a feasible employees combination(a combination of personnel that can meet the constraints of task k of project j)，m∈{1,…,M}，M is the feasible number of employees combination， for virtual process M=1(that is only one model of virtual processes, its workload and resource requirements is 0)；

• Tjk* represents fixed working hours for task k of

project j，which is used as planned task duration;

Kijk represents the time coefficient to complete task k of project j by personnel i, and its value is greater indicates that the corresponding working efficiency is higher；

Φjkmz represents the sum of consumption rate for resource z in mode m, which means employees combination for task k of project j；Φjkz represents the sum of consumption rate for resource z by the employees combination assigning task k of project j.

G represents the mission-critical set of muiti-project，N is the total number of mission-critical of muiti-project ， g0 and gN+1 are virtual process ，

{ }1,2, ,nG g n N= = . (2) 0～1 decision variables sjkm is equal to 1 represents that task k of project j in

mode m，otherwise is equal to 0; yjkmt is equal to 1 represents that personnel i is

assigned to task k of project j in the period t，otherwise is equal to 0.

(3) Other variables tjk represents the start time for task k of project j； stj represents the start time of task aj,N+1，also the

end time of project j; djkm represents execution time (day) for task k of

project j in mode m； L(Sjk) represents employees combination that

assigned task k of project j； xijkt is the indicator variable，if person i is allocated

task k of project j in period t，so its value is taken 1; Otherwise 0.

(4) Objective function

11=0

minN

st

g tt

t y+

×∑ (10)

1 1 1 1min

NjZ NP M

jkm jkm jkmz zz j k m

d s p= = = =

× ×Φ ×∑∑∑∑

(11)

(5) Constraint conditions

( )jk

jkm ijk jk jki L S

d K N T ∗

∈

× ≥ ×∑ i=1,2,…,NR

j=1,2,…,NP k=1,2,…,Nj m=1,2,…,M (12)

jkm jkz jkzd L×Φ ≤ z=1,2,…,Z (13)

1 11

NjNP

ijktj k

x= =

≤∑∑ t=0,1,…,st (14)

- 1015 -

yj010=1 (15)

11

M

jkmtm

y=

≤∑ (16)

1

NR

ijkt jki

X N=

=∑ (17)

i 1 1 1

NjNR NP

ijkt ij k

x N= = =

≤∑∑∑

(18)

1 0 1 0( )

j jst stM M

jvmt jkm jkmtm t m t

ty t d y= = = =

≥ +∑∑ ∑∑ ajk∈Pjv (19)

Formula (10) means that the primary target function is multi-project total duration minimization. multi-project total duration is the time period from the start time of the first task to the end of the last task when multiple projects is executed; Formula(11) means that the secondary objective is to minimize the total cost; Formula(12) represents the total amount of work done by each mission personnel combinations not be less than the task workload. To be sure, the obtained actual number of hours according to the formula is often non integer. In order to ensure that the task must be firstly completed, generally as the upper limit integral solution of decimal number; Formula(13) represents resources consumed by personnel can not exceed resource constraints of the task; Formula(14) represents that each person is assigned at most to an activity at the same time period; Formula (15) represents the project starts at t=0; Formula (16) represents each activity can only select one mode (i.e. cannot change or interrupt the execution mode); Formula (17) provides the number of employees assigned to task k of project j; Formula (18) provides the maximum number of employees that can be allocated at the same time; Formula (19) represents that ensure the order relation between activities (i.e. the start time of an activity is not less than the end time of its all predecessor activities).

3 Algorithm based on dynamic programming

General idea of the algorithm: Firstly, use the enumeration method to solve the feasible solution set of human resource allocation for each task of multi-project. Then determine the initial multi-project network diagram based on multi-project priorities and divide the multi-project into several stages in according with multi-mission-critical projects using dynamic programming ideas; Thirdly solve the start time and optimal solution of human resource allocation for each task to find the optimal solution for each stage of multi-project. Finally determine the optimal dispatch plan and schedule of multi-project.

(1) Solving the feasible solution set of human resource allocation for each task of multi-project

According to the mathematical model previously

established, I cited the set I for each activity using enumeration method (the set is a collection of human resource allocation which only meet the human resource constrained), According to the mathematical model previously established, I cited the use of a collection of enumeration method for each activity (the activity is assigned to only meet staff needs combinations (12-13) of the number of human resources solution set conditions), and then solve the10 sets for each activity (personnel combinations feasible solution set which satisfies all the constraints).

(2) Multi-project human resource optimization based on dynamic programming

Dynamic programming is a mathematical method to solve the multi-stage decision process optimization problems. This paper applies the basic idea of dynamic programming to solve the optimal allocation issues of human resources.

① Determine the initial total project network diagram

According to the given relationship between predecessor and successor of each project task, estimate duration to render the single project network diagram. And combined with the company's strategic target, determine the start time of each project as the goal to resources demand equilibrium in any stage of the total project. Then get the preliminary total project schedule which expressed by the total project network diagram.

② Divide the total project into several stages Critical path is determined according to the network

diagram of total project. And then divide the total project into several interrelated stages according to the key mission in critical line of the total project. Namely translate the total project problem into a group of the same type of sub problems.

③Solve the shortest duration and the lowest cost scheme at various stages of the total project

General idea of solving is: Obtain all the staff-time combination set I0 of each task which meets the model’s constraints. In each stage, taking the shortest time limit of the stage as the main target, and screen the subset of feasible solution I1 from I0; Then taking the lowest cost of each stage as secondary target, and screen personnel combination solution I2 which makes the duration shortest and the cost lowest in each stage. In order to firstly achieve the main goal of the shortest total project duration, should compress mission-critical duration of the total project as much as possible. When staffing, firstly configure personnel whose time coefficient is the highest and resources for the key task of the overall project. And then consider the key task of single project, finally consider the noncritical tasks.

Based on the above ideas, the specific steps of each stage solving the optimal solution is：Firstly select the solution of personnel combination which meets that the critical task’s duration of total project is shortest; Secondly select the solution of personnel combination which meets that the noncritical tasks’ durations of total

- 1016 -

project are not greater than the critical task’s duration of total project; Finally according to the combination of the first two steps, get several groups of feasible solutions in this stage, and then by comparison choose the staffing solutions of lowest cost.

④ Integrated to determine the optimal human resources allocation scheme of total project (i.e. multi- project staffing management plan)

Consolidate the schemes of shortest duration and lowest cost in each stage of the total project. Then get the optimal human resources allocation programs of total project.

(3) Determine the start execution time for each task in every stages of the multi-project

Determine the start execution time for each task in every stages of the multi-project, according to the optimal human resource allocation in each task of the multi-project gotten from step (2) and the maximum allocation staff in a period prescribe.

(4) Determine the optimal multi-project scheduling plan

According to the result of step (2) and (3), we can get the optimal dispatch plan and schedule of multi-project(the optimal human resource allocation and start time in each task of the multi-project).

4 Simulation example

Taking human resources allocation of multi-project

of a company as example, verify the human resources allocation model of dynamic multi-project which is proposed in this paper. The basic reference datas of Project A, B are shown in Table 4, the unit price of resource1 is 3 yuan each, the unit price of resource2 is 5yuan each. There are a total of 12 people in human resources. And provide that the number of the people allocated in each stage of the multi-project is no more than 7 people. Because the calculating process is very complicated, so use the method of MATLAB[21-22] to solve.

Fig.2 The total project divided phases diagram We can calculate the staff time coefficient of 12

people performing nine tasks of project A and B by formula 9 and show in table 5.

Tab.3 The basic situation of project A, B The project

Task Fixed working

hours

Resource1 constraints

Resource2 constraints

Number of

provisions A A1 7 170 140 2

A2 9 175 165 2 A3 12 315 370 3 A4 10 240 220 2

B

B1 19 380 340 2 B2 12 360 300 2 B3 12 260 220 2 B4 14 500 410 3 B5 13 280 240 2

Tab.4 The resource consumption

Staff 1 … 12 resource 1

consumption rate 10 … 11

resource 2 consumption rate 11 … 13

Tab.5 Time coefficient of each staff relative to each task

Time coefficient Personnel 1 … Personnel 12 A1 1.1556 … 0.4720 A2 1.4928 … 1.4896 A3 1.4625 … 1.0029 A4 0.8724 … 0.9124 B1 1.2143 … 0.6939 B2 1.0909 … 1.1250 B3 1.0211 … 1.4857 B4 0.9774 … 0.7552 B5 0.9625 … 1.2727

We can find the portfolio feasible solution of 11

people performing task A1 of project A using mixed integer programming model in table 6. And we can also get the portfolio feasible solutions of remaining 8 tasks in the same way.

Figure 2 is multi-project stages dividing network diagram. And the key process A2-B1-B3-B5 established based on multi-project network diagram can be divided into 4 stages.

We can determine the start time of every tasks based on multi-project human resource optimal scheduling program and the fact that project A is better than project B and there are 7 persons can be assigned in one stage. The actual number of participants in the project for this program is 9; The total duration is 46; The least total cost is 16736, shown in table 7.

Comparing Figure 3 and Figure 4, the optimized period of project A, B is 46 days. The period before optimizing schedule is 53 days. The total construction period is shortened 7 days. So multi-project human resources allocation method based on dynamic programming method with strong operability, improves the rationality of project personnel allocation, reduces the project cost, eases the resource conflicts, and provides a more scientific and efficient method for multi-project human resources allocation in enterprises.

- 1017 -

Tab.6 Staff-time combination solution set table to task A1 of project A Task Personnel

combinations Start time The actual task

time Resource 1

consumption Resource 2

consumption The total cost

of the task A1 1，2 0 6 120 114 930 A2 6，10 0 8 152 128 1096 A3 1，4，6 9 11 319 330 2607 A4 8，10 26 10 210 180 1530 B1 8，10 9 17 190 190 1520 B2 2，12 9 11 273 273 2184 B3 3，9 26 11 253 209 1804 B4 1，6，12 20 17 493 408 3519 B5 1，2 27 10 200 190 1550

Total 46 2187 2033 16736 Tab.7 Multi-project optimal scheduling plan

Task Solution No. Personnel combinations

The actual task time

Resource 1 Consumption

Resource 2 Consumption

The total cost of the task

A1 1 1，2 6 120 114 930 … … … … … … 11 9，12 6 102 138 996

Fig.3 Personnel-task gantt of total project before optimized

Notes: The black bar indicates the key line, gray bar indicates the noncritical path.

Fig.4 Personnel-task gantt of total project after optimized

5 Conclusion

While most scholars have focused on studying how to optimize the number of human resources to meet the needs of the project, this article focuses on how to improve the quality of multi-project human resources, puts forward improved cube model and propose the method to estimate staff time coefficient from the angle of human and post matching. Then use improved TOPSIS method based on the entropy weight to evaluate the priority of multi-project and use backpacks principle to build human resource allocation model for resource-constrained. Next use dynamic programming to optimize and solve, finally obtain the optimized multi-project personnel distribution scheme and multi-project schedule. And solve and verify the feasibility of the proposed method according to an example. The result proves that the multi-project human resource allocation method based on dynamic

programming is feasible. The proposed algorithm provides new ideas for solving the multi-project human resources.

References [1]Bellenguez O, Neron E. Lower bounds for the multi-skill project scheduling problem with hierarchical levels of skills [J]. PATAT, 2005: 229-243. [2]Bellenguez O. Methods to solve multi-skill project scheduling problem [J]. 4OR-A Quarterly Journal of Operations Research, 2008, 6(1): 86-88. [3]Perron L. Planning and scheduling teams of skilled workers [J]. Journal of Intelligent Manufacturing, 2010(21): 155-164. [4]Heimerl C, Kolisch R. Scheduling and staffing multiple projects with a multi-skilled workforce [J]. OR Spectrum, 2010, 32: 343-368. [5]Cai X, Li K N. A genctic algorithm for scheduling

- 1018 -

staff of mixed skills under multi-criteria [J]. European Journal of Operational Research, 2000: 125: 359-369. [6]Hegazy T, Shabeeb A, Elbcltagi E, Cheema T. Algorithm for scheduling with multi-skilled constrained resources [J]. Joural of Construction Engineering and Management, 2000(6): 414-421. [7]Drezeta L, Billaut J. A project scheduling problem with labour constraints and time-dependent activities requirements [J]. International Journal of Production Economics, 2008, 112: 217-225. [8]Vails V, Perez A, Quintanilla S. Skilled workforce scheduling in service centres [J]. European Journal of Operational Research, 2009, 193: 791-804. [9]Jain V, Grossmann I. Algorithms for Hybrid MILP/CP Models for a class of optimization problems [J]. INFORMS Journal on Computing, 2001, 13(4): 258-276. [10]Li H, Womer K. Scheduling projects with multi-skilled personnel by a hybrid MILP CP benders decomposition algorithm [J]. Journal of Scheduling, 2009(12): 281-298. [11]Li H, Womer K. Project scheduling with multi-purpose resources: A combined MILP/CP decomposition approach [J]. International Journal of Operations and Quantitative Management, 2006, 12(4): 305-325. [12]Vairaktarakis G L. The value of resource flexibility in the resource-constrained job assignment problem [J]. Management Science, 2003, 49(6): 718-732. [13]Bellenguez O. Méthodes de Résolution pour un problem de gestion de project multi-compétence. Ph.D. dissertation, Université Francqis Rabelais Tours, 2006. [14]KardrouY, Najid N M. Tabu search algorithm for the MRCPSP with multi-skilled labor, CPI’ 2007, Rabat,

Maroc. [15]Kardrou Y, Najid N M. A new heuristic to solve RCPSP with multiple execution modes and multi-skilled labor [C]// Proceedings of IMACS Multicomference on “Computational Engineering in Systems Applications” (CESA), October, 2006, Beijing. [16]Drezet L, Billaut J. A project scheduling problem with labour constraints and time-dependent activities requirements [J]. European Journal of Operational Research, 2008, 112: 217-225. [17]Wu W, Sun S. A project scheduling and staff assignment model considering learning effect [J]. International Journal Advanced Manufacture Technology, 2006, 28: 1190-1195. [18]Heimerl C, Kolisch R. Scheduling and staffing multiple projects with a multi-skilled workforce [J]. DOI 10.1007/s00291-009-0169-4, 2009. [19]Fu Fang. Research on human resource scheduling based on immune genetic algorithm and column generation [J]. Chinese Journal of Management Science, 2010, 18(2): 120-126. [20]Wang Shating. Study on dynamic optimization allocation of the enterprise multi-project human resources [D]. Xi’an: Northwestern Polytechnical University, 2011. (in Chinese) [21]Jia Qiuling, Yuan Dongli, Luan Yunfeng. System simulation, analysis and design based on matlab7.x/ Simulink/state flow [M]. Xi’an: Northwestern Polytechnical University Press, 2012. (in Chinese) [22]Zhang Kun. Matlab from entry to the master [M]. Beijing: Electronic Industry Press, 2012. (in Chinese)