Optimization of Project Resource Based on Concept Lattice and AHP by Co. SEP

International Journal of Management Science and Engineering Research Volume 2 Issue 1, June 2015 doi: 10.14355/ijmser.2015.0201.03

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Optimization of Project Resource Based on Concept Lattice and AHP Li Cun-bin, Liu Yun-qi*, Li Shu-ke School of Economic & Management, North china Electric Power University 2 Beinong Road, Huilongguan Town, Changping District, Beijing, 102206, China *

liu_yunqi@163.com

Abstract Multi-project resources optimization is the goal of the project management, resource optimization is a kind of optimal allocation of resources, which is operated among similar resource projects .Based on the theory of formal concept lattice, this article conducts clustering analysis for multiple projects to find out the projects of higher resource similarity, then uses AHP method to evaluate their priority in order to determine optimal resources allocation scheme. Finally, the method is validated by an example. This method will help the implementation of resources optimization work for the project manager. Keywords Multi-Project Management; Resource Optimization; Concept Lattice; AHP

Introduction With the rapid development of social economy, the scale of modern construction enterprise is continuous expansion [1], Enterprise business projects are on the increase, most of which are running in multi project management environment [2]. According to previous researches, an investigation about construction contractors from small to mid-size showed that 84% of construction contractors executed their projects in a multi-project environment [3]. Construction enterprises managing multiple projects at the same time can be seen everywhere. The core and key of multi project management is how to realize the most effective and reasonable allocation with the limited resources so as to optimally meet the demand for resources of different projects and maximize eventually enterprise benefit. The rationality of resources allocation will greatly affect the success rate of each project, which is a quite important in multi project management .In a multi-project environment, scheduling problems with resource constraints are much more complicated than those in a single project. The Resource Constrained Multi-project Scheduling Problem (RCMPSP) is a NP-hard optimization problem [4], and is one of the most important topics in the field of project management [5]. The solution largely depends on the task modes and task scheduling schemes [6]. In order to increase resource utilization, managers try to allocate all available resources for all planned projects at the beginning of a new cycle [7]. In this article, we introduced the related theory knowledge at the first; next, an optimization model was constructed; finally, the model was verified through applying in an example. Methodology Concept lattice (Galois) is a concept hierarchy structure which is proposed according to the binary relation, for information retrieval and extraction rule. The process of from formal context to generation of concept lattice is a kind of conceptual clustering process [8-10]. The concept lattice can be represented in graphical methods as labeled line graph (labelled line diagram), which is called the Hasse diagram. Concept lattice Hasse diagram can be stratified according to the number of attributes contained in the nodes, attributesâ&#x20AC;&#x2122; numbers of all formal concepts are arranged in order, then the hierarchy numbers of the formal concept node can be represented as ordinal number of formal concept attribute sorting. If the number of attributes for concept lattices acts as hierarchical basis, then the hierarchy number of formal concept C is expressed as Lm (C ) , the hierarchy number of the top node is 1. Generally speaking, the bigger layer number of formal concept is, the more numbers of attributes it contains, the

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International Journal of Management Science and Engineering Research Volume 2, Issue 1, June 2015

less numbers of objects there will be. With the increase of the layer number of nodes, knowledge content which is used to express is more clear and specific. Formal context is usually defined as a triad: K := (G, M , R) , where G is defined as: a set of all objects, M is defined as: a = B( K ) ( B(G, M , R), ≤) is the set of all attributes, R is defined as: binary relation between G and M , i.e. R ⊆ G × M . Assume concept lattice of formal context K , g ∈ G is an object of concept lattice, A1' , A2' , , An' ⊆ G is a set of objects, M 1' , M 2' , , M n' ⊆ M

is a set of attributes,

( A , B ), ( A , B ), , ( A , B ) , ' 1

' 1

' 2

' n

g ∈ A1' , A2' , , An'

, If meet the constraint that

( Ai' , Bi' )

is minimum formal concept of

namely A is minimum in A , A , , A While B is maximum in B , B2' , , Bn' , ' i

' 1

' 2

' n

' i

C ' ( g ) = ( Ai' , Bi' )

' 1

is defined

as the attribute formal concept of object g . where, n ≤ N ( N is the numbers of formal concept of concept lattice B( K ) ) , i ∈ [1, n] . Assume A1' , B1' , A2' , B2' , C1' , C2' is a non-empty formal concept subset of B( K ) , C ' ( g1 ) and C ' ( g 2 ) are two formal concepts of B( K ) , which represents respectively the attribute formal concept of C ( g2 ) ≤ B '

is,

' 1

' 2

' 1

sup(C )

, that is , A is super concept set of C ( g1 ) , B is super concept set of ' 1

is the sub concept set of of C and the infimum ' 1

' 2

C ( g1）, B '

inf(C )

' 2

' 1

is the sub concept set of

and

C ( g 2 ) , C= A ∩ B '

' 1

C ( g 2）, C= A ∩ B '

' 2

, where,

C ' ( g1 ) ≤ A1'

≤ C ( g1）, B ≤ C ( g 2）, '

' 2

that

. Then the supremum

meet respectively: If ∀X ∈ C , always be X ≤ sup(C ) ; if ∀X ∈ C2' , always be ' 1

' 1

Y ≥ inf(C2' ) .

Construction of the Model Similarity Analysis of Multi Project Resource When conducting resource adjustment between projects, policy makers consider firstly similar project and resource, such adjustment of resources is relatively easy. The similarity of the project j and the project i is denoted by S ji (0 ≤ S ji ≤ 1) , I.e. one similar degree can be described like that the two projects needed resources, the higher S ji value represents greater similarity, the smaller S ji value represents the smaller similarity. This article adopts the research methods of literature [11], establishing relation between similarity calculation of concept node and of objects by delimiting the characteristics of formal concept, introducing supremum and infimum parameters of the characteristics of formal concept, considering comprehensively the semantic distance, semantic coincidence degree, node depth in concept lattice and hierarchy factors and so on, using the semantic similarity algorithm based on formal concept analysis to achieve similarity calculation between project resources based on knowledge context. Similarity of knowledge is defined as: different project resources contained exists the corresponding concepts, and these concepts exist the same relation. Similarity analysis of resource knowledge context in the multi-project, as shown in Fig. 1. Project 1 context knowledges

Project 3 context knowledges

Project 2 context knowledges

Similar attributes Similarity relationships

Project n context knowledges

FIG. 1 SIMILARITY ANALYSIS OF PROJECT KNOWLEDGE CONTEXT

Formal context

can be represented by a two-dimensional matrix, each row represents a project in G ' , each

column represents an attribute of M ' . At the intersection of the row g and the column m , if object g requires resources

(project g requires resources m ), then value 1 is assigned; otherwise, value 0 is assigned.

The constructing process of concept lattice is a process of conceptual clustering, an improved algorithm was used in this article based on Godin algorithm [12]. The improved algorithm reduces the traversal times when

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determining the father -son relationship among the generated new lattice nodes. When calculating similarity of two objects g1 and g 2 in the concept lattice, that is, the group needs to solve attributes form concept of the objects C ' ( g1 ) and C ' ( g 2 ) ,

using the supremum of formal concept C ' ( g1 ) and

of formal concept expressed as:

C ( g1 ) and C ( g 2 ) '

Sim( g1 , g 2 ) ,

C ' ( g2 )

instead of attribute intersection, infimum

instead of attribute union. Similarity between g1 and g 2 of concept lattice can be

The formula is as follows: Sim( g1 , g 2 ) =

(1)

Lm (sup(C1' )) Lm (inf(C2' ))

where C1' is the intersection of the super concept of C ' ( g1 ) and C ' ( g 2 ) ; C2' is the intersection of the sub concept of C ' ( g1 ) and

C ' ( g 2 ) ; sup

and inf represent respectively supremum and infimum of formal concept;

represents

hierarchy number of formal concept; The value range of Sim( g1 , g 2 ) is[0,1]. Assessment Model of Multi Project Priority For resource allocation between multi projects, the decision maker will evaluate the project priorities to determine the sequence of each project for solving parallel projects of various types and priorities [13]. The higher grade of the project should be given more priority to complete it and should be properly priority arranged in the allocation of project resources [1, 14]. With reference to the part of literatures, evaluation index system of project priority is established, which sets the 3 dimensions, 13 indicators and 5 grade standard in this article, as shown in table 1. This article will adopt the subjective weigh method, through expert grade the importance of indexes, the weight of each index which is calculated by using the eigenvalue method. The article applies fuzzy theory to construct evaluation model of multi project priority. Experts are divided into ( g1, g 2, g3 ) three classes according to the degree of understanding of the project indicators, the evaluation of the index is that various expert k given is converted into fuzzy number, each index value is obtained by weighting average value of evaluation results that three kinds of experts gave, finally fuzzy value of any project I m can be expressed as: g1 g2 g3   ai1k ai2 k ai3k  ∑ ∑ ∑  A A A A + I m = ω1 ⋅ ∑= ωi ⋅  e1 ⋅ k 1 = + e2 ⋅ k 1 = + e3 ⋅ k 1 g1 g2 g3   i =1     g1 g2 g3   bj2 k ∑ bj1k B ∑ ∑ bj3k   4 + ω2 ⋅ ∑= ω Bj ⋅  e1B ⋅ k 1 = + e2 ⋅ k 1 = + e3B ⋅ k 1 g1 g2 g3   j =1     g1 g2 g3   ch2 k ∑ ch1k C ∑ ∑ ch3k   4 C C k 1= C k =1 k 1 =  ω3 ⋅ ∑ ωh ⋅  e1 ⋅ + e2 ⋅ + e3 ⋅ g1 g2 g3   h =1     = = (i 1, 2,3, 4,5; j 1,= 2,3, 4; h 1, 2,3, 4) 5

(2)

where ω1 , ω2 , ω3 respectively represents weights of three dimensions; ωi , ω j , ωh respectively represents weights of each index of each dimension;

e1A , e2A , e3A

respectively represents the weigh that three kinds of experts assigned on

the each evaluation index; ai1k , bj1k , ch1k respectively represents evaluation value of indicators in three dimensions that the

expert of the first category experts gave .

The projects that have the resource similarity are sorted according to the evaluation scores, the high priority project can get more resources from the low priority project, in order to ensure the high priority project can be completed as planned and to achieve the purpose of multi project resource optimization. Example Analysis A wind power construction enterprise was selected as a case study, in which the enterprise has more than 100 projects at the same time in the construction, a total of 10 major categories of project resources. This article uses the

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Matlab to calculate data and simulate optimization process in multi projects. Similarity Calculating of Multi Projects Step 1: In multi projects, object set of projects-resources context 100; All project resources include

a  j 10

G ' = {1, 2,3, ,100} ,

categories, that is, attribute set M

that is, the number of projects is

= {a, b, c, d , e, f , g , h, i, j} ,

Which can construct

the corresponding formal context. Project-resources knowledge context matrix is shown in Table 2. TABLE 1 PROJECT PRIORITY ASSESSMENT SYSTEM, STANDARD AND FUZZY NUMBER EXPRESSION

first grade indices

Second grade indices

evaluation criterion

project contract sum A1 Project dimension A

very small

small

medium

big

very big

project profit per unit time A2

very low

low

medium

high

very high

project operation stability A3

very low

low

normal

high

very high

project technical difficulty and complexity A4

very low

low

normal

high

very high

Project team management ability A5

very weak

weak

medium

strong

very strong

owner payments ability B1

very weak

weak

medium

strong

very strong

owner support level B2

very low

low

medium

high

very high

owner impact on project B3

very small

small

normal

big

very big

project schedule and quality requirements B4

very low

low

normal

high

very high

project strategic integrity C1

very low

low

normal

high

very high

owner dimension B

enterprise dimension C

Project brand image C2

very bad

bad

good

better

much better

Project collaboration. C3

very bad

bad

good

better

much better

project accumulated experience C4

very little

little

ordinary

much

(0,0,2.5)

(0,2.5,5)

(2.5,5,7)

(5,7,10)

(7.5,10,10)

Fuzzy number expression

TABLE 2 PROJECT-RESOURCES KNOWLEDGE CONTEXT

Resource Category a b c d e f g h i j

1 1 0 0 1 1 0 1 0 0 1

2 1 0 1 1 0 0 0 1 0 1

3 0 0 1 0 0 0 0 1 0 0

4 0 0 0 0 0 0 0 0 0 0

5 1 0 0 1 1 0 1 0 0 0

Object Set Elements 6 7 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0

8 1 0 0 0 0 1 0 0 0 0

9 0 1 0 1 1 0 1 0 1 0

10 1 0 1 1 0 0 0 0 0 0

… … … … … … … … … … …

100 0 0 1 0 0 0 1 0 0 1

Step 2: According to improvement algorithm of the incremental construction concept lattice Godin algorithm, by using project-resources knowledge context in Table2, visualization concept lattice is generated, which is displayed by corresponding Hasse diagram, as shown in Fig. 2. Step 3: Based on project-resources knowledge context in Table2, using improvement algorithm of Godin algorithm to calculate resource similarity between projects, resource similarity matrix was captured, as shown in Table 3. Priority Assessing of Multi Projects Step 1: Assessing each project in accordance with the method, evaluation value of multi project priority can be obtained, results are those as shown in table 4. Step 2: According to the data in Table 4 the project priority relation value Pji can be drawn, as shown in Table 5. Analysis and Optimization of Project Resources As it can be seen from table 3 that project 5, project 2, project 3 has as higher resource similarity as project 1; from

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table 5 it can be seen that the priority of project 1 is higher than those of project 2, project 3 and project 4. If the project1 in the construction process needs more resources, it can get from the project 5, project 2, project 3 etc.. When the project3 needs more resources, although there are a lot of projects’ resources which are similar to project 3, the project 3 can’t get more resources from other projects due to its relatively low priority. This case can provide a resource optimization scheme for multi project management and can facilitate the adjustment of resources in the actual construction process. (1,2,3,4,5,6,7,8,9, 10,11,…,100;Ø) (2,3,9,10, 11,…;c)

(1,2,5,6,7,8, 9,10,…;a,d)

(1,2,3,4,…;j)

(2,5,6,7,9,1 0,…;a,d) (2,9,10,…; a,c,d) (5,6,7,9…; a,d,e,g)

(9,10,11…; b,c) (2,5,6,7,9,1 0,…;a,d)

(7,…;a,d ,e,g)

(2,3,…;c,h)

(2,…;a,c, d,h,j) (Ø;a,b,c,d,e,f,g,h,i,j)

FIG. 2 PROJECT-RESOURCES FORMAL CONCEPT LATTICE HASSE DIAGRAM TABLE 3 RESOURCE SIMILARITY BETWEEN PROJECTS BASED ON KNOWLEDGE ATTRIBUTES

Project

…

100

1.00

0.43

0.14

0.50

…

0.14

0.43

1.00

0.43

0.29

0.44

…

0.29

0.14

0.43

1.00

0.14

0.11

…

0.14

0.29

0.14

1.00

0.11

…

1.00

0.50

0.44

0.11

1.00

…

0.11

…

1.00

…

100

0.14

0.29

0.14

1.00

0.11

…

1.00

TABLE 4 EVALUATION VALUE OF PROJECT PRIORITY

Project

…

100

Value

0.82

0.77

0.15

0.56

0.32

…

0.47

TABLE 5 PROJECT PRIORITY RELATION

Pji

VALUE

Project

…

100

…

-1

…

-1

…

-1

…

-1

…

-1

…

100

-1

…

Conclusion With the development of the enterprise, the changing of the enterprise external environment, and the increasing of enterprises project tasks, more and more enterprises face many challenges of projects concurrently. In the parallel

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implementation of multiple projects, effective solutions to the multi-project resource allocation problem in terms of money, time, manpower and other resources between the projects can ensure the projects are completed as planned according to quality and schedule requirement and can improve enterprise economic benefits and customer satisfaction. In view of the enterprise multi-project resource optimization configuration problems, this article introduced formal concept lattice into clustering analysis for multiple projects which have similar resources, and then used AHP method to evaluate their priority to get resources optimization method. The method was verified through an example in the real project management, which has strong applicability. ACKNOWLEDGMENT

This research is partially funded by the National Natural Science Foundation of China (71271084), the Fundamental Research Funds for the Central Universities (2014XS55; 2015XS37) and the Project for The Beijingâ&#x20AC;&#x2122;s EnterpriseAcademics-Research Co-Culture Post-Graduate. REFERENCES

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