A refined selection method for project portfolio optimization considering project interactions

A Refined Selection Method for Project Portfolio Optimization Considering Project Interactions

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A Refined Selection Method for Project Portfolio Optimization Considering Project Interactions Hechuan Wei, Caiyun Niu, Boyuan Xia, Yajie Dou, Xuejun Hu PII: DOI: Reference:

S0957-4174(19)30670-0 https://doi.org/10.1016/j.eswa.2019.112952 ESWA 112952

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Expert Systems With Applications

Received date: Revised date: Accepted date:

9 April 2019 24 June 2019 12 September 2019

Please cite this article as: Hechuan Wei, Caiyun Niu, Boyuan Xia, Yajie Dou, Xuejun Hu, A Refined Selection Method for Project Portfolio Optimization Considering Project Interactions, Expert Systems With Applications (2019), doi: https://doi.org/10.1016/j.eswa.2019.112952

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Highlights • The project co-utilization network indicating project interactions is constructed. • The project interaction is used as a regularization item of the value function. • A refined selection strategy is proposed to get the optimal one in the Pareto set.

1

A Refined Selection Method for Project Portfolio Optimization Considering Project Interactions Hechuan Weia , Caiyun Niub , Boyuan Xiab , Yajie Doub,∗, Xuejun Huc a College

of Information Science and Engineering, Northeastern University, ShenYang, Liaoning, 110004, P.R. China b College of Systems Engineering, National University of Defense Technology, Changsha, Hunan, 410073, P.R. China c Business School, Hunan University, Changsha, Hunan, 410073, P.R.China

Abstract This paper proposes two innovative approaches to address the two challenges of project portfolio selection: (i) determining how project interactions influence the final values of project portfolios, and (ii) selecting the best solution from non-dominated project portfolios. First, based on the dependence relationship between projects and technologies, we construct a project co-utilization network indicating project interactions using the co-citation network method. We also consider project interaction as a regularization item of the value function to indicate the influence of project interactions on the value of project portfolios. Second, we propose a refined selection strategy. All the projects are ranked in the order of their appearance in the ranked association rules. Project portfolios having no highly ranked projects are deleted successively until only one of them remains. Finally, a case of multi-objective project portfolio selection is studied and an optimal project portfolio recommended. A comparison shows the availability and advantages of the proposed approaches. Keywords: project portfolio, project interaction, multi-objective, non-dominated solutions, refined selection, value and risk

✩ Fully

documented templates are available in the elsarticle package on CTAN. author Email addresses: [email protected] (Hechuan Wei), [email protected] (Caiyun Niu), [email protected] (Boyuan Xia), [email protected] (Yajie Dou), [email protected] (Xuejun Hu) ∗ Corresponding

Preprint submitted to Elsevier

September 16, 2019

1

1. Introduction

2

Project portfolio selection has gained increased interest and attention in the

3

fields of public administration (Voss & Kock , 2013), industrial firms, enterprises

4

(Martinsuo et al. , 2014), and military (Zhang et al. , 2016) during the past

5

decades. It focuses on selecting the project proposals under limited resources to

6

maximize the benefits of stakeholders with multiple evaluation criteria (Paquin

7

et al. , 2016). However, practitioners face two challenges in selecting the op-

8

timal project portfolio. First, the wide interactions between projects influence

9

the real value and risk of a project portfolio. Second, since project portfolio

10

optimization always has multiple objectives, common multi-objective optimiza-

11

tion methods are efficient to obtain the non-dominated solutions, but they raise

12

the problem of how to further select the optimal project portfolio from these so-

13

lutions. Therefore, a refined selection method for project portfolio optimization

14

based on project interactions has great research significance.

15

Project portfolio research and development (R&D) in the existing literature

16

can be summarized as follows: (i) dividing the projects into tasks, with focus on

17

how they can be implemented effectively (Carazo et al. , 2010). (ii) assessing

18

the value of projects in order to determine funding policies aimed at maximizing

19

their total value (Medaglia et al. , 2007). (iii) analysing how project synergies

20

affect the value and expected performance of projects (Liesi¨ o & Salo , 2012).

21

In this paper, project portfolio selection focuses on project programming rather

22

than project engineering. Therefore, this study focuses on which projects should

23

be selected for implementation, rather than how to complete the projects.

24

The portfolio idea in the financial domain was first applied to address the

25

asset allocation problem. A major breakthrough was the publication of the

26

mean-variance method, proposed by Markowitz in 1952. This method has been

27

regarded as the modern portfolio theory (Kolm et al. , 2014). The financial

28

portfolio problem gives great importance to the trade-off between return and

29

risk, and is therefore inconsistent with the essence of project portfolio selection,

30

which is to select the optimal portfolio from among several project proposals

3

31

within resource constraints and maximize stakeholder benefits (Petit & Hobbs

32

, 2010). Additionally, for the financial portfolio, the proportion or weight of

33

alternative stocks can be distributed and modified with discretion, but project

34

portfolio selection is analogous to the Boolean problem, where each project can

35

have only selected/unselected options (Eilat et al. , 2006). Thus, financial

36

portfolio selection consists of continuous linear programming, whereas project

37

portfolio selection adopts integer programming. Obviously, the decision-making

38

methods and models of financial portfolios cannot be directly applied to non-

39

financial project portfolio selection problems.

40

Furthermore, project interaction or dependence draws less attention in the

41

project portfolios R&D. According to the existing literature, Ghasemi et al.

42

(Ghasemi et al. , 2018) proposed a Bayesian network methodology to model and

43

analyse the portfolio risks, considering project interdependencies and the cause-

44

effect relationship between risks as interaction factors. Takami et al. (Takami

45

et al. , 2018) used the hesitant fuzzy weighted averaging operator to aggregate

46

the project interaction hesitant fuzzy information that is considered to affect the

47

final portfolio values. Eilat et al. (Eilat et al. , 2006) considered a combination

48

of the extended data envelopment analysis model and balanced scorecard ap-

49

proach to evaluate the individual R&D projects and alternative R&D portfolios

50

with interactions. Ying et al. (Ying & Liu , 2017) discussed how uncertainty

51

and interaction impact the return and staff allocation of project portfolios from

52

the critical value criterion perspective. In addition, because the exact possibility

53

distributions of uncertain parameters are often not available, the variable para-

54

metric possibility distributions are used to characterize the uncertainty interac-

55

tion parameters. Liesi¨ o et al. (Liesi , 2008) used a robust portfolio modelling

56

method to account for project interdependencies by incorporating synergies and

57

the portfolio positioning requirements.

58

From the literature, project portfolio optimization draws more attention

59

from researchers than project interaction. This can be classified into single

60

objective and multi-objective optimization. The latter is more widely stud-

61

ied in the literature. Several studies confuse multi-objective optimization with 4

62

multi-criterion decision since they directly transform multiple objectives to sin-

63

gle objective through the weighting operation. In fact, numerous concepts

64

and algorithms are specially defined for multi-objective optimizations. The

65

widely adopted algorithms include the non-dominated sorting genetic algorithm

66

(NSGA) (Deb & Jain , 2014), non-dominated sorting and local search (Chen

67

et al. , 2015), strength Pareto evolution algorithm (SPEA) (Shankar & Baviskar

68

, 2018), niched Pareto genetic algorithm (Tongur & lker , 2016), Pareto-archived

69

evolution strategy (Rostami & Neri , 2016), multi-objective evolutionary algo-

70

rithms based on decomposition (Zhang & Hui , 2007) and so on. These algo-

71

rithms have been successfully used in diverse areas. However, since the decision

72

makers always expect only one solution, these algorithms have the common

73

problem of being dedicated to finding the best Pareto solutions set, instead of

74

the best solution. Therefore, the Pareto set should be further refined to find

75

the optimal solution.

76

Motivated by the aforementioned challenges and also following the literature

77

discussed, this study tries to go deeper into the project interactions and propose

78

a new refined selection strategy for project portfolio optimization. The purposes

79

of the paper are to: (i) define a representative project interaction model and

80

determine how project interactions influence the final value of a project portfo-

81

lio, and (ii) construct a refined selection method to obtain the optimal solution

82

from obtained non-dominated project portfolios. By applying the above meth-

83

ods, the finally recommended project portfolio can convincingly benefic after

84

implementation.

85

The study has three main contributions. First, it determines project inter-

86

action degree by adopting the project co-utilization model transformed from

87

co-citation network, rather than by subjective experience. This model is proven

88

to be effective, operable, and interpretable. Second, the method can directly

89

provide decision makers with the optimal solution without asking them to deter-

90

mine the optimal solution from the non-dominated set, and thus saves decision

91

cost and effort. Third, it proposes a clear, concise, and reasonable integrated

92

framework that addresses the project portfolio optimization problem consider5

93

ing project interactions and can be applied directly to enterprise practices.

94

The remainder of the paper is structured as follows. Section 2 introduces

95

the value and risk calculation models, which take account of project interaction,

96

modelled using the co-utilization network. Section 3 studies the optimal project

97

portfolio selection method by illustrating how to obtain the optimal solution

98

through a refined selection strategy. Section 4 presents an application of the

99

proposed models and approaches to verify their feasibility and effectiveness.

100

101

102

103

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2. Value and Risk Model Construction Considering Projects Interactions 2.1. Notations Definition The section introduces and defines notations that will be used later in the study.

105

(1) n, the number of project proposals.

106

(2) m, the number of concerned technologies.

107

(3) s, the number of company strategies.

108

(4) C = [c1 , c2 , ..., cn ] , the vector of project costs.

109

(5) b, the total budget.

110

(6) X, the solution space of total project portfolios.

111

(7) Xf , the set of feasible solutions.

112

(8) Xe , the set of efficient project portfolios.

113

(9) P = {p1 , ..., pn }, the set of alternative projects.

114

(10) xi = [xij ]1×n , j = 1, 2, ..., n, a project portfolio solution.

115

(11) x+ i = {i|xij 6= 0, j = 1, ..., n}, the set of nonzero elements of xi .

116

(12) T P = [tij ]m×n , the dependence relation matrix of technologies and

117

T

projects.

118

(13) P U = [puxy ]n×n , the project co-utilization network.

119

(14) R = [r1 , r2 , ..., rn ] , the vector of projects revenue.

120

(15) A = [aij ]n×t , the alignment degree matrix of projects with company

121

T

strategies.

6

122

(16) V (xi ), the value of a project portfolio solution xi .

123

(17) E = [e1 , e2 , ..., et ] , the weight vector of company strategies.

124

(18) α, the trade-off parameter for revenue and alignment of company strate-

125

T

gies.

126

(19) RI (xi ), the risk index of a project portfolio solution xi .

127

(20) w = [wj ], j = 1, 2, ..., m, the weight vector of technologies.

128

(21) T RL, technology readiness level.

129

(22) γ1 , γ2 , the parameters that determine the effects of exceeding part on

130

the fitness values.

131

(23) r1 , r2 , and r3 , individuals randomly selected in the current population.

132

(24) r, rbest , the current individual and the optimal individual in the popu-

133

134

135

136

lation at current moment. (25) r0 , r00 , the new individual after mutation operation and the crossover operation. (26) p, the probability that each code bit of r0 be replaced by the corre-

138

sponding bit of parent r.   (27) Cro = cro1 , cro2 , ..., cro|r| , the crossover operator.

139

2.2. Project Interaction Model based on Co-citation Network

137

140

Technologies are interpreted as project supporters, while projects are re-

141

garded as technology utilizers. This paper denotes the technology-project de-

142

pendence relations by a matrix T P = [tij ]m×n , where tij = 1 indicates technol-

143

ogy ti is required for project pj . This kind of dependence relations is analogous

144

to the citation network. A project co-utilization network can be constructed

145

using the co-citation method.

146

In a co-citation network, if two papers are cited together, a co-citation rela-

147

tion exists between the two papers. By mapping this to the project co-utilization

148

network, two projects are treated as having a co-utilization relation if they re-

149

quire the same technology. The amount of technology simultaneously utilized

150

by two projects are set as the weight of the co-utilization edge between the

151

projects. Therefore, the project co-utilization network P U = (puxy )n×n can be 7

152

constructed, where puxy represents the amount of technology required simulta-

153

neously by projects x and y. Following the adjacency network, if technology

154

k simultaneously points to project t and project l, then tkt tkl = 1; otherwise,

155

156

157

158

tkt tkl = 0. Therefore, the co-utilization amount of project t and l can be calPn T culated as putl = k=1 tkt tkl . One can prove that P U = (T P ) (T P ), and that the diagonal elements conform to Equation 1. In fact, putt is equal to the amount of technology required by project t.

putt =

n X

tkt 2 =

k=1

n X

tkt

(1)

k=1

159

The following example illustrates this transformation. Assume a T P3×4

160

network, with the adjacency matrix indicated as T P ; see Equation 2. Then, the

161

co-utilization network can be calculated as P U , which is a symmetric matrix.

162

For the transformation process, see Figure 1.



1   TP =  1  0

1

1

1

0

1

0





2 2 1 1



       2 3 1 2  T   1  , P U = (T P ) T P =     1 1 1 0    1 1 2 0 2 0

(2)

Figure 1: Transformation from T P network to P U network

163

Projects 1 and 2 co-utilize technologies 1 and 2. Projects 1 and 3 co-utilize

164

technology 1. Projects 1 and 4 co-utilize technology 2. Projects 2 and 3 co-

165

utilize technology 1. Projects 2 and 4 co-utilize technologies 2 and 3. Projects

166

3 and 4 have no technology co-utilization relations.

8

167

2.3. Value Determination Model

168

Typically, the value of a project mainly consists of two aspects, (i) the rev-

169

enue of the project itself, (ii) the alignment degrees with company strategies.

170

In this paper, the project revenue is indicated by Net Present Value; the align-

171

ment degrees with company strategies is determined by experts experience. The

172

revenue of project pi is quantitatively denoted by ri . The alignment degrees of

173

project pi with t company strategies are denoted as aij , j = 1, 2, ...t. The overall

174

value of a project portfolio solution xi can be expressed as Equation 3.

ξ

V (xi ) = [α × (xi · R) + (1 − α) × (xi · A · E)] + [xi (P U − λ)xi T ] where : α ∈ [0, 1], xi = [xi1 , ..., xin ], xij = 0/1, R = [r1 , r2 , ..., rn ]T ,

A = [aij ]n×t , t P E = [e1 , e2 , ..., et ] , ei = 1, i=1   pu11 0 0 0      0 pu22 · · · 0  . λ=    0 0 ··· 0    0 0 0 punn T

(3)

175

where xi · R indicates the total revenue of the project portfolio xi ; xi · A · w de-

176

notes the total alignment degree with company strategies of xi ; w is the weight

177

information of t company strategies; α is a trade-off parameter determining the

178

importance degree of ”revenue” and ”alignment with company strategies” to

179

180

181

182

183

project portfolio value. Matrix λ is the diagonal elements of P U . Notation
ξ xi (P U − λ) xi T is the regularization item, reflecting the project interaction
effect on the project portfolio values. xi (P U − λ) xi T represents the regular-

ization coefficient indicating the total number of interactions of project portfolio xi ; ξ indicates the exponential item of xi (P U − λ) xi T .

9

184

2.4. Risk Determination Model

185

Risk is determined on two aspects: the technology readiness level (TRL), a

186

widely used measure for readiness degree of a single technology ranging from 1

187

to 9 (Crosbie et al. , 2018), and the co-utilization relation on technologies. An

188

example for the latter is that risk will be relatively high if all the projects in a

189

project portfolio co-utilize an immature technology. To construct a risk calculation model for a project portfolio solution xi , two preconditions should be met. (i) The risk value range should lie in the interval [0, 1]. (ii) When more technologies with low T RL are required by projects in xi , the risk should be higher. Therefore, considering the project interactions, the risk index of project portfolio xi can be defined as Equation 4. s  2 m P 9−T RLj RI (xi ) = wj 9 j=1

where : wj =

P

(4)

tij

+ i∈x i

kxi ·T P k1 , j

= 1, 2, ..., m

190

Equation 4 is defined by computing the gap between the ideal value and the

191

real T RL value of a project portfolio. Only the selected projects in xi and the

192

technologies required by those projects are considered. T RLj is the T RL of

193

technology j. The weight wj , j = 1, 2, ..., m is set under the rule that the more

194

times a technology is required, the greater the weight. kxi · T P k is the L1 norm

195

of the vector xi · T P , that is the sum of elements in the vector. Next, Equation 4 can be proved to meet the preconditions mentioned above. First, the value of RI (xi ) , xi ∈ X can be proved to lie in the range [0, 1] according to Equation 5.   2   T RLj ∈ [1, 9] ⇒ 9−T RLj ∈ [0, 1] ⇒ 9−T RLj ∈ [0, 1]  9 9 m P m P P  tij = kxi · T P k1 ⇒ wj = 1   i=1 j=1 j∈x+ i  2 m P 9−T RLj ⇒ wj ∈ [0, 1] 9 j=1 s  2 m P 9−T RLj wj ∈ [0, 1] ⇒ RI (xi ) = 9 j=1

10

(5)

196

197

198

199

200

Next, it can be proved that in Equation 6, for two weight vectors w1 =     1 2 1 , where wa1 + wb1 = and w2 = w12 , ..., wa2 , ..., wb2 ., , , wm w1 , ..., wa1 , ..., wb1 ., , , wm
P P wa2 + wb2 , wa1 > wa2 and i6=a,b wi1 = i6=a,b wi2 , the RI xi , w1 is bigger than
RI xi , w2 if technology b is more mature than technology a, that is T RLa < T RLb .


2
2  RI xi , w1 − RI xi , w2      2 P 2  m m P  9−T RLj  2 9−T RLj  wj1 w = −  j 9 9   j=1 j=1  


2 2 2  1 9−T RLa   + wb1 9−T9RLb − wa2 9−T9RLa − wb2  = wa 9

2
2
= 9−T9RLa × wa1 − wa2 + 9−T9RLb × wb1 − wb2   
   ∵ wa1 + wb1 = wa2 + wb2   

2
2     = wa1 − wa2 × 9−T9RLa − 9−T9RLb   
  >0 wa1 > wa2 , T RLa < T RLb

⇒ RI xi , w1 > RI xi , w2 201

3.


9−T RLb 2 9

(6)

Project Portfolio Optimization Based on A Refined Selection

202

Strategy

203

This section describes a process for obtaining the optimal project portfolio.

204

The process has three steps: (i) obtaining the non-dominated project portfolio

205

solutions, (ii) association rules mining and ranking, (iii) refined selection on non-

206

dominated solutions. The detailed flows in Figure 2 illustrate the framework

207

of the whole process to make it more understandable and operational. The

208

following three subsections illustrate the three steps in detail, respectively.

209

3.1. Obtaining the Non-dominated Set

210

Project portfolio solution xi is a subset of n project proposals, where xij =

211

0 or 1. While xij = 0 denotes that project pj in P is not selected in solution

212

xi , xij = 1 means that pj is selected in solution xi . xi is feasible if and only

213

214

if the total cost of it is less than b. Therefore, the feasible set Xf is defined as P n o n Xf := {xi ∈ X |xi C ≤ b } := xi ∈ X j=1 xij × cj ≤ b . 11

Figure 2: The framework for project portfolio optimization with refined selection strategy 215

Considering the presence of multiple evaluation indicators for project port-

216

folios, solution that performs best on all criteria may not exist. Therefore, the

217

corresponding definitions for evaluating multi-objectives are used for reference.

218

First, the domination relationships between project portfolios can be defined

219

as Equation 7. The dominated solutions need not to be considered, and can be

220

directly deleted from the decision space. Definition 1: Project portfolio solution xj is dominated by xi , when:    V (xi ) ≥ V (xj ) and RI (xi ) ≤ RI (xj )   xi  xj := V (xi ) 6= V (xj ) or RI (xi ) 6= RI (xj )     x ,x ∈ X i

j

(7)

f

Definition 2: From definition 1, the efficient project portfolio set Xe can be

defined as Equation 8.  /xj ∈ Xe , xj  xi . s.t. xj 6= xi Xe := xi ∈ Xe ∃

(8)

221

Next, the method to obtain the project portfolio Pareto set is discussed.

222

The project portfolio solution space shows an exponential increase with the 12

223

increase in number of proposed projects. For m projects, it generates 2m − 1

224

solutions (Xia et al. , 2017). Therefore, an efficient multi-objective algorithm

225

is needed to solve the optimization problem. The NSGA is a type of widely

226

used multi-objective intelligent algorithm that can retain the elites in offsprings.

227

While the differential evolution (DE) algorithm is a good genetic operator for

228

maintaining population diversity. Thus, the DE algorithm is embodied in the

229

NSGA framework to obtain non-dominated solutions by fusing the advantages

230

of NSGA and DE. The precise steps are as follows.

231

(1) Population initialization. The chromosome form of individuals is defined as

232

xi = {xi1 , xi2 , ..., xin }, xij ∈ [0, 1], from which the initial populations are

233

generated. (2) Fitness and penalty functions construction and calculation. Fitness functions are set with focus on the two objectives of maximizing value and minimizing risk. The penalty function is set to avoid exceeding the budget by adding a penalty item to the fitness functions, as shown in Equation 9. f1 = −V (xi ) + γ1 × max {0, xi · c − b} f2 = R (xi ) + γ2 × max {0, xi · c − b}   x = (x , ..., x ) , x ∈ {0, 1}m i i1 im i s.t.  x ∈X i e

(9)

234

where V (xi ) and R(xi ) are the value and risk of the project portfolio xi ,

235

respectively. The objective of the algorithm is to minimize the values of f1

236

and f2 . In addition, if the cost exceeds the budget, the amount exceeding

237

the budget will be added to the two fitness functions, where γ1 and γ2

238

are the parameters determining how the amount exceeding the budget will

239

affect the fitness values. (3) Mutation operation. Assume that r1 , r2 , and r3 are individuals randomly selected in the current population, and r and rbest stand for the current and optimal individuals in the population, respectively. The new individual r0

13

will be generated according to Equation 10. r0 = r3 + rand ∗ (r2 − r1 )

r0 = rgebest + rand ∗ (r2 − r1 )

(10)

r0 = r + rand ∗ (rgebest − r1 ) + rand ∗ (r2 − r1 ) (4) Crossover operation. This is executed on the newly generated individuals to ensure the diversity of species according to Equation 11; where, r00 represents the new individual after the crossover operation. |r| is the number of elements in r. Cro is the crossover operator guaranteeing that each code bit of r0 has a probability of p being replaced by the corresponding bit of parent r. r00 = (1 − Cro) ∗ (r0 ) + Cro ∗ (r)      Cro = cro1 , cro2 , ..., cro|r|     1, if (rand < p) where :  croi =    0, if (rand ≥ p) 

(11)

240

(5) Selecting the elites from offsprings. The elite individuals will be selected

241

using the crowding comparison operator, a non-dominated sorting method.

242

(6) Repeating step 2 to 5 until the termination condition is met.

243

3.2. Association Rules Mining on the Non-dominated Set

244

3.2.1. Association Criteria

245

Finding the patterns of association rules can help in decision making. The as-

246

sociation rules from the non-dominated project portfolios represent the projects

247

frequently appearing simultaneously in the Pareto set; that is the combination

248

of these projects tends to perform better. Therefore, before the refined selec-

249

tion on the non-dominated project portfolios, the association rules need to be

250

obtained in the Pareto set.

251

To effectively find the association rules set, the standards for identifying

252

the association rules need to be defined. Support degree, confidence degree

253

and promotion degree are three commonly used evaluation criteria for mining

254

association rules. To introduce these criteria in detail, the variables Z1 and Z2

14

255

are taken for example. In addition, to make the criteria more understandable,

256

an example is provided after each criterion introduction. (1) Support degree is the proportion of several items that appear simultaneously in the data set to the total amount of the data set. This can also be explained as the association probability of certain items. The calculation of support degree is shown in Equation 12, where f requent (Z1 Z2 ) denotes the frequency of items appearing simultaneously in Z1 and Z2 , and |AllSamples| indicates the total number of items in the data set. In general, the items with high support degree may not form an association rule, but those with low support degree certainly do not form an association rule. SD (Z1 , Z2 ) = P (Z1 Z2 ) =

f requent (Z1 Z2 ) |AllSamples|

(12)

257

Example 1: If 1000 customers go to the mall to purchase items, of which

258

150 customers purchase ballpoint pens and notebooks at the same time,

259

then the support degree of the association rule (ballpoint pen, notebook) is

260

150/1000 × 100% = 15%. (2) Confidence degree means the probability of the appearance of an item when another item appears, or the conditional probability of an item. The confidence degree of Z1 ← Z2 can be expressed as Equation 13. CD (Z1 ← Z2 ) = P (Z1 |Z2 ) =

f requent (Z1 Z2 ) f requent (Z2 )

(13)

261

Example 2: In the above example, if 65% of the customers who purchase

262

ballpoint pen also purchase notebook, then, the confidence level of the as-

263

sociation rule (ballpoint pen → notebook) is 65%. (3) The lift degree represents the ratio of the probability of P (Z1 |Z2 ) to the probability of Z1 , as shown in Equation 14. It reflects the degree of association between Z1 and Z2 . A lift degree greater than 1 indicates that Z1 ← Z2 is an effectively strong association rule. However, Z1 ← Z2 is judged as an invalidly strong association rule when the lift degree is less than or equal to 1. As a special case, when Z1 and Z2 are independent, the lift degree will 15

be equal to 1. LD (Z1 ← Z2 ) =

P (Z1 |Z2 ) C (Z1 ← Z2 ) = P (Z1 ) P (Z1 )

(14)

264

Example 3: In the above example, the lift degree of the association rule

265

(ballpoint pen → notebook) is 65%/25% = 2.6, indicating that it is a effec-

266

tively strong association rule.

267

The refined selection strategy traverses the ranked association rules to re-

268

fine non-dominated solutions until only one solution remains. There should be

269

enough association rules, otherwise, the rules may have been completely tra-

270

versed before the optimal solution obtained. Therefore, when implementing the

271

association rule mining algorithm, users should set minimum limits of support

272

and confidence degrees to slightly lower values to guarantee to obtain enough

273

association rules.

274

3.2.2. Association Rules Ranking based on TOPSIS

275

The technique for order preference by similarity to ideal solution (TOPSIS)

276

is adopted to obtain the most frequent association rule. The TOPSIS method

277

aggregates the performance of support degree, confidence degree and lift degree.

278

The basic idea of TOPSIS is to compare the distance of all candidates with

279

the positive and negative ideal points (Lin & Yeh , 2012). Assuming that

280

m frequent association rules exist, with the three criteria vectors denoted as

281

SD = [si ]T , CD = [ci ]T , and LD = [li ]T , where i = 1, 2, ...m, respectively, the

282

283

284

285

specific steps of the method are presented as follows. h i (1) The criteria matrix M = [mij ]m×3 = SD CD LD is normalized to  construct the decision matrix M 0 = m0ij m×3 , where aij is determined as Equation 15.

,v um uX 0 mij = mij t m2ij , i = 1, · · · , m; j = 1, 2, 3 i=1

16

(15)

(2) Determine the positive ideal point pip and negative ideal point nip according to Equation 16. pip = [max {m0i1 } , max {m0i2 } , max {m0i3 }] , i = 1, 2, ..., m nip = [min {m0i1 } , min {m0i2 } , min {m0i3 }] , i = 1, 2, ..., m

(16)

(3) The distances between the criteria of the frequent set mi = [mi1 , mi2 , mi3 ] and the ideal points (both positive and negative) are calculated according to Equation 17. d∗i d0i

= =

s


2 m0ij − pip , i = 1, · · · , m

3 P

sj=1 3 P

m0ij

j=1


2

(17)

− nip , i = 1, · · · , m

(4) The relative distances to the negative ideal point are calculated in Equation 18; here tvi∗ is also called the TOPSIS value. tvi∗ = d0i 286




d0i + d∗i , i = 1, · · · , m

(5) tvi∗ describes the importance of the corresponding frequent items, according

287

to which the most frequent association rule can be obtained as

288

arg (maxi=1,2,...m tvi∗ ).

289

290

(18)

3.3. Refined Selection Strategy on Non-dominated Project Portfolios From the ranked association rules, the refined selection strategy is to delete

291

the worst project portfolios with no highly ranked projects from the non-dominated

292

solutions step-by-step until only one project portfolio remains. The detailed

293

steps are as follows.

294

(1) Define the counter t = 1, empty set U , and input the non-dominated port-

295

296

297

298

299

folio solutions Nd0 . (2) Scan the tth association rule, and store the projects appearing in this rule as Ut . (3) Screen Nd0 by deleting all the project portfolios that do not contain Ut − U . Let U = U + Ut . 17

300

(4) Examine the remaining project portfolio NdL . If only one project portfolio

301

exists in NdL , then stop the algorithm and output NdL ; otherwise, let t = t+1,

302

and repeat steps 2 to 4.

303

This strategy is based on the principle that the more frequently a project is

304

selected in the non-dominated set, the more it should be retained. Therefore,

305

the strategy follows the inverse idea that project portfolios without any project

306

should be deleted.

307

Consider the following example illustrating the processes of the refined se-

308

lection strategy. First, 12 association rules are abstracted from the total set in

309

the case study, as shown in Table 1 . The projects will be ranked in the order in

310

which they first appears in the 12 rules. That is the earlier a project appears in

311

the association rules, the higher it would be ranked. Therefore, as shown Table

312

1, the project ranking is [3,4], 6, 13, and 16, indicating that projects 3 and 4

313

are the most important ones. Table 1: Abstracted association rules Association rules

Support degree

Confidence degree

Lift degree

TOPSIS values

Rank

Project 3 → Project 4

1

1

1

0.883607

1

Project 6→ Project 3

0.851852

1

1

0.779176

2

Project 3→ Project 6

0.851852

0.851852

1

0.777655

3

Project 13→ Project 3

0.814815

1

1

0.739088

4

Project 3→ Project 13

0.814815

0.814815

1

0.73698

5

Project 13→ Project 6

0.740741

0.909091

1.067194

0.654444

6

Project [6,13]→ Project 3

0.740741

1

1

0.65408

7

Project 6→ Project 13

0.740741

0.869565

1.067194

0.653932

8

Project 3→ Project [6,13]

0.740741

0.740741

1

0.650578

9

Project [6,16]→ Project 13

0.703704

1

1

0.610265

10

Project 6→ Project 16

0.703704

0.826087

0.969754

0.60742

11

Project 3→ Project [6,16]

0.703704

0.703704

1

0.605923

12

314

Next, the non-dominated project portfolios are refined. First, the project

315

portfolios that do not include projects 3 and 4 are deleted, then those that do

316

no include project 6 are deleted, and finally those that do not include project 13

317

are deleted. The deletion process will continue until only one project portfolio

318

remains.

18

319

4. Case Study

320

The following case study comprising project portfolio selection illustrates the

321

feasibility and efficiency of the proposed model and approach. The stakeholders

322

are assumed to demand from the researchers a recommendation of the optimal

323

project portfolio.

324

4.1. Data Instruction

325

In the case study, the data sets in Table 2 and 3 refer to the context in

326

(Martino , 1995). Table 2 provides 16 project proposals with information of

327

the expected costs, revenues and the scores of alignment degrees (between 0-1)

328

with 6 company strategies: (i) improving market share, (ii) expanding products

329

ranges, (iii) occupying the new products market, (iv) reducing supply chain

330

risk, (v) increasing turnover by more than 10%. The interactions between the

331

projects on technology and T RL information are tabulated in Table 3. The total

332

budget is set to 600 ∗ (104 ) $. There are 216 − 1 = 65535 alternative project

333

portfolio solutions, within which the recommended solution will be generated. Table 2: Criteria of project proposals Alignment degree with company strategies

Project

Project name

Cost (104 $)

Revenue (104 $)

S1

S2

S3

S4

S5

P1

Digital signal processor

48

113

0

0.06

0.25

0.22

0

P2

Digital image processor

38

77

0.19

0.06

0

0.01

0

P3

Speech synthesizer

40

78

0.20

0

0.07

0

0.07

P4

Low-voltage computer chip

43

67

0.11

0

0.07

0

0.07

P5

High-efficiency solar cell

35

19

0

0.04

0

0.01

0

P6

Digital-to-analog converter

25

54

0

0

0

0.09

0.1

P7

Analog-to-digital converter

26

67

0.02

0.09

0

0

0.09

P8

Frequency converter module

41

73

0

0

0.06

0

0.24

P9

Conformal phased-array antenna

53

76

0

0

0.34

0.14

0

P10

Radio frequency mixer

81

148

0

0.24

0.11

0.14

0

P11

Signal converter

97

114

0.20

0

0

0.07

0.16

P12

High-speed encryption circuit

51

105

0.01

0.21

0

0

0.06

P13

Lightweight dc-to-dc converter

89

34

0

0.05

0.02

0.05

0

P14

High-energy-density battery

78

88

0.17

0

0.03

0.14

0

P15

Improved coding algorithm

97

105

0.11

0

0

0.07

0.20

P16

Optical materials for solar cells

90

111

0

0.24

0.08

0.02

0

19

Table 3: Utilization relations between projects and technologies

334

Project

T1

T2

T3

T4

T5

T6

T7

T8

T9

P1

1

0

1

0

0

0

1

1

1

P2

1

0

0

0

0

1

0

1

0

P3

0

0

0

0

0

0

1

0

0

P4

0

1

1

1

1

1

1

0

0

P5

0

1

0

0

0

0

0

0

1

P6

1

0

1

0

0

1

1

1

0

P7

1

0

1

0

0

1

1

1

0

P8

1

0

1

0

0

0

1

0

1

P9

1

0

1

0

0

0

0

0

0

P10

1

0

1

0

0

0

1

0

1

P11

1

0

1

0

0

0

1

0

1

P12

0

0

0

0

1

0

1

0

1

P13

1

1

1

0

0

0

1

0

0

P14

0

1

0

0

0

0

1

0

0

P15

0

0

0

0

0

0

1

0

0

P16

0

1

0

0

0

0

0

0

1

TRL

8

9

5

7

8

5

6

7

6

4.2. Constructing the Co-utilization Network of Projects

335

From the utilization relation in Table 3, the technology-project network T P

336

is transformed into the project co-utilization network P U , as shown in Figure

337

3 (The work is carried out with the widely used Gephi, a network visualization

338

tool (Nycz et al. , 2016)).

(a) Technology-project network

(b) Project co-utilization network

Figure 3: The transformation from the T P network to P U network

339

In (a) of Figure 2, an edge indicates that the linked project is in need of

340

linked technology. The edges have no weight information. In (b) of Figure

341

2, the thickness of an edge represents the co-utilization amount of the linked 20

342

projects. Projects 3, 5, 12, 14, 15, and 16 have weaker co-utilization relations

343

than the other projects.

344

The detailed co-utilization information is shown in Table 4; the numbers in

345

the table represent the co-utilization amount of the projects in the corresponding

346

rows and columns. The diagonal elements give the quantity of technologies re-

347

quired by the corresponding project. Besides the diagonal elements, 200 project

348

interaction relationships form the non-zero items in the project co-utilization

349

matrix. Table 4: Project co-utilization matrix

350

351

PU

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

P13

P14

P15

P16

P1

5

2

1

2

1

4

4

4

2

4

4

2

3

1

1

1

P2

2

3

0

1

0

3

3

1

1

1

1

0

1

0

0

0

P3

1

0

1

1

0

1

1

1

0

1

1

1

1

1

1

0

P4

2

1

1

6

1

3

3

2

1

2

2

2

3

2

1

1

P5

1

0

0

1

2

0

0

1

0

1

1

1

1

1

0

2

P6

4

3

1

3

0

5

5

3

2

3

3

1

3

1

1

0

P7

4

3

1

3

0

5

5

3

2

3

3

1

3

1

1

0

P8

4

1

1

2

1

3

3

4

2

4

4

2

3

1

1

1

P9

2

1

0

1

0

2

2

2

2

2

2

0

2

0

0

0

P10

4

1

1

2

1

3

3

4

2

4

4

2

3

1

1

1

P11

4

1

1

2

1

3

3

4

2

4

4

2

3

1

1

1

P12

2

0

1

2

1

1

1

2

0

2

2

3

1

1

1

1

P13

3

1

1

3

1

3

3

3

2

3

3

1

4

2

1

1

P14

1

0

1

2

1

1

1

1

0

1

1

1

2

2

1

1

P15

1

0

1

1

0

1

1

1

0

1

1

1

1

1

1

0

P16

1

0

0

1

2

0

0

1

0

1

1

1

1

1

0

2

4.3. Non-dominated Solutions For multi-objective optimization, the DE algorithm is embodied in the NSGA-

352

II framework, to obtain the non-dominated solutions. The advantage of the pro-

353

posed NSDE is illustrated by comparing the algorithm with three other mature

354

algorithms, NSGA-II, NSGA-III, and SPEA2. The population of the four algo-

355

rithms is set to 200, with the maximal number of iterations set to 100. Through

356

sensitive analysis, the parameters of the four algorithms are determined as those

357

with the best performance, as shown in Table 5. From the value function in

358

Equation 3, parameter ξ plays an important role in determining the project

359

portfolio values. Therefore, nine cases of ξ = 1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, 5 are

360

tested on each algorithm, as shown in Figure 4. 21

Table 5: Project co-utilization matrix Algorithms

NSDE

NSGA-II

NSGA-III

SPEA-II

Crossover probability

0.7

0.5

0.5

0.1

Mutation probability

/

0.1

0.1

0.2

Figure 4: The 9 non-dominated sets

361

The term “Xi” represents the notation of ξ. x and y are value and risk,

362

respectively. A comparison shows that the Pareto set of SPEA2 is more concen-

363

trated and those of the other three NSGA series algorithms are more diversified.

364

Further, the NSDE shows the widest distribution of the Pareto set due to diver-

365

sity of the DE operator. Therefore, the NSDE is better for population diversity.

366

In addition to the diversity, the hypervolume metric value is adopted to measure

367

the non-dominated set performance. The hypervolume metric values of the four

368

algorithms on the nine cases are calculated as shown in Figure 5. The results

22

369

show that the NSDE is the best for ξ = 1/5, 1/4, 1/2, and performs at least as

370

good as any other algorithm on other cases.

Figure 5: The hypervolume metric value of four algorithms on 9 cases

371

4.4. Analysis of the Pareto Set

372

A statistical analysis is carried out on a single Pareto set to illustrate the

373

relative importance of projects. In the following step, the non-dominated solu-

374

tions generated by NSDE with ξ = 1/2 is further studied as an example, with

375

200 Pareto set individuals shown in Figure 6.

376

The rectangular area is divided into 200 × 16 rectangles according to the

377

number of project proposals. Each rectangle shows whether a project candidate

378

is selected in the non-dominated set. If project P − i is selected in the j th non-

379

dominated project portfolio, the corresponding rectangular block indicated by

380

ji is coloured black, or is left blank.

381

Then, the projects in the Pareto set selected are counted, and the relative

382

importance degree is calculated, as shown in Figure 7. From the single project

383

aspect, projects 3, 4 and 14 are the most important. However, this type of

384

ranking does not consider the combination of projects. Therefore, to determine

385

the optimal project portfolio, the association rules should be mined from the

386

Pareto set. 23

Figure 6: The non-dominated project portfolios generated by NSDE with ξ = 1/2

387

4.5. Refined Selection on The Non-dominated Solutions

388

There are some repetitive project portfolios in the 200 individuals. After

389

deleting the repetitive items, only 42 non-dominated project portfolios remain,

390

as shown in Figure 8.

391

The association rules of the 42 project portfolios can be found with the

392

Apriori mining algorithm (Pei et al. , 2002). Finally, 18,405 association rules

393

are mined from the 42 project portfolios with the support degree, confidence

Figure 7: The relative importance of projects when xi = 1/2

24

Figure 8: Project portfolios of the non-dominated set after deleting repetitive solutions

394

degree, and lift degree information. The TOPSIS algorithm is then applied on

395

the degrees to determine the rule rankings, as shown in Figure 8. In (b) of

396

Figure 9, the area graph indicates the ranked TOPSIS value distributions of all

397

the association rules. Most of the TOPSIS values are located in the interval

398

of [0.3, 0.5] and the higher the TOPSIS value, the higher the association rule

399

ranks. Furthermore, the support degrees, confidence degrees, lift degrees, and

400

TOPSIS values are exhibited in (a) of Figure 9, where the three coordinate axes

401

represent the three degrees and the area of the circle indicates the corresponding

402

TOPSIS values. The degrees are found distributed in the left-hand side of the

403

space, indicating that the support and confidence degrees are correlative. That

404

is the confidence degree tends to be higher than the support degree in the case.

405

Next, regarding the example’s processes in the 3.3 part, the 42 non-dominated

406

project portfolios are refined. First, the projects are ranked according to their

407

first appearance in the association rules, as Table 6 shows. The association rule

408

rankings correspond to the rules position when the projects first appear. For

409

example, project 6 first appears in the 603th association rule.

410

Now, following the project rankings, the project portfolios without projects

25

(b) Area graph of TOPSIS values

(a) Degree distributions

Figure 9: The association rules information

Table 6: Project co-utilization matrix Project

The association rule ranking position

Project

when a project first appears

The association rule ranking position when a project first appears

3

1

2

1755

14

1

5

3928

15

3

13

4139

16

13

12

5433

4

51

8

/

7

181

9

/

6

603

10

/

1

1511

11

/

411

[3, 14], 15, 16, 4, 7, 6, 1, 2, 5, 13, and 12 are successively deleted from the 42

412

non-dominated project portfolios. The detailed process is shown in Figure 10.

413

The deletion process terminates when the algorithm tries to delete the project

414

portfolios that do not include project 8, because otherwise no project portfolio

415

will remain. Finally, the project portfolios of [P1, P2, P3, P4, P5, P6, P7, P12,

416

P14, P15, P16] will be recommended as the optimal solution. The value and

417

risk of this project portfolio are 453.7215 and 0.4736, respectively.

418

The advantage of the proposed refined selection strategy can be shown by

419

comparing the results with those obtained using the weighted goal programming

420

methodology. The two objectives are combined to a single objective, as Equation

26

Figure 10: The refined selection processes

421

19 shows, which is actually an integer programming problem. i) min z = δ · (− value(x value∗ ) + (1 − δ) ·   0<δ<1  x = (x , ..., x ) , x ∈ {0, 1}n

i

i1

in

risk(xi ) risk∗

(19)

i

∗

422

Variable δ is set to 0.5. The value and value∗ are respectively the target

423

goals of value and risk. They are divided by the value and risk to standardize

424

the different goals following the percentage normalization method (Sarkar et al.

425

, 2018).

426

Using the lingo tool, the minimum value of z is obtained as −0.3857, when

427

value = 382.8549 and risk = 0.5026. The optimal solution is [P2, P3, P4,

428

P5, P7, P8, P9, P12, P13, P16]. A comparison shows that for both value and

429

risk, the optimal solution is dominated by the refined selection solution, because

430

382.8549 < 453.7215 and 0.5026 > 0.4736.

431

In conclusion, the case study includes 16 projects proposals. The project

27

432

co-utilization network shows 200 project interaction relations. Projects 7 and

433

6 have the biggest interaction degree of 5, meaning that the two projects si-

434

multaneously depend on five technologies. An improvement in any one of the

435

five technologies will promote the value of the projects and project portfolios

436

containing the two projects.

437

Then, by setting parameter ξ to 1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, 5, respectively, 9

438

non-dominated sets are obtained. The bigger ξ is, the closer the frontier is to

439

the upper left-hand corner. Thus, the value of ξ determines the solution space

440

shape, and hence the Pareto frontier shape.

441

As for the 200 non-dominated solutions considered, after deleting the repet-

442

itive items in the non-dominated solutions, 42 project portfolios remain and

443

18,405 association rules are mined, providing information of the support degree,

444

confidence degree, and lift degree. The three criteria are ranked according to

445

the multi-criteria decision method TOPSIS. The TOPSIS values show a normal

446

distribution characteristic that most rules locate in the interval of [0.3, 0.5].

447

Then, the 42 project portfolios are refined using the successively deleting

448

strategy. First, the projects are ranked in the order of their appearance in the

449

ranked association rules. Projects 3 and 14 rank first in all projects. In fact, all

450

the project portfolios contain the two projects, indicating that they are truly

451

important. Second, from the project rankings, the 42 non-dominated projects

452

are refined step-by-step, until only one project portfolio remains, that is [P1,

453

P2, P3, P4, P5, P6, P7, P12, P14, P15, P16]. The value and risk of this project

454

portfolio are 453.7215 and 0.4736, respectively.

455

5. Discussion and Conclusions

456

The result in case study indicates that: (i) The project interaction degrees

457

have influence on the project portfolio values. The exponential coefficient of

458

regularization item has a significant influence on the value of project portfolio

459

and the shape of non-dominated project portfolios. (ii) The four multi-objective

460

algorithms are all efficient to obtain non-dominated set. The NSDE algorithm

28

461

has a slight advantage in both diversity and performance. (iii) The refined

462

optimal solution shows better performance compared to the result obtained

463

using the goal programming method. The result prove the advantage of the

464

proposed refined selection strategy. Further studies should focus on proving the

465

optimality of the refined selection solution by logical reasoning. In addition,

466

the current project interaction only includes project dependence relation with

467

technology. More factors in real world should be considered, such as market

468

and resource, to make the provided methods more applicable.

469

In conclusion, the study shows that the two proposed innovative approaches

470

on project interaction and refined selection are feasible and reasonable to tackle

471

the two challenges on project portfolio selection: (i) modelling the influences of

472

project interactions on the final value of project portfolios, and (ii) selecting the

473

best solution from the non-dominated project portfolios.

474

As for the value and risk models, based on the dependence relations between

475

projects and technologies, the project co-utilization network representing the

476

interaction relations is constructed. Then, the project interaction degree is set as

477

a regularization item, and is added to the original value function to indicate the

478

influence of interactions on the project portfolio values. Second, the risk model

479

is constructed based on the T RL and the condition that the more immature the

480

technologies a project portfolio depends on, the higher the risk.

481

As for the optimal project portfolio selection, based on the obtained non-

482

dominated project portfolios, the association rules are mined and ranked by

483

the TOPSIS method. Next, the refined selection strategy is adopted to delete

484

the worst project portfolios that have no highly ranked projects from the non-

485

dominated solutions step-by-step until only one project portfolio remains, which

486

will be recommended as the optimal solution.

487

488

Finally, the paper provides a case study, which verifies the feasibility and advantage of the proposed models and approaches.

29

489

Declaration of Competing Interest

490

The authors declare that they have no known competing financial interests or

491

personal relationships that could have appeared to influence the work reported

492

in this paper.

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Acknowledgements

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The work was supported by the National Key R&D Program of China under

495

Grant SQ2017YFSF070185, the National Natural Science Foundation of China

496

under Grant 71971213, 71901214 and 71901215.

497

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Credit author statement Hechuan Wei: Conceptualization, Data curation, Writing - original draft, Formal analysis. 581 Caiyun Niu: Revision, Language improving. Boyuan Xia: Software, Validation, Methodology, Visualization. Yajie Dou: Funding acquisition, Supervision. Xuejun Hu: Project administration, Resources, Writing - review & editing.