Research Article Research on a Risk Assessment Method …downloads.hindawi.com/journals/mpe/2016/9191360.pdf · [ ], like analytic hierarchy process about risk assessment [ , ], hazard

Research ArticleResearch on a Risk Assessment Method consideringRisk Association

Zhan Zhang,1 Kai Li,1 and Lei Zhang2

1School of Business Administration, Northeastern University, Shenyang, China2School of Economics and Management, Beijing Jiaotong University, Beijing, China

Correspondence should be addressed to Lei Zhang; zhanglei [email protected]

Received 26 August 2016; Revised 12 October 2016; Accepted 10 November 2016

Academic Editor: Yan-WuWang

Copyright © 2016 Zhan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Regarding risk assessment problems with multiple associated risks, a risk assessment method (RAM) is proposed in this paper.According to the risk-associated assessment information offered by expert panel, a comprehensive associated matrix is constructedto identify the influence relationship among risks so as to determine the hierarchical structure of risks. Then, based on thedetermined divided or undivided risk hierarchical structure as well as the possibility and loss of risks provided by expert panel,each value at risk (VAR) is calculated through knowledge related to probability theory. Finally, the feasibility and efficiency of theproposed method are demonstrated through a calculating case.

1. Introduction

Risk assessment is quantifying the probable degree of influ-ence or loss brought by a certain event or thing [1]. Riskassessment is an important segment of risk management, forinstance, in financial risk management, credit risk manage-ment, engineering risk management, and other aspects; itplays a significant decision-support role for risk managers toadopt reasonable risk prevention measures and strategies [2–4]. Over the years, many scholars at home and abroad haveattached great importance to researches on RAM and therehave already been some outstanding research achievements[5–10], like analytic hierarchy process about risk assessment[5, 6], hazard degree calculation method [7, 8], risk matrixmethod [9–11] and artificial neural network [12, 13], andso on. Though all the RAMs mentioned above have solvedvarious kinds of risk assessment problems from differentperspectives, most of them do not take risk-associated sit-uations into account. However, in reality there are usuallyconnections among risks. For example, Baranoff and Sager[14] analyzed the associated situation between resource riskand production risk in life-insurance companies. Abdellaouiet al. [15] find that associated risks are valued differentlythan corresponding reduced simple risks. Thus, research on

RAM in consideration of risk association has academic andpractical application values. Now, there are few researchesof this aspect. Liao et al. [16] adopted Bayesian Networkapproach to evaluate IT outsourcing risk meanwhile consid-ering the association between IT outsourcing risk factors andIT outsourcing risks; Buyukozkan and Ruan [17] proposed aChoquet integrals-based software development risk evalua-tion approach regarding the association among developmentenvironment risk, code constraint risk, and engineeringrisk during the development of software. However, theseRAMs mainly focus on specific risk assessment problems inrisk-associated situations with no universal RAM proposed.Therefore, based on previous researches, we establish out-sourcing risk hierarchy, introduce the interaction betweendifferent levels of risk, and give a RAM in consideration ofmultiple risk-associated situations.

There are four sections in this paper. In Section 1 weelaborate the research background and the problems thatneed to be explored or studied and then clarify the objectivesand significance of the proposed method. In Section 2 wedescribe the RAM considering risk association in detail. InSection 3 we use a calculating case to prove feasibility andefficiency of the proposed method. Finally in Section 4 wesummarize the conclusion and the main contributions of this

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 9191360, 7 pageshttp://dx.doi.org/10.1155/2016/9191360

2 Mathematical Problems in Engineering

paper and also the limitations and further research work tobe carried out.

2. Theories and Methods

2.1. Description of Problems. In risk assessment problems, theoccurrence of a certain risk may lead to another risk, that is,the association among risks. The following symbols are usedto express the set and quantity of a risk assessment problemin consideration of risk-associated situations:

(i) 𝑅 = {𝑅1, 𝑅2, . . . , 𝑅𝑚}: the set of risks, where 𝑅𝑖 meansthe risk number 𝑖, 𝑖 = 1, 2, . . . , 𝑚

(ii) 𝐸 = {𝐸1, 𝐸2, . . . , 𝐸𝑛}: the set of the expert panel, where𝐸𝑘 means the expert number 𝑘, 𝑘 = 1, 2, . . . , 𝑛(iii) 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤𝑛): the weight vector of experts,

where 𝑤𝑘 means the importance or weight of theexpert 𝐸𝑘; it satisfies 0 ≤ 𝑤𝑘 ≤ 1, ∑𝑛𝑘=1 𝑤𝑘 = 1, 𝑘 =1, 2, . . . , 𝑛

(iv) 𝐴𝑘 = [𝑎𝑘𝑖𝑗]𝑚×𝑚: risk-associated matrix offered bythe expert 𝐸𝑘, where 𝑎𝑘𝑖𝑗 means the evaluation valueoffered by the expert𝐸𝑘 for the direct influence degreeof the risk 𝑅𝑖 on the risk 𝑅𝑗, in which five-pointscale is adopted. 0 means “no influence,” 1 means“weak influence,” 2 means “rather weak influence,” 3means “medium influence,” 4 means “rather stronginfluence,” and 5 means “strong influence”

(v) 𝜆𝑘: the thresholds offered by the expert 𝐸𝑘 for com-prehensive influence degree. Setting up thresholds isto reject the influence relationship that is insignificantand has little comprehensive influence degree accord-ing to expert panel. 0 ≤ 𝜆𝑘 ≤ 1, 𝑘 = 1, 2, . . . , 𝑛

(vi) 𝑃𝑘𝑖 : the probability of occurrence of the risk 𝑅𝑖 offeredby the expert 𝐸𝑘, 𝑘 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑚

(vii) 𝑙𝑘𝑖 : the loss after the occurrence of the risk 𝑅𝑖 offeredby the expert 𝐸𝑘, 𝑘 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑚

(viii) 𝑃𝐺𝑖 : the comprehensive probability of the occurrenceof the risk 𝑅𝑖. 𝑖 = 1, 2, . . . , 𝑚

(ix) 𝐿𝐺𝑖 : the comprehensive loss of the risk 𝑅𝑖, 𝑖 =1, 2, . . . , 𝑚(x) 𝑧𝑖: the VAR of the risk 𝑅𝑖, 𝑖 = 1, 2, . . . , 𝑚We propose the detailed procedure of RAM in consid-

eration of risk association. Firstly, several risk-associatedmatrixes offered by expert panel are combined as group risk-associated matrix, while several thresholds of comprehensiveinfluence degree offered by expert panel are combined asgroup threshold; secondly, risk relationship identificationis performed through methods like matrix transform andthere are usually two situations, divided and undivided riskhierarchical structures in the result of risk relationship iden-tification. If the undivided risk hierarchical structure is set assituation A, each value at risk would be calculated accordingto the occurrence probability and loss of each risk offeredby the expert panel; if the divided risk hierarchical structure

is set as situation B, the expert panel are required to offerthe occurrence probability of bottom risk and conditionalprobability according to the risk hierarchical structure. Fur-thermore, the comprehensive probability of the occurrence ofeach riskwould be calculated through conditional probabilityformula and total probability formula specifically so as tocalculate each value at risk in consideration of loss of eachrisk.

2.2. Risk Relationship Identification. According to the descrip-tion of problems above, identification methods of relation-ship among several risks (or risk hierarchical structures) arelisted below.

Firstly, 𝑛 risk-associated matrixes (𝐴1, 𝐴2, . . . , 𝐴𝑛) arecombined as a group-associated matrix 𝐴𝐺 = [𝑎𝐺𝑖𝑗 ]𝑚×𝑚,among which the calculating formula of 𝑎𝐺𝑖𝑗 is

𝑎𝐺𝑖𝑗 ={{{{{{{

𝑛∑𝑘=1

𝑎𝑘𝑖𝑗𝑤𝑘, 𝑡𝑖𝑗 ≤ 𝑛2

0, 𝑡𝑖𝑗 > 𝑛2

𝑖, 𝑗 = 1, 2, . . . , 𝑚. (1)

In formula (1), 𝑡𝑖𝑗means the number of experts who scorethe direct influence degree of the risk 𝑅𝑖 on the risk 𝑅𝑗 as 0.Furthermore, the group-associatedmatrix𝐴𝐺 is transformedinto regulated group-associatedmatrix𝐵𝐺 = [𝑏𝐺𝑖𝑗 ]𝑚×𝑚, amongwhich the calculating formula of 𝑏𝐺𝑖𝑗 is

𝑏𝐺𝑖𝑗 = 𝑎𝐺𝑖𝑗max𝑖 {∑𝑛ℎ=1 𝑎𝐺𝑖ℎ} 𝑖, 𝑗 = 1, 2, . . . , 𝑚. (2)

Secondly, comprehensive influence matrix with indirectassociation 𝐶 = [𝑐𝑖𝑗]𝑚×𝑚 is set up and the calculating formulais

𝐶 = 𝐵𝐺 (𝐼 − 𝐵𝐺)−1 . (3)

Meanwhile, 𝑛 thresholds (𝜆1, 𝜆2, . . . , 𝜆𝑛) of comprehen-sive influence degree are combined as a group threshold 𝜆𝐺,and the calculating formula is

𝜆𝐺 =𝑛∑𝑘=1

𝑤𝑘𝜆𝑘. (4)

According to the group threshold 𝜆𝐺, the comprehensiveinfluence matrix 𝐶 = [𝑐𝑖𝑗]𝑚×𝑚 is transformed into 𝜆𝐺-cutmatrix 𝐶𝜆 = [𝛼𝑖𝑗]𝑚×𝑚, where

𝛼𝑖𝑗 = {{{1, 𝑐𝑖𝑗 ≥ 𝜆𝐺0, 𝑐𝑖𝑗 < 𝜆𝐺 𝑖, 𝑗 = 1, 2, . . . , 𝑚. (5)

Assume 𝐷 = [𝑑𝑖𝑗]𝑚×𝑚 as 0-1 comprehensive associatedmatrix, in which

𝑑𝑖𝑗 = {{{𝛼𝑖𝑗 + 1, 𝑖 = 𝑗𝛼𝑖𝑗, 𝑖 = 𝑗 𝑖, 𝑗 = 1, 2, . . . , 𝑚. (6)

Mathematical Problems in Engineering 3

Matrix 𝐷 reflects the preference of the expert panel forinfluence degree among each risk with indirect influencerelationship. The influence of the risk 𝑅𝑖 on itself is 1.

According tomatrix𝐷, hierarchical structure of each riskis divided. Assume

𝐻𝑖 = 𝐻𝑖 ∩ 𝐽𝑖 𝑖 = 1, 2, . . . , 𝑚, (7)

where 𝐻𝑖 = {𝑅𝑗 | 𝑑𝑗𝑖 = 1} and 𝐽𝑖 = {𝑅𝑗 | 𝑑𝑖𝑗 = 1}. 𝐻𝑖 meansthe risk set corresponding to the element valued 1 in the listnumber 𝑖 in thematrix𝐷; 𝐽𝑖means the risk set correspondingto the element valued 1 in the row number 𝑖 in the matrix𝐷.

If formula (7) works for a certain 𝑖, 𝑅𝑖 should be regardedas the bottom element in 𝐷 and the list number 𝑖 and therow number 𝑖 in 𝐷 should be deleted to form a new matrix.During the process, if the bottom element cannot be found,risk assessment should be performed regarding situation A.If there is bottom element, searching bottom element in thenew matrix should be performed until all the elements inthe matrix are deleted. Then the hierarchical structure ofmatrix𝐷 should be built up according to the order of deleting.Finally, risk assessment should be performed for situation B.

2.3. Risk Assessment. RAMs for situations A and B are listedbelow.

Situation A. As for the situation that risk hierarchical struc-ture cannot be divided, each risk is regarded as elements of thesame layer to process. Firstly, the probability of occurrence ofrisk offered by the expert panel (𝑃1𝑖 , 𝑃2𝑖 , . . . , 𝑃𝑛𝑖 ) is combinedas comprehensive probability of occurrence of risk 𝑃𝐺𝑖 , andthe calculating formula is

𝑃𝐺𝑖 = 𝑛∑𝑘=1

𝑤𝑘𝑃𝑘𝑖 𝑖 = 1, 2, . . . , 𝑚. (8)

Secondly, the loss of risks (𝑙1𝑖 , 𝑙2𝑖 , . . . , 𝑙𝑛𝑖 ) offered by theexpert panel is combined as comprehensive loss of risks 𝐿𝐺𝑖 ,and the calculating formula is

𝐿𝐺𝑖 =𝑛∑𝑘=1

𝑤𝑘𝑙𝑘𝑖 𝑖 = 1, 2, . . . , 𝑚. (9)

Finally, on the basis of the comprehensive probability ofoccurrence of risk 𝑃𝐺𝑖 and comprehensive loss of risks 𝐿𝐺𝑖 , thevalue at risk is calculated, and the calculating formula is

𝑧𝑖 = 𝑃𝐺𝑖 𝐿𝐺𝑖 𝑖 = 1, 2, . . . , 𝑚. (10)

Situation B. As for the situation that risk hierarchical struc-ture can be divided, 𝑅𝑓 = {𝑅𝑖1, 𝑅𝑖2, . . . , 𝑅𝑖𝑚} is assumed asthe set of risk on the layer number 𝑓, and 𝑅𝑖𝑓 is set as therisk number 𝑖 on the layer number 𝑓, 𝑓 = 1, 2, . . . , 𝑚; 𝑃𝑘1 =(𝑝𝑘11 , 𝑝𝑘12 , . . . , 𝑝𝑘1𝑑 ) is the probability vector of the occurrenceof the risk on the first layer (the bottom layer) offered bythe expert 𝐸𝑘, where 𝑝𝑘1𝑗 is the probability of occurrence of

the risk 𝑅𝑗 offered by the expert 𝐸𝑘; 𝑑𝑓 is the number ofrisks on the layer number 𝑓. 𝑅𝑗,𝑓−1 is assumed as the risknumber 𝑗 on the layer number 𝑓 − 1 related to the risk𝑅𝑖𝑓, 𝑗 = 1, 2, . . . , 𝑑𝑖𝑓

𝑓−1;𝑑𝑖𝑓𝑓−1

is the number of risks on the layernumber 𝑓 − 1 related to the risk 𝑅𝑖𝑓. 𝑅𝑖𝑓 = 𝑇 is assumed asthe occurrence of the risk 𝑅𝑖𝑓, and 𝑅𝑖𝑓 = 𝐹 means that therisk 𝑅𝑖𝑓 does not happen. In consideration of two conditions,

𝑅𝑗,𝑓−1 = 𝑇 and 𝑅𝑗,𝑓−1 = 𝐹, there will be 2𝑑𝑖𝑓𝑓−1 conditionalprobabilities related to 𝑅𝑖𝑓 = 𝑇.

Furthermore, assuming that 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 | 𝑅1,𝑓−1 =𝑇, 𝑅2,𝑓−1 = 𝑇, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇) is the probability of occur-rence of the risk 𝑅𝑖𝑓 offered by the expert 𝐸𝑘 with the risks𝑅1,𝑓−1, 𝑅2,𝑓−1, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

happening simultaneously, 𝑘 =1, 2, . . . , 𝑛; 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 | 𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝐹, . . .,𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇) is the probability of occurrence of the risk𝑅𝑖𝑓 offered by the expert 𝐸𝑘 with the risks 𝑅1,𝑓−1, 𝑅3,𝑓−1,𝑅4,𝑓−1, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

happening simultaneously but no risk

𝑅2,𝑓−1, 𝑘 = 1, 2, . . . , 𝑛; . . . , 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 | 𝑅1,𝑓−1 = 𝐹, 𝑅2,𝑓−1 =𝐹, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝐹) is the probability of occurrence of therisk 𝑅𝑖𝑓 without 𝑅1,𝑓−1, 𝑅2,𝑓−1, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

offered by the

expert 𝐸𝑘, 𝑘 = 1, 2, . . . , 𝑛; for convenience, 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 |𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝑇, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇), 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 |𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇), . . . , 𝑃𝑘(𝑅𝑖𝑓 =𝑇 | 𝑅1,𝑓−1 = 𝐹, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝐹) are

recorded as 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝑇, . . . , 𝑇), 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 |𝑇, 𝐹, . . . , 𝑇), . . . , 𝑃𝑘(𝑅𝑖𝑓 = 𝑇 | 𝐹, 𝐹, . . . , 𝐹) for abbreviation.

Firstly, the probability vectors 𝑃11 = (𝑝111 , 𝑝112 , . . . , 𝑝11𝑑 ),𝑃21 = (𝑝211 , 𝑝212 , . . . , 𝑝21𝑑 ), . . . , 𝑃𝑛1 = (𝑝𝑛11 , 𝑝𝑛12 , . . . , 𝑝𝑛1𝑑 ) arecombined as the comprehensive probability vector 𝑃1 =(𝑝11 , 𝑝12 , . . . , 𝑝1𝑑) and the calculating formula is

𝑝1𝑗 =𝑛∑𝑘=1

𝑤𝑘𝑝𝑘1𝑗 𝑗 = 1, 2, . . . , 𝑑1. (11)

Secondly, the conditional probabilities 𝑃1(𝑅𝑖𝑓 = 𝑇 |𝑇, 𝑇, . . . , 𝑇), 𝑃2(𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝑇, . . . , 𝑇), . . . , 𝑃𝑛(𝑅𝑖𝑓 = 𝑇 |𝑇, 𝑇, . . . , 𝑇) are combined as the comprehensive conditionalprobability 𝑃𝐺(𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝑇, . . . , 𝑇); the conditional prob-abilities 𝑃1(𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝐹, . . . , 𝑇), 𝑃2(𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝐹,. . . , 𝑇), . . . , 𝑃𝑛(𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝐹, . . . , 𝑇) are combined asthe comprehensive conditional probability 𝑃𝐺(𝑅𝑖𝑓 = 𝑇 |𝑇, 𝐹, . . . , 𝑇); . . .; the conditional probabilities 𝑃1(𝑅𝑖𝑓 = 𝑇 |𝐹, 𝐹, . . . , 𝐹), 𝑃2(𝑅𝑖𝑓 = 𝑇 | 𝐹, 𝐹, . . . , 𝐹), . . . , 𝑃𝑛(𝑅𝑖𝑓 = 𝑇 |𝐹, 𝐹, . . . , 𝐹) are combined as the comprehensive conditionalprobability 𝑃𝑛(𝑅𝑖𝑓 = 𝑇 | 𝐹, 𝐹, . . . , 𝐹), and the calculatingformulas are as follows:

𝑃𝐺 (𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝑇, . . . , 𝑇)= 𝑛∑𝑘=1

𝑤𝑘𝑃𝑘 (𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝑇, . . . , 𝑇) , (12a)


𝑃𝐺 (𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝐹, . . . , 𝑇)= 𝑛∑𝑘=1

𝑤𝑘𝑃𝑘 (𝑅𝑖𝑓 = 𝑇 | 𝑇, 𝐹, . . . , 𝑇) ,...

(12b)

𝑃𝐺 (𝑅𝑖𝑓 = 𝑇 | 𝐹, 𝐹, . . . , 𝐹)= 𝑛∑𝑘=1

𝑤𝑘𝑃𝑘 (𝑅𝑖𝑓 = 𝑇 | 𝐹, 𝐹, . . . , 𝐹) . (12b𝑑𝑖𝑓𝑓−1)

According to formulas (11)–(12b𝑑𝑖𝑓𝑓−1), total probabilityformula is adopted to calculate the comprehensive probability𝑃𝐺𝑖 of occurrence of each risk specifically (from the bottomlayer); for instance, the formula to calculate the comprehen-sive probability of occurrence of risk on the layer number 𝑓is

𝑃𝐺𝑖 = 𝑃𝐺 (𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝑇, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇, 𝑅𝑖𝑓 = 𝑇) + 𝑃𝐺 (𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1= 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇, 𝑅𝑖𝑓 = 𝑇) + ⋅ ⋅ ⋅+ 𝑃𝐺 (𝑅1,𝑓−1 = 𝐹, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝐹, 𝑅𝑖𝑓 = 𝑇) ,

(13)

where

𝑃𝐺 (𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝑇, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇, 𝑅𝑖𝑓 = 𝑇) = 𝑃𝐺 (𝑅1,𝑓−1 = 𝑇)𝑃𝐺 (𝑅2,𝑓−1= 𝑇) ⋅ ⋅ ⋅ 𝑃𝐺 (𝑅𝑖𝑓 = 𝑇 | 𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1= 𝑇, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇) ,

(14a)

𝑃𝐺 (𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇, 𝑅𝑖𝑓 = 𝑇) = 𝑃𝐺 (𝑅1,𝑓−1 = 𝑇)𝑃𝐺 (𝑅2,𝑓−1= 𝐹) ⋅ ⋅ ⋅ 𝑃𝐺 (𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇)𝑃𝐺 (𝑅𝑖𝑓= 𝑇 | 𝑅1,𝑓−1 = 𝑇, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝑇) ,

(14b)

...𝑃𝐺 (𝑅1,𝑓−1 = 𝐹, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝐹, 𝑅𝑖𝑓 = 𝑇) = 𝑃𝐺 (𝑅1,𝑓−1 = 𝐹)𝑃𝐺 (𝑅2,𝑓−1= 𝐹) ⋅ ⋅ ⋅ 𝑃𝐺 (𝑅𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝐹)𝑃𝐺 (𝑅𝑖𝑓= 𝑇 | 𝑅1,𝑓−1 = 𝐹, 𝑅2,𝑓−1 = 𝐹, . . . , 𝑅

𝑑𝑖𝑓

𝑓−1,𝑓−1

= 𝐹) .

(14b𝑑𝑖𝑓𝑓−1)

Finally, according to formulas (11)–(14b𝑑𝑖𝑓𝑓−1), the com-prehensive probability 𝑃𝐺𝑖 of occurrence of each risk iscalculated. Furthermore, according to formula (10), eachvalue at risk 𝑧𝑖 is calculated.3. The Calculating Case for Situations A and B

In the following part, 2 calculating cases for situations A andB are used to explain the RAM proposed above.

Case 1 for Situation A. In order to improve its compet-itive ability, Liaoning GTE Biopharmaceutical Companywants to outsource its clinical experiment business. Beforeoutsourcing, the company needs risk assessment on thisoutsourcing activity. The company sets up a panel ofexperts including 5 experts (𝐸1, 𝐸2, . . . , 𝐸5), and accordingto experience and business proficiency of each expert, theweight vectors of experts provided by the company are𝑤 = (0.25, 0.2, 0.3, 0.1, 0.15). Through relevant analysis andarrangement of feedback suggestions from questionnaires,the group of experts determine 2 kinds of outsourcing risks:bad book (𝑅1) and contract modification (𝑅2). These 2 riskscannot be divided to a hierarchical structure. It should becalculated by using formulas (8), (9), and (10). The 5 expertsgive the value as Tables 1 and 2 show.

By using formulas (8), (9), and (10), we can get

𝑃𝐺1 = 𝑛∑𝑘=1

𝑤𝑘𝑃𝑘1= 0.25 ∗ 0.36 + 0.2 ∗ 0.24 + 0.3 ∗ 0.28 + 0.1

∗ 0.32 + 0.15 ∗ 0.24 = 0.29,𝐿𝐺1 =

𝑛∑𝑘=1

𝑤𝑘𝑙𝑘1= 0.25 ∗ 20 + 0.2 ∗ 22 + 0.3 ∗ 25 + 0.1 ∗ 22

+ 0.15 ∗ 22 = 22.3,𝑧1 = 𝑃𝐺1 𝐿𝐺1 = 0.29 ∗ 22.3 = 6.467,𝑃𝐺2 = 𝑛∑

𝑘=1

𝑤𝑘𝑃𝑘2= 0.25 ∗ 0.32 + 0.2 ∗ 0.34 + 0.3 ∗ 0.25 + 0.1

∗ 0.25 + 0.15 ∗ 0.35 = 0.3005,


Table 1

𝐸1 𝐸2 𝐸3 𝐸4 𝐸5𝑃(𝑅1 = 𝑇) 0.36 0.24 0.28 0.32 0.24𝑃(𝑅2 = 𝑇) 0.32 0.34 0.25 0.25 0.35

Table 2

𝑙𝑘𝑖 𝐸1 𝐸2 𝐸3 𝐸4 𝐸5𝑅1 20 22 25 22 22𝑅2 28 30 28 29 28

𝐿𝐺2 =𝑛∑𝑘=1

𝑤𝑘𝑙𝑘2= 0.25 ∗ 28 + 0.2 ∗ 30 + 0.3 ∗ 28 + 0.1 ∗ 29

+ 0.15 ∗ 28 = 28.5,𝑧2 = 𝑃𝐺2 𝐿𝐺2 = 0.29 ∗ 22.3 = 28.8005.

(15)

The risk value of contract modification is much biggerthan the value of bad booking.The company should considerthis result to design the outsourcing plan.

Case 2 for Situation B. On the same background, throughrelevant analysis and review of feedback suggestions fromquestionnaires, the panel of experts determines 5 outsourcingrisks: bad book (𝑅1), contract modification (𝑅2), decreasedservice quality (𝑅3), hidden costs (𝑅4), and damaged com-pany image (𝑅5).The 5 risk-associatedmatrixes offered by theexperts are as follows:

𝐴1 =

𝑅1 𝑅2 𝑅3 𝑅4 𝑅5𝑅1𝑅2𝑅3𝑅4𝑅5

[[[[[[[[[

00000

20030

30010

20001

05000

]]]]]]]]]

,

𝐴2 =


[[[[[[[[[

00000

00030

42010

30001

15000

]]]]]]]]]

,

𝐴3 =


[[[[[[[[[

00000

20030

30011

40004

05000

]]]]]]]]]

,

𝐴4 =


[[[[[[[[[

00000

00030

50040

30001

05010

]]]]]]]]]

,

𝐴5 =


[[[[[[[[[

00000

00030

51040

20001

05010

]]]]]]]]]

.

(16)

Firstly, according to formula (1), the 5 risk-associatedmatrixes (𝐴1, 𝐴2, . . . , 𝐴5) are combined as the group-associated matrix 𝐴𝐺 = [𝑎𝐺𝑖𝑗 ]5×5:

𝐴𝐺 =


[[[[[[[[[

00000

00030

3.700

1.750

2.90000

05000

]]]]]]]]]

. (17)

Secondly, according to formula (2), the group-associatedmatrix 𝐴𝐺 is transformed into regulated group-associatedmatrix 𝐵𝐺 = [𝑏𝐺𝑖𝑗 ]5×5; namely,

𝐵𝐺 =


[[[[[[[[[

00000

000

0.450

0.5600

0.270

0.440000

00.76000

]]]]]]]]]

. (18)


R1

R2

R3

R4

R5

Figure 1: Hierarchical structure of risks.

According to formula (3), group comprehensive influencematrix 𝐶 = [𝑐𝑖𝑗]5×5 is constructed; that is,

𝐶 =


[[[[[[[[[

00000

0.19800

0.450

0.67900

0.270

0.440000

0.150.760

0.3420

]]]]]]]]]

. (19)

Furthermore, according to the situation of the company,the thresholds about comprehensive influence matrix pro-vided by 5 experts separately are 𝜆1 = 0.1, 𝜆2 = 0.1, 𝜆3 =0.15, 𝜆4 = 0.1, 𝜆5 = 0.1. According to formula (4), thegroup threshold is 𝜆𝐺 = 0.115. According to formula (5),the comprehensive influencematrix𝐶 is transformed into 0-1comprehensive influence matrix𝐷 = [𝑑𝑖𝑗]5×5; that is,

𝐷 =


[[[[[[[[[

10000

11010

10110

10010

11011

]]]]]]]]]

. (20)

According to matrix 𝐷 and formula (7), the hierarchicalstructure of 5 kinds of risks is divided as shown in Figure 1.

According to the hierarchical structure of risks shownin Figure 1, combined with the historical data and reality ofthe market, the panel of experts offered the probability ofoccurrence of the bottom risk 𝑅1, the conditional probabilityof occurrence of the risks on the second layer 𝑅2 and 𝑅4,and the conditional probability of occurrence of the risks onthe third layer 𝑅3 and 𝑅5, as shown in Table 3. The loss ofrisks offered by experts (unit: ten thousand Yuan) is listed inTable 4. Furthermore, according to formulas (11)–(14.2𝑑𝑖𝑓𝑓−1),the comprehensive probability of occurrence of risks 𝑃𝐺𝑖 iscalculated, shown in the second list of Table 5. For example,the probability of occurrence of the risk 𝑅2 is 𝑃𝐺2 = 𝑃𝐺(𝑅1 =𝑇)𝑃𝐺(𝑅2 = 𝑇 | 𝑅1 = 𝑇) + 𝑃𝐺(𝑅1 = 𝐹)𝑃𝐺(𝑅2 = 𝑇 |𝑅1 = 𝐹) = 0.284. According to formula (9), the loss of risks

Table 3: Probability and conditional probability of occurrence ofrisks offered by experts.

𝐸1 𝐸2 𝐸3 𝐸4 𝐸5𝑃(𝑅1 = 𝑇) 0.36 0.24 0.28 0.32 0.24𝑃(𝑅2 = 𝑇 | 𝑅1 = 𝑇) 0.42 0.44 0.45 0.45 0.45𝑃(𝑅2 = 𝑇 | 𝑅1 = 𝐹) 0.22 0.20 0.23 0.23 0.22𝑃(𝑅4 = 𝑇 | 𝑅1 = 𝑇) 0.33 0.35 0.33 0.32 0.36𝑃(𝑅4 = 𝑇 | 𝑅1 = 𝐹) 0.15 0.12 0.11 0.15 0.14𝑃(𝑅3 = 𝑇 | 𝑅4 = 𝑇) 0.40 0.38 0.42 0.40 0.40𝑃(𝑅3 = 𝑇 | 𝑅4 = 𝐹) 0.20 0.22 0.20 0.23 0.21𝑃(𝑅5 = 𝑇 | 𝑅2 = 𝑇, 𝑅4 = 𝑇) 0.44 0.42 0.44 0.43 0.44𝑃(𝑅5 = 𝑇 | 𝑅2 = 𝑇, 𝑅4 = 𝐹) 0.40 0.38 0.37 0.38 0.37𝑃(𝑅5 = 𝑇 | 𝑅2 = 𝐹, 𝑅4 = 𝑇) 0.33 0.32 0.28 0.26 0.28𝑃(𝑅5 = 𝑇 | 𝑅2 = 𝐹, 𝑅4 = 𝐹) 0.18 0.15 0.20 0.16 0.16

Table 4: Loss of risks offered by experts.

𝑙𝑘𝑖 𝐸1 𝐸2 𝐸3 𝐸4 𝐸5𝑅1 20 22 25 22 22𝑅2 28 30 28 29 28𝑅3 56 56 54 55 54𝑅4 64 65 66 66 64𝑅5 80 78 78 80 82

Table 5: Probability of occurrence of risks, loss of risks, and valueat risk.

𝑃𝐺𝑖 𝑙𝐺𝑖 𝑧𝑖𝑅1 0.29 22.4 6.5𝑅2 0.284 28.5 8𝑅3 0.245 55 13.5𝑅4 0.19 65 12.4𝑅5 0.253 79.3 20

(𝑙1𝑖 , 𝑙2𝑖 , . . . , 𝑙5𝑖 ) is combined as the comprehensive loss of risks𝐿𝐺𝑖 , as shown in the third list of Table 3. Finally, accordingto formula (10), each value at risk 𝑧𝑖 is calculated as shownin fourth list of Table 3. Thus, the calculating results of riskassessment provide decision support for risk management ofthe biopharmaceutical company.

4. Conclusion

A risk assessment method in consideration of risk-associatedsituations is provided in this paper. Based on various kindsof evaluating information about risks offered by the groupof experts, this method calculates each value at risk throughidentification of hierarchical structure of risks by adoptingknowledge related to probability theory. According to calcu-lation analysis, the proposedmethod is feasible and proved tohave certain application value. For structured risk assessmentproblems, the proposed method is general. However, therequirements of different industries and different types ofbusiness for service outsourcing are not the same. So themethod we give in this paper may not be directly applicable


to all industries or all types of enterprises, especially in somespecial industries. As service outsourcing progresses, someof the expected outsourcing risks will change, and on theother hand, new and unpredictable outsourcing risks willemerge. In order to ensure the smooth realization of serviceoutsourcing, the enterprise risk control during the wholeservice outsourcing process is worth studying. Also furtherresearch is required for large calculation amount of the riskoccurrence.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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