2015美赛特等奖原版论文集33900

1、For offi ce use only T1 T2 T3 T4 Team Control Number 33900 Problem Chosen C For offi ce use only F1 F2 F3 F4 2015 Mathematical Contest in Modeling (MCM) Summary Sheet Analyzing Employee Turnover Using Dynamic Network Summary In this paper, we build a dynamic network model and perform various simulat

2、ions to study the behavior of Human Capital Network (HCN) under disturbance such as churns. We detail the structure of HCN factors and continuing eff ects of turnover and dy- namics of HCN utilizing given information and run simulations to examine the contagion of turnover and analyze the sustainabi

3、lity of HCN. We interpret and present our results along with robustness analysis and strengths and weaknesses of our model. For Task 1, we build two networks (Personnel Relationship Network and Organiza- tional Network), which together form our Human Capital Network. We extract important information

4、 from problem material and make assumptions when necessary to deduce the detailed structure and assign employees with the offi ces. We propose an evaluation function of assignments and use Genetic Algorithm to fi nd the optimal assignment. The assign- ment provides us with new information about empl

5、oyee relationship(co-offi ce relationship etc.), together with the supervisory relationship form the edges in HCN and complete our network building. For Task 2, we fi rst detail the turnover in terms of factors and impacts and evaluate the cost and productivity of HCN. We then dig into the dynamics

6、in terms of turnover process, promotion process and recruitment process using probabilistic method and discrete time system simulation. The results show that organizational churn has both direct and indirect eff ects on the organizations productivity. The direct eff ect is because it causes loss of

7、employee and the indirect is because it will make other employee more likely to turnover. For Task 3, we utilize our dynamic network model with our discrete time simulating system to calculate the budget. Our result is a budget requirement of 19.14 for recruiting and 10.87 for training in the next 2

8、 years, adding up to 30.01. For Task 4, we also run simulations and results show that ICM cant sustain its 80% full status. For Task 5, our results show that promotion increases the number of high-level em- ployee and improve the quality of staff because only those qualifi ed can be promoted. Thus p

9、romotion increases the productivity. We also make further discussions on Multi-layer Networks and HR issues which com- plete Task 6, 7 and other requirements that the problem material brought forth. Team # 33900i Contents 1Introduction1 1.1Problem Statement. . . . . . . . . . . . . . . . . . . . . .

10、 . . . . . . . . . . . . .1 1.2Related Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2Assumptions and Notations1 2.1Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2.2Notations . . . . . . . . . . . . . . . . . . . . . . .

11、 . . . . . . . . . . . . . . . . . .2 3The Human Capital Network Model2 3.1 Assigning Employees to Offi ces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 3.1.1Principles of Employee Assignment . . . . . . . . . . . . . . . . . . . . . .2 3.1.2Evaluation of Employee Assignment. . . . . .

12、 . . . . . . . . . . . . . . .3 3.1.3Optimization Using Genetic Algorithm . . . . . . . . . . . . . . . . . . . .3 3.2Building The Human Capital Network . . . . . . . . . . . . . . . . . . . . . . . . .4 3.2.1Building The Network of Personnel Relationship. . . . . . . . . . . . . .4 3.2.2Building th

13、e Organizational Network . . . . . . . . . . . . . . . . . . . . .5 3.3Evaluating Turnover Intention of Employee . . . . . . . . . . . . . . . . . . . . . .6 3.3.1Factor of Turnover Contagion . . . . . . . . . . . . . . . . . . . . . . . . .6 3.3.2Factor of Employee Centrality . . . . . . . . . . .

14、. . . . . . . . . . . . . .7 3.3.3Factors of Salary and Promotion Prospect . . . . . . . . . . . . . . . . . .7 3.3.4Evaluation using AHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 3.4Productivity and Costs of HCM . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 4Dynamics of

15、Human Capital Network9 4.1Process of Turnover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 4.2Process of Promotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 4.3Process of Recruitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16、. .10 5Simulation Results and Analysis11 6Sensitivity Analysis14 7Improvements using Multi-layered Networks15 8Further Discussions on HR16 9Strengths and Weaknesses16 9.1Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 9.2Weaknesses . . . . . . . . . . .

17、. . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 References17 Team # 33900Page 1 of 18 1Introduction 1.1Problem Statement Among various factors that infl uence a teams performance, the members of the team and the chemistry between them, i.e. teamwork, is an important one. The approach to

18、this is to maintain a team of talented and well-trained people and keep them in proper positions. However, in modern companies employee turnover 5 often takes place and disturb the environments. This paper off ers a model of employee turnover and its impact based on the Human Capital Network. The HC

19、N is extracted as a hierarchy structure of staff offi ces and divisions as nodes and a list of 370 employees as nodes, then the turnover intention is analyzed by the AHP method, and divided as salary, promotion, importance and impacts from others. The importance of the nodes is given by the closenes

20、s centrality and the LeaderRank value of nodes departments. Salary and promotion are impacts indicators in consideration for diff erent levels. The impact is conducted through links and damps as time elapse. The promotion process and recruit process are also predicted by stochastic process methods,

21、which make a part in the total dynamic process. 1.2Related Work In recent years, there has been tremendous interest in studying Team Science and de- veloping appropriate model of team and team performance. Salas et al. reviewed key discoveries and developments in the fi eld of team performance over

22、the past 50 years 2. They concluded that team performance can be modeled and which are our motivation to model the productivity.Stokols et al. argued about 3 and Kivel et al. studied the Multilayer Networks. which are incorporated into our paper for improvement of our model. Many works have inspecte

23、d the nature of rumor propagation etc. which is similar toNOT good.turnover propagation.However, the methods they used are mainly mean-fi eld equations for homogeneous networks and . for heterogenous networks. the underlying mechanism of how network structure infl uences the rumor spreading is still

24、 an open question. There have been extensive study But limited work has been done in analyzing the turnover process using a mathematical model and quantifying (?)its impact on the overall team performance. We are concerned in this paper with the. 2Assumptions and Notations 2.1Assumptions We make the

25、 following basic assumptions in order to simplify the problem. Each of our assumptions is justifi ed and is consistent with the information given in 1. Turnover behavior of an employee is contagious. So the worker connected to other employees who have churned is more likely to churn 1. Similar indiv

26、iduals are more likely to aff ect each others behavior. This is Team # 33900Page 2 of 18 an adopted assumption from previous work of Feeley et al. 15. 2.2Notations We use a list of symbols (cf. Table 1) for simplifi cation of expression. Table 1. Notations (in the order of model) Symbol Defi nitionN

27、otes Human Capital Network Model Ggraph of Personnel Relationship Network of ICMcf. Figure 2 ggraph of Organizational Network of ICMcf. Figure 3 Nnumber of nodes dijtopological distance between node i and j Rtemployee turnover rateRt= NELDY 1 (NEBY 2+NEEY3)/2 100% 5 Tturnover intention CCCloseness C

28、entralityCCi= 1 di = N PN j=1dij Piaverage annual salary(payment) of employee irepresents employees ability Li level of hierarchy of the offi ce of employee iLCEO= 5 and LBranch= 1 The Second Model Emu stuff ed Cfrozen 3The Human Capital Network Model In this model, the HCN is extracted as a hierarc

29、hy structure of staff offi ces and divisions as nodes and a list of 370 employees as nodes, then the turnover intention is analyzed by the AHP method, and divided as salary, promotion, importance and impacts from others. The importance of the nodes is given by the closeness centrality and the Leader

30、Rank value of nodes departments. 3.1 Assigning Employees to Offi ces Due to limited given information on the personnel relationship of ICM in detail, we assume some basic rules of allocating offi ces to complete our model. We apply these rules and form an AHP-based evaluation model, then we use Gene

31、tic Algorithm to fi nd the solution. 3.1.1Principles of Employee Assignment Our principles for allocating offi ces is listed as follows: 1NELDY:Number of Employees who Left During the Year. 2NEBY:Number of Employees at the Beginning of the Year. 3NEEY:Number of Employees at the End of the Year. Team

32、 # 33900Page 3 of 18 Employees with higher level of position should occupy offi ces of higher centrality and higher hierarchy in the structure diagram given in 1. Supervisors and executives should distribute as broadly as possible to ensure widespread supervisory and to maximally utilize the manager

33、 resource. Offi ces of same level should have similar constitute for equality. Each executive should be matched with an administrative clerk as his/her secretary. 3.1.2Evaluation of Employee Assignment We fi rst quantify these principles as follows: 1. Principle of centrality: E1= P Pi CCi 2. Princi

34、ple of hierarchy: E2= P Pi Li 3. Principle of equality: E3= P Dj 4. Principle of uniformity: E4= P Sj Piis the annual salary of an employee, which is a representative of his ability. Djis diff erences between offi ce j and other offi ces of the same level. Sjis the number of same type of employees i

35、n offi ce j . We hereby give the evaluation function: Eval = 1E1+ 2E2+ 3E3+ 4E4(1) Because that E3and E4are too large, we choose 1= 0.5, 2= 0.5, 3= 0.05, 4= 0.05 after some experiments with random assignments. We plug these parameters back into Equation 1 to obtain the evaluation function. 3.1.3Opti

36、mization Using Genetic Algorithm We got an evaluation function in Section 3.1.2. Next we are going to fi nd its optimum, which in correspondence is the best assignment solution.It is easy to show that this problem is a Combinatorial Optimization problem, for which search heuristics such as Genetic A

37、lgorithm and Tabu Search are proven to be more effi cient. We notice that the assignment itself is naturally a representation of the gene code, so we choose Genetic Algorithm to fi nd the maximum of the evaluation function (cf.Equation 1).Thus the evolution process such as gene crossover can be easi

38、ly achieved by swap employees in diff erent offi ces. We implement the algorithm using MATLAB, initialize 500 random solutions and begin iteration. The algorithm produce a satisfactory result (cf. Figure 1). We can see from Figure 1 that the algorithm not only assigned offi ces according to staff hi

39、erarchy (i.e. superior offi ces for staff of higher position like senior manager) but also incorporated uniformity and equality. Team # 33900Page 4 of 18 CEO SM JM AC*2 FACILITIES SM AC EE*2 RESEARCH SM AC EE*2 CIO* SM AC EE*5 VP SM JM AC*2 SALES SM AC EE*2 HR SM AC EE*2 DIRECTOR ES IS*2 IE DIRECTOR

40、 IS*3 IE DIRECTOR IS*3 IE DIRECTOR IS*3 IE DIRECTOR IS*3 IE DIRECTOR IS*3 IE CFO* SM AC EE*5 Manager Offices* JM AC IS EE*4 Branch Offices* ES IE*6 PRODUCT* JM AC ES EE*4 BRANCH* IS IE*6 Figure 1. Organizational structure of ICM with all the employees assigned. Each block represents an offi ce (with

41、out *) or two divisions (with * and thicker border). Bold text is abbreviation of positions(SM for Senior Manager etc.).Blocks with the same color and style share same constitute. Some blocks (Manager Offi ces and Branch Offi ces) are replaced by legend symbols for simplifi cation. 3.2Building The H

42、uman Capital Network We assigned the staff with proper offi ces in the above mentioned. Next we are going to build the network using this newly deducted information. 3.2.1Building The Network of Personnel Relationship Now that there are a grant total of N(N 6 370) employees in ICM, which is correspo

43、nding to the N node in graph G, and we use vi(1 6 i 6 N) to denote the ith node in G. In order to simplify the problem,we make a natural assumption that any pair of nodes viand in G is connected if and only if the three following conditions are satisfi ed: viand vjare in the same Branch or division;

44、 viand vjare in the same level; viand vj are respectively leaders of two offi ce whose superior department are the same. Its worth noting that the relationship between any two persons in ICM varies, thus G is a weighted graph,the weight of edge eijreveals the level of relationship between node vi an

45、d node vj.Let N dimensional matrix A = (wij)NNbe the adjacency matrix of graph G ,we can conclude from previous discussion that: wij= ? if viand vjsatisfy condition 2 1if viand vjsatisfy condition 1 or 3 The graph G is illustrated in Figure 2. Team # 33900Page 5 of 18 SM0 AC19 AC18 JM18 IE145 IS1 EE

46、3 EE2 IE146 IS2 EE5 EE4 IE144 IS0 EE1 EE0 IE148 IS4 EE9 EE8 IE147 IS3 EE7 EE6 IE149 IS6 IS5 ES24 SM1 JM19 AC20 AC21 SM3 EE84 EE85 AC23 SM2 EE82 EE83 AC22 SM4 EE86 EE87 AC24 SM5 EE88 EE89 AC25 SM9 SM8 AC29 EE102 EE103 EE100 EE101 EE106 EE107 EE104 EE105 EE108 EE109 AC28 AC26 AC27 SM7 SM6 EE93 EE95 EE

47、94 EE91 EE90 EE92 EE97 EE96 EE99 EE98 EE33 EE32 EE31 EE30 IS12 IS11JM4 JM5 AC5 AC4 EE26 EE27 EE28 EE29 IS20 AC13 AC12 EE59 EE58 IS19JM12 JM13 EE64 EE65 EE60 EE61 EE62 EE63 EE48 EE49 EE42 EE43 EE46 EE47 EE44 EE45 IS16 IS15 JM8 JM9 AC9 AC8 AC11 AC10 EE55 EE54 EE57 EE56 EE51 EE50 EE53 EE52 IS18 IS17JM1

48、0 JM11 IS8 IS7 EE11 EE10 EE13 EE12 EE15 EE14 EE17 EE16 JM0 JM1 AC1 AC0 IS22 IS21 EE73 EE72 EE71 EE70 AC15 AC14 JM14 JM15 EE68 EE69 EE66 EE67 IS9 IS10 EE19 EE18 JM2 JM3 AC3 AC2 EE20 EE21 EE22 EE23 EE24 EE25 EE37 EE36 EE35 EE34 EE39 EE38 EE40 EE41 IS14 IS13JM6 JM7 AC7 AC6 IS23 IS24 EE79 EE78 EE77 EE76

49、 EE75 EE74 EE80 EE81 AC17 AC16 JM16 JM17 ES20 ES21 IE131 IE130 IE128 IE129 IE124 IE125 IE126 IE127 IE120 IE121 IE122 IE123 ES19 ES18 IE115 IE114 IE117 IE116 IE111 IE110 IE113 IE112 IE119 IE118 IE108 IE109 IE142 IE143 IE140 IE141 IE133 IE132 IE137 IE136 IE135 IE134 IE139 IE138 ES22 ES23 ES13 ES12 IE8

50、2 IE83 IE80 IE81 IE73 IE72 IE77 IE76 IE75 IE74 IE79 IE78 ES11 ES10 IE60 IE61 IE62 IE63 IE64 IE65 IE66 IE67 IE68 IE69 IE71 IE70 ES17 ES16 IE106 IE107 IE104 IE102 IE103 IE100 IE101 IE99 IE98 IE97 IE96 IE105 ES15 ES14 IE86 IE87 IE84 IE85 IE88 IE89 IE95 IE94 IE91 IE90 IE93 IE92 IE28 IE29 IE24 IE25 IE26

51、IE27 IE35 IE34 IE33 IE32 IE31 IE30 ES4 ES5 IE15 IE14 IE17 IE16 IE13 IE12 IE19 IE18 IE20 IE21 IE22 IE23 ES3 ES2 IE59 IE58 IE51 IE50 IE53 IE52 IE55 IE54 IE57 IE56 ES9 ES8 IE48 IE49 IE39 IE38 IE37 IE36 IE46 IE47 IE44 IE45 IE42 IE43 IE40 IE41 ES7 ES6 IE11 IE10 IE9 IE8 IE1 IE0 IE3 IE2 IE5 IE4 IE7 IE6 ES1

52、 ES0 CEO VP top-level mid-level bottom-level mid-top mid-bottom Branch L Director 2 Internet Links within an office/division Links between office heads Inter-hierarchy links Figure 2. Personnel Relationship Network of ICM. The color denotes offi ce hierarchy. Small clusters of offi ces and three typ

53、es of edges are labeled. Some edges are omitted for conciseness. 3.2.2Building the Organizational Network Also we can get the Organizational Network. The procedures of forming g can be imple- mented as follows: 1. add all staff offi ces and divisions to g as nodes. 2. add hierarchy chains as directe

54、d edges, from lower hierarchy to higher. 3. for all nodes pairs, if the two nodes have the same hierarchical father, add a bi- directed edge between them. 4. add a sink node to graph, then connect all nodes except the sink node to the sink node with a bi-directed edge, which is in convenience for le

55、ader rank. Graph g is a directed graph (cf. Figure 3). The calculated LeaderRank Value is illustrated by the node size. Note that LeaderRank Value is not always consistent with hierarchy. Team # 33900Page 6 of 18 CEO Director 2 Director 6 Director 5 Director 3 Director 1 Director 4 Research HR Sales

56、 Marketing Facilities CIO VP CFO Information Internet Networks Regional Plant Blue World Wide Production Manager Plant Green Program Manager Branch C Branch K Branch J Branch E Branch D Branch L Branch I Branch F Branch A Branch G Branch B Branch H top-level mid-level bottom-level mid-top mid-bottom

57、 Figure 3. Organizational Network of ICM. The color denotes offi ce level. The size of the node denotes the LeaderRank Value. Note that the LeaderRank Value of Production Manager is higher than that of VP despite of their hierarchical relations. 3.3Evaluating Turnover Intention of Employee Cotton et

58、 al. found that factors such as pay and employment perception are the most highly reliable correlates of turnover 6. The factors can be divided into External Factors and Internal Factors. We adopt his conclusion and further specify them into three basic elements: salary, level of position, promotion

59、 prospect. We also introduce the infl uence of other peoples turnover behavior, summing up to four elements denoted by C1, C2, C3, C4 for quantifi cation. In order to fi nd the employee with the highest turnover probability, we use AHP to evaluate the contribution of each factor to turnover intention. 3.3.1Factor of Turnover Contagion Hereinbefore we assumed that if one employee left, other employees close to him are infl uenced and are more likely to churn (cf. Section 2.1). We quant