2015美賽特等獎原版論文集 37075

For office use only T1 T2 T3 T4 Team Control Number 37075 Problem Chosen C For office use only F1 F2 F3 F4 2015 Mathematical Contest in Modeling MCM Summary Sheet Organizational Churn A Roll of the Dice Network science is essential in many interdisciplinary studies due to its potential to deal with complex systems Since the organization of ICM forms a network structure network science can be utilized to analyze dynamic processes within the company e g the diffusion effects of organizational churn In this paper we construct a Human Capital network according to the hierarchical structure of ICM and create a simple yet effective model to capture the dynamic processes which includes organizational churn promotion and recruitment For organizational churn we propose and implement our probabilistic churn model inspired by Bayesian learning principles which estimates and updates the likelihood of individual churn using the Beta Binomial distribution Then we develop three promotion measures based on working experience inclination to churn and closeness centrality Moreover we propose several means of controlling the recruitment rate from the HR manager s perspective and further define some key concepts for evaluation such as dissatisfaction and productivity Through extensive simulations we show that our model is flexible enough to encompass most features of the current situation and yield convincing productivity and cost results We further extend our model to scenarios with higher churn rates and discover an interesting fact that higher churn rates lead to lower productivity cost ratios In an extreme case with no recruitment we discover differentiated HR health degeneration among different offices over two years through visualization Ultimately we incorporate methods from team science and approaches from multilayer networks in our context to combine Human Capital network with other network layers and discuss how to improve our estimation on organizational churn In summary our model is powerful and reliable for various types of human capital dynamic processes Nevertheless there are some existing problems such as simulation volatility which introduces extra computational costs 更多數(shù)學(xué)建模資料請關(guān)注微店店鋪 數(shù)學(xué)建模學(xué)習(xí)交流 Organizational Churn A Roll of the Dice Contents 1Introduction2 2Fundamental Assumptions2 3Preliminaries3 3 1Constructing Human Capital Network 3 3 2Terms and Mathematical Notations 4 4Models5 4 1Modeling Staff Churn 5 4 2Modeling HR Manager s Reactions 7 4 3Model Functions 8 5Simulations9 5 1Task 1 Simulations under Current Situation 9 5 2 Task 2 Defi ning Productivity and Testing Churn Infl uences 10 5 3Task 3 Budget Calculation 11 5 4Task 4 Changing Churn Rate 13 5 5Task 5 Pure Promotion and HR Health 14 5 6Comparing among Strategies 15 6Task 6 Extension Team Science and Multilayers16 6 1Incorporating Team Science 16 6 2Incorporating Multilayer Networks 17 7Sensitivity Analysis18 8Strengths and Weaknesses19 1 Team 37075Page 2 of 20 8 1Strengths 19 8 2Weaknesses 19 9Conclusion20 1Introduction Network science has gained its popularity in management science Modeling issues on human resource organization is at root modeling on its networks In this problem we need to consider a specifi c phenomena churn in ICM company To fulfi ll this we decompose the problem into several steps Build up a human capital network structure using information provided Use it as framework for further analysis Design a model capturing the mechanism of churn effect and design reasonable reactions of the HR manager Estimate organizational productivity and costs Analyze the sustainability of the network under different churn rates and estimate its effects Set up measures for company health and test effects of various changes Point out heuristics for the HR manager accordingly Incorporate ideas from team science into the model and point out the possibilities of analyzing from multilayer view Implement sensitivity test and analyze model strengths and weaknesses 2Fundamental Assumptions No staff naturally retire or get fi red Each staff member makes a decision whether to leave or not The staff members have latent characteristics unknown to the HR manager and us which might infl uence the decision process Beyond the visible organizational structure there exists a Human Capital network A staff member s monthly decision to leave or to stay acts as a piece of infor mation and fl ows through the Human Capital network Individuals digest received information through a learning process This learning mechanism will affect their decisions The HR manager can affect the number of people in the positions via different combined uses of promotion and recruitment A combined use of promotion and recruitment in this paper is called a strategy Team 37075Page 3 of 20 3Preliminaries 3 1Constructing Human Capital Network First we merge the table and graph given in the problem by assigning levels of positions to entries based on several reasonable assumptions Every senior junior manager has a clerk in his offi ce for administrative tasks The level of position of a staff member tend to be higher if his offi ce is closer to the CEO in the organizational graph The level of position of a manager cannot be lower than someone whose offi ce belongs to a lower tier in the organization graph Thus we can get the following allocation table for the 370 positions HH HH H H Tier XXXX XXXX XXXPosition level 1234567Total 1CEO20000024 2 Research10002014 CIO120080314 CFO120080314 HR01002014 VP20000024 Facilities10002014 Sales Marketing10002014 3 Networks0110110114 Information0110110114 Program Manager011065114 Production Manager1100100214 Plant Blue011065114 Plant Green011065114 Regional011065114 World Wide011065114 Internet011065114 4Director066060624 5Branch001125121200168 Total1020252511015030370 1 Senior Manager 2 Junior Manager 3 Experienced Supervisor 4 Inexperienced Supervisor 5 Experienced Employee 6 Inexperienced Employee 7 Administrative Clerk Table 1 The distribution of staff in different positions We begin to build the human capital network in ICM Defi ne V G v1 v2 v370 as the set of all positions Each node denotes one position Defi ne E G as the set of edges in the network vi vj E G if at least one of the following holds i and j are in the same offi ce Here one entry in the organization graph is consid ered as an offi ce whether it consists of two divisions or only four staff members Team 37075Page 4 of 20 i is the head of an offi ce and j is the head of the directly related upper offi ce or the opposite Here the staff member in the highest level of position within an offi ce is considered as the head of the offi ce such as the junior manager in Networks offi ce and the experienced supervisor in Branch offi ce i and j are both senior managers G V G E G defi nes the graph of Human Capital network We then visualize this network in Figure 1 Figure 1 Information Network in ICM Utilizing this network as a frame we delve into our main part of modeling 3 2Terms and Mathematical Notations In order to be clear and consistent through the paper we now settle down some terms and mathematical notations Level level of positions such as managers supervisors or employees Abbreviations we assign each level an abbreviation SE Senior Executive JE Ju nior Executive ES Experienced Supervisor IS Inexperienced Supervisor EE Experienced Employee IE Inexperienced Employee AC Administrative Clerk t time is discrete and the minimum time interval is one month t the set of people who leave the company at the end of t t the set of people who are recruited the company at the beginning of t Team 37075Page 5 of 20 t the set of people who work in the company at the beginning of t after recruit ment It s obvious that the relation t 1 t S t 1 t holds f t the mapping from t to V G which maps individual i t to his position f t i V G at time t f t 1is the inverse mapping d u v the distance between two nodes u v V G defi ned by the length of the shortest path connecting u and v in the graph d t ij thedistancebetweentwoindividualsi j t att defi nedbyd f t i f t j 4Models We construct our analysis by modeling the dynamic processes of staff churn promotion and recruitment Our probabilistic model for staff churn inspired by Bayesian learning principles which estimates and updates the likelihood of individual churn using the Beta Bernoulli distribution Next we develop three promotion measures Moreover we propose several means of controlling the recruitment rate 4 1Modeling Staff Churn 4 1 1Preliminaries In recent studies Bayesian learning has been used to analyze information aggregation in social networks 1 in which individuals modify their decision based on previous out comes of other individuals in the network For the sake of explaining our intuition consider a simple Bayesian learning process Suppose an random variable u 0 1 is drawn from a Bernoulli distribution where p is unknown u Bernoulli u p pu 1 p 1 u 1 Assume an observer wants to estimate the parameter p by drawing multiple u0s The individual has a prior estimation f p on p which is described as a Beta distribution1 f p Beta p p 1 1 p 1 B 2 where B is the normalization constant When seeing an outcome of u 1 the observer updates his prior according to the Bayes law2 f p p 1 1 p 1 p p 1 p 1 which can be viewed as increasing by 1 Similarly the observer increases by 1 if an outcome of u 0 is seen A simple analysis will show that if the number of observations reaches infi nity p whereas the Beta distribution in this case reduces to a Dirac delta function x p indicating that the observer s estimation converges to the correct p regardless of the original prior 1The Beta distribution is chosen because it is the conjugate prior of the Bernoulli distribution For more information on conjugate distributions please refer to 2 2We ignore the normalization constants for simplicity Team 37075Page 6 of 20 4 1 2Modeling the Churn Rate In light of this we introduce a novel method to model the churn rate which is concep tually similar to the above Bayesian learning process Specifi cally we view leaving the position as a decision making process suppose an individual i decides whether to leave or to stay in a particular month t based on a random variable u t i 0 1 where u t i 0 indicates to leave and u t i 1 indicates to stay u t i isdrawnasfollows First weassumetwohyperparameters t i and t i fori and draw p t i Beta t i t i then we draw u t i Bernoulli p t i fi nally we determine i is to stay if u t i 1 to leave otherwise Integrating out the random variable p t i we notice that the distribution of u t i is a specifi cation of the Beta Binomial distribution3with mean and variance 2 which has some nice properties for modeling the churn process on the onehand wecaneasilyestimatei sprobabilitytoleave whichisequalto t i t i t i on the other hand an increase in idecreases i s tendency to leave while an increase in iincreases the tendency to stay However three problems remain How to determine the prior iand i How to update the hyperparameters How to take the network structure into account We will explain these problems in the following paragraphs Determining the PriorGiven a churn rate p we can easily model the effect of a churn rate of p per year by setting p 12 We further observe that the variance of the Beta distribution is 2 1 so that larger leads to smaller variance indicating better estimation of p and more knowledge to the company status Thus it is safe to assume that people on high level positions have a larger compared to others and their decisions are less volatile Updating t i and t i We notice that in ICM an individual is more likely to churn if he is connected to other individuals who have churned This can be described as a learning process for the individual each month he observes the decision made by other individuals in last month For every observation of to stay the individual increases his for every observation of to leave the individual increases his We normalize the update values so that every month an individual s increases by 1 Information ReductionThe impact of churn information vary upon the distances be tween the source and destination From an individual s perspective the resignation of someone in the same department should have a greater impact than that of someone from another department We take this into account by reducing the update value of the hyperparameters Empirically we reduce the update by d2if the information takes at least d steps to transmit 3For simplicity we call this the Beta Bernoulli distribution Team 37075Page 7 of 20 4 1 3An Algorithm for the Churn Model To summarize we introduce an algorithm for this process For every individual i Sample the churn result for month t using hyperparameters i tand i t and de termine whether to stay or to leave If i decides to stay initialize two variables and for update For every individual j in t t individuals who stays update 1 d t ij For every individual j in t update 1 d t ij Update t 1 i t i and t 1 i t i 4 2Modeling HR Manager s Reactions After modeling the churn process we need to consider the strategic process of fi lling the vacancy from the perspective of the HR manager which combines promotion strategies and recruitment strategies Since recruiting a higher level position usually requires more time and money compared to promoting a low level staff and then recruiting new staff for that vacancy a rational HR manager would always prefer promoting to recruiting whenever possible This allows us to consider these two aspects separately 4 2 1Promotion Models We summarize some basic rules for promotion The HR manager does not read the annual evaluation report thus does not know anything about the matching between staff and positions In this way he will not consider changing staff within one level His fi rst choice is to promote someone that has reached the experience requirement to fi ll the vacancy If no person is qualifi ed and recruiting resources are permitted he will then post recruitment need for this position on the next If during the time the recruitment need is posted he fi nds that there is one person s experience has reached the requirement He will directly promote the fi rst such person and cancel the recruitment post Hewillnotpromoteaclerkbecauserecruitinganinexperiencedemployeeischeaper and less time consuming than recruiting a clerk In other words a naive manager never promotes a clerk Under current situations the HR manager has no knowledge about the capabilities of an employee nor their probability to leave Therefore to make the promotion process fair the HR manager should choose the employee on the lower level with the longest working experiences Hence we have the following strategy Team 37075Page 8 of 20 Experience OrientedFor a vacancy on level l l 6 select the employee on level l 1 with longest working experiences the employee should also satisfy the promotion requirements If nobody is available start recruiting If the HR manager happens to learn the churn model previously mentioned he can make inference on the probability of churn of an individual thus introducing a slight improvement over the experience oriented model Dissatisfaction Oriented For a vacancy on level l l 6 select the employee with the largest or the highest churn probability among all the employees on level l 1 who satisfy the promotion requirements If nobody is available start recruiting The HR manager can also take the Human Capital network structure into considera tion by promoting the employee with the largest centrality Centrality Oriented For a vacancy on level l select the employee with the largest close ness centrality tends to be greater when the employee is in the middle of the network from the qualifi ed employees on level l 1 If nobody is available start recruiting 4 2 2Recruitment Models We make the following assumptions on the recruiting strategies The HR manager has a maximum possible effort to recruit He cannot post more recruitment than this maximum because of his ability and resource limits The maximum effort is not affected when there is vacancy in HR offi ce When the number of vacant positions is higher than the maximum effort he ranks the vacant positions from higher level to lower level and only try to recruit the positions with the highest levels In other words he will prioritize recruiting a manager over recruiting an employee He can only renew his recruitment post over a length of period e g quarterly or semi annually Thus the HR manager has two direct means to increase the recruitment rate he can either increase the resource limits so that more people will be recruited in a fi xed time period or simply increase the frequency of the renewal of his recruit post Also the HR can control the promotion rate by setting different thresholds for promotion which is also an indirect method of controlling recruitment 4 3Model Functions Till now our models have already encompassed a large variety of mentioned features in ICM company including The information web captures how churn diffuses among staff members The risk of churn can be identifi ed in early stage by observing each staff member s The higher is the more likely the staff member chooses to leave Team 37075Page 9 of 20 The resignation of a staff member will increase the parameter of other employees thus increasing their chance of resigning We cover the fact that churn rates for middle managers are higher than other levels of positions by allowing different priors and for different levels The HR manager can choose recruitment effort recruitment time period and pro motion threshold to control the recruitment fl ow Matching between staff members and positions is one aspect that our model cur rently does not encompass However it can be incorporated by adding more assump tions about staff s skill classifi cations We will not highlight it in this paper 5Simulations Added Assumptions for Simulations In o