2015年美赛O奖论文C题Problem_C_37075
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.
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
5.1Task1:Simulations under Current Situation (9)
5.2Task2:De?ning Productivity and Testing Churn In?uences (10)
5.3Task3:Budget Calculation (11)
5.4Task4:Changing Churn Rate (13)
5.5Task5:Pure Promotion and HR Health (14)
5.6Comparing among Strategies (15)
6Task6:Extension-Team Science and Multilayers16
6.1Incorporating Team Science (16)
6.2Incorporating Multilayer Networks (17)
7Sensitivity Analysis18 8Strengths and Weaknesses19
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 speci?c phenomena,churn,in ICM company.To ful?ll this,we decompose the problem into several steps:
?Build up a human capital network structure using information https://www.wendangku.net/doc/e09954121.html,e 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?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 in?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?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".
3
Preliminaries 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 of?ce for administrative tasks.
?The level of position of a staff member tend to be higher if his of?ce is closer to the CEO in the organizational graph.
?The level of position of a manager cannot be lower than someone whose of?ce belongs to a lower tier in the organization graph.
Thus we can get the following allocation table for the 370positions:H H H H H H Tier X X X X X X X X X X X Position level 1234567Total
1
CEO 20000024
2Research 10002014
CIO 120080314
CFO 120080314
HR 01002014
VP 20000024
Facilities 10002014
Sales Marketing 10002014
3Networks 0110110114
Information 0110110114
Program Manager 011065114
Production Manager 1100100214
Plant Blue 011065114
Plant Green 011065114
Regional 011065114
World Wide 011065114
Internet 011065114
4Director 066060624
5Branch 001125121200168
Total 1020252511015030370
*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.De?ne V (G )={v 1,v 2,...v 370}as the set of all positions.Each node denotes one position.De?ne E (G )as the set of edges in the network.(v i ,v j )∈E (G )if at least one of the following holds:
?i and j are in the same of?ce.Here one entry in the organization graph is consid-ered as an of?ce,whether it consists of two divisions or only four staff members.
?i is the head of an of?ce and j is the head of the directly-related upper of?ce or the opposite.Here the staff member in the highest level of position within an of?ce is considered as the head of the of?ce,such as the junior manager in Networks of?ce and the experienced supervisor in Branch of?ce.
?i and j are both senior managers.
G={V(G),E(G)}de?nes the graph of Human Capital network.We then visualize this network in Figure1.
Figure1: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.
?Θ(t):the set of people who are recruited the company at the beginning of t.
?Γ(t ):the set of people who work in the company at the beginning of t after recruit-ment.It’s obvious that the relation Γ(t +1)=Γ(t ) Θ(t +1)\?(t )holds.
?f (t ):the mapping from Γ(t )to V (G ),which maps individual i ∈Γ(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 ),de?ned by the length of the shortest path connecting u and v in the graph.
?d (t )
ij :the distance between two individuals i,j ∈Γ(t )at t ,de?ned by d (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.
4.1
Modeling 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 )=p u (1?p )1?u (1)
Assume an observer wants to estimate the parameter p by drawing multiple u s .The individual has a prior estimation f (p )on p ,which is described as a Beta distribution 1
f (p )=Beta(p ;α,β)=p α?1(1?p )β?1
B(α,β)(2)
where B(α,β)is the normalization constant.When seeing an outcome of u =1,the observer updates his prior according to the Bayes’law 2:f (p )～(p α?1(1?p )β?1)·p ～p α(1?p )β?1,which can be viewed as increasing αby 1.Similarly,the observer increases βby 1if an outcome of u =0is seen.A simple analysis will show that if the number of observations reaches in?nity,α/β→p ,whereas the Beta distribution in this case reduces to a Dirac delta function δ(x ?p ),indicating that the observer’s estimation converges to the correct p ,regardless of the original prior.
1
The 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.
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.Speci?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 =1indicates to stay .
u (t )i is drawn as follows:First,we assume two hyperparameters α(t )i and β(t )i for i ,and
draw p (t )i ～Beta(α(t )i ,β(t )i );then we draw u (t )i ～Bernoulli(p (t )i );?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 speci?cation of the Beta-Binomial distribution 3with mean α/(α+β)and variance (αβ)/((α+β)2),which has some nice properties for modeling the churn process:on the
one hand,we can easily estimate i ’s probability to leave ,which is equal to β(t )i /(α(t )i +β(t )i );
on the other hand,an increase in αi decreases i ’s tendency to leave ,while an increase in βi increases the tendency to stay .However,three problems remain:How to determine the prior αi and βi ?How to update the hyperparameters?How to take the network structure into account?We will explain these problems in the following paragraphs.Determining the Prior Given a churn rate p ,we can easily model the effect of a churn rate of p per year by setting β/(α+β)=p/12.We further observe that the variance of the Beta distribution is αβ(α+β)2(α+β+1)
,so that larger (α+β)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 (α+β)compared to others,and their decisions are less volatile.
Updating α(t )i and β(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 α;for every observation of "to leave",the individual increases his β.We normalize the update values,so that every month,an individual’s (α+β)increases by 1.
Information Reduction The 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 d 2if the information takes at least d steps to transmit.
3For simplicity,we call this the Beta-Bernoulli distribution.
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 hyperparametersαi,t andβi,t,and de-termine whether to stay or to leave;
?If i decides to stay,initialize two variables?αand?βfor update;
?For every individual j inΓ(t)\Θ(t)(individuals who stays),update?α=?α+1
d(t)
ij
;
?For every individual j in?(t),update?β=?β+1
d(t)
ij
;
?Updateα(t+1)
i =α(t)
i
+?α
?α+?β
,andβ(t+1)
i
=β(t)
i
+?β
?α+?β
4.2Modeling HR Manager’s Reactions
After modeling the churn process,we need to consider the strategic process of?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?rst choice is to promote someone that has reached the experience requirement to?ll the vacancy.If no person is quali?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?nds that there is one person’s experience has reached the requirement.He will directly promote the?rst such person and cancel the recruitment post.
?He will not promote a clerk because recruiting an inexperienced employee is cheaper 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:
Experience Oriented For a vacancy on level l(l<6),select the employee on level l+1with 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 largestβ/α(or the highest churn probability)among all the employees on level l+1who 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 quali?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 of?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?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 identi?ed in early stage by observing each staff member’s β/α.The higherβ/αis,the more likely the staff member chooses to leave.
?The resignation of a staff member will increase theβparameter 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 priorsαandβfor different levels.
?The HR manager can choose recruitment effort,recruitment time period,and pro-motion threshold to control the recruitment?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 classi?cations.We will not highlight it in this paper.
5Simulations
Added Assumptions for Simulations
In order to get reasonable simulation results,we set some parameters.This,on the one hand,offers ease for simulations,while on the other land,does not lose its closeness to reality.Our added assumptions for parameters are listed below:
?The required experience for seven levels are48months,48months,24months,24 months,12months,0month respectively(from higher level to lower level).
?The time period for updating the recruitment post is6months.
?Theαandβfor different levels of positions are144,120,64,48,32,24,24(from higher level to lower level).
?The maximum recruiting effort for the HR manager is9%of370(average of8%-10%)for most cases.Any change in this parameter will be mentioned.
?All the data given related to recruiting time,recruiting cost,annual salary and training cost are deterministic.
5.1Task1:Simulations under Current Situation
In this section,we?rst present our basic simulation results of current situation.Figure 2shows the churn rate in different level of positions in50simulations.The churn rate of middle-manager(JE,ES and IS)is roughly30%and the churn rate for other positions (SE,EE,IE and AC)is around15%.The overall churn rate of the company is relatively stable at18%,which can accurately depict the current situation of ICM.
Another feature of ICM is that churn rate is steadily increasing.4The simulation result in Figure3does exhibit similar trend.We show the overall churn rate in next?ve 4Although our model should maintain the same churn rate in expectation,we can make slight,tractable modi?cations to simulate the steadily increasing process.Given our currentαandβ,if we want next
month’s churn rate to increase by?p,we can achieve that by multiplying?βby1+?p·(α+β)2+(α+β)
β.
Hence,the expectedβ/(α+β)for next month increases by?p
years.In spite of a couple of outliers,the major trend can be easily seen in the boxplot. The median value of churn rate has shown a slow but steady increase,from18%in the ?rst year to20%in the?fth year.
Figure2:Churn Rates of Different Levels
Figure3:Overall Churn Rates in Next Five Years
5.2Task2:De?ning Productivity and Testing Churn In?uences
To de?ne a metric to measure this company’s organizational productivity,we start from the individual level.This metric should incorporate the following three aspects:
Position Level People in different levels surely make different contribution to the over-all performance of a company.We reasonably assume the relative average annual salary of i’s level,S(t)i can properly re?ect his actual level contribution.
Training experience Experience in his current position contributes to one’s productiv-ity.We use the training cost the company spent on the individual since he began working in the current position as a proxy of his experience,denoted by T(t)i?t(t)i. Dissatisfaction An individual more unsatis?ed with current situation tends to work less inef?ciently,leading to lower productivity.We calculate i’s dissatisfactionσ(t)i as:
σ(t) i =
t?1
τ=t?t(t)
i
+1
e?(τ?(t?1))
j∈?(τ)
1
d
ij
We normalizeσ(t)i to the form of1
1+αln(1+σ(t)
i )
.αis a parameter re?ecting the in?u-
ence of dissatisfaction.We setα=0.1in the following calculation in this part.We will carry on sensitivity analysis regarding to this parameter in later section.People tend to forget past event at a fast speed[?].So the exponential term in this expression can depict the decline of memory.
Incorporating the three components,the productivity of an individual i at time t is:
p(t) i =
1
1+αln(1+σ(t)
i
)
(S(t)
i
+T(t)
i
?t(t)
i
)
To calculate organizational productivity,we use weighted sum of individual’s pro-ductivity.The weight w(t)i is determined by information network structure.An individ-ual in an important position weighs more.Here we use closeness centrality to re?ect this importance.So organizational productivity is de?ned by:
P(t)=
i∈Γ(t)w(t)
i
?p(t)
i
=
i∈Γ(t)
p(t)
i
v∈V(G)\f(t)?1(i)
d(v,f(t)?1(i))
370?1
Using this measure,we can track the dynamic change of organizational productivity. We can distinguish two kinds of effect associated with an individual’s resignation: Direct Effect The productivity of resigned individuals.
Indirect Effect The loss of productivity caused by the increase of dissatisfaction of the remaining staff in this company after the resignation.
DE(t)=
i∈?(t)w(t)
i
?p(t)
i
IE(t)=
i∈Γ(t)\?(t)w(t)
i
?(
1
1+αln(1+e?1σ(t)
i
)
?
1
1+αln(1+σ(t+1)
i
)
)(S(t)
i
+T(t)
i
?t(t)
i
)
We now calculate the loss of organizational productivity associated with a person’s resignation and decompose it into direct and indirect effect.We run our simulations over next?ve years for50times and average the results for presentation in Figure4.
It is obvious that direct and indirect effect on organizational productivity closely fol-low the number of resigned staff.An individual’s resignation will cause about20-unit direct effect and25-unit indirect effect.Total organizational productivity is2000-2500 units monthly,so a person’s resignation costs2%of total organizational productivity.
Another observation is the ratio between indirect and direct effect.In the simulations, the former is larger than the later with a stable ratio.The parameterαcan re?ect how a person view the resignation of another individual and we will return to it later.
5.3Task3:Budget Calculation
We now consider the budget of the company related to human capital.The budget con-sists of three components:staff salaries,recruitment cost and training cost.To make
Figure4:Effect on Organizational Productivity
calculation easier,as we assumed before,all individuals in the same position level have the same salary and training cost,and recruitment cost is uniform in each position level. All costs are incurred uniformly distributed across the corresponding time span.
We run simulations of the company and show the calculated cost below:
Figure5:Budget Calculation for Next Five Years
The costs presented in Figure5are calculated every six months.On average the recruitment cost and training cost is approximately7σand37σevery six months respec-tively.However,they are only a small fraction of total cost,as we have not reported salary cost here,which accounts for the largest part.
5.4Task4:Changing Churn Rate
In this part,we run simulations of ICM under different churn rates.We assume an increase in average churn rate from current18%to25%and35%.To achieve this goal, we can simply adjust theβ-to-αratio:for a churn rate of25%per year,we letβ/α= 0.25/12=0.02083;for35%per year,we letβ/α=0.35/12=0.02916.
We run simulations?fty times under each churn rate and take average.Figure6 shows our results.5The line charts re?ect the evolution of staff number,monthly orga-nizational productivity and monthly cost over next20years under different churn rates. The trends of three company’s characteristics are quite clear.The company sustains80% of its positions,namely around300staff members,under18%churn rate over a long time span.However,its staff number declines dramatically during the?rst six years if churn rate increases to25%or35%.Then it gradually becomes stable and settles down at approximately60%and40%full status of positions.
Figure6:Evolution of the Company in Next Twenty Years Under Different Churn Rates The trend also applies to organizational productivity and cost.Under high churn rate,more people leave the company,leading to both higher direct and indirect effect. Organizational productivity decreased by around30%and50%.Although high churn rates incur higher recruitment cost,it is minor compared with the large decline on salary and training cost resulted from fewer staff.Cost decreases by nearly20%and40%.
To sustain in current percentage of status,HR manager should change the recruit-ment strategy.In our baseline simulations of current situation,we assume the time pe-riod for updating recruitment post is6months and recruitment effort is9%.Now we need to change these two parameters to?ll in the gap quickly.We change updating in-terval to4months and recruitment effort to8.3%under25%churn rate and3months and9%under35%churn rate to sustain80%full status of positions.Figure7shows the distribution of the results from ten simulations:
Under new promotion strategies,ICM can roughly sustain80%capacity.Its monthly cost presents no signi?cant change.However,organizational productivity varies a lot. It decreases by more than10%under high churn rate,even if the staff number remains 5Due to limited space,we cannot provide full statistical results to prove that churn rates of our simu-lations are indeed25%and35%.However,we can estimate that is the case using the following method: We observe that the recruitment rate is stable for different churn rates,thus the stable percent of employees with a35%churn rate should be around80×18/35～42percent.This is apparent in the top graph in Figure 6.
Figure7:Comparison of Different Churn Rate under New Promotion Strategy
the same.If we use organizational productivity per unit cost as an indicator of the com-pany’s ef?ciency on using money,we can see from the last boxplot that the company is running more ef?ciently under lower churn rate.
5.5Task5:Pure Promotion and HR Health
Having set up several scenarios to evaluate the impact of different churn rates,we now apply our methodology to help us identity the impact on human resource health with no external recruiting.Apparently,with a alarming churn rate,recruiting nobody will de?nitely result in a decrease in the number of employees,cost,and productivity,so our emphasis will be on the number of employees in different of?ces.We modify our model to eliminate external recruitment(thus allowing only churn and promotion),and estimate the percentage of the employees in the of?ces.For better estimation,we assume a uniformly random distribution on the years of experience for the employees,so that one-fourth of the employees in each level are quali?ed to promote.
We visualize our results in Figure8.We found that the of?ces in tiers1and2sustains good health,whereas of?ces in lower tiers suffer signi?cant reduction in HR health.This conclusion is rather counter-intuitive at?rst sight,since middle managers tend to have a higher churn rate,but quite plausible given that low-level positions are not supplied by recruitment,whereas high-level positions can be sustained by promotion.We report that the number of remaining employees after2years is around204people,which is close to the theoretical value6.
6The theoretical value is370×0.85×(1?(325×0.18+45×0.30)/370)2=204.01
(c)After18months(d)After24months
Figure8:HR health in different periods.Darker colors indicate a of?ce with higher percentage of employees.White indicates a50percentage.
5.6Comparing among Strategies
Now we aim to improve the current situation by adopting some alternative strategies. Currently,when deciding which one to promote among all quali?ed candidates,HR manager will choose the one with the longest working time(Experience).We consider two other ways discussed before:selection by centrality within the information network (Centrality)and selection by likelihood to leave(Likelihood).We still use closeness cen-trality de?ned in previous sections.
Different Strategies Year2Year4Year6Year8Year10 Experience Staff Number313.0310.7304.9298.8292.9
Centrality Staff Number311.1307.5302.0295.7290.0 Increase(%)-0.60%-1.04%-0.94%-1.01%-0.99%
Likelihood Staff Number311.1309.1308.4303.7300.6 Increase(%)-0.62%-0.52% 1.16% 1.64% 2.61%
Experience Productivity2574.92706.22698.52651.72612.8
Centrality Productivity2589.32789.02878.62897.02916.8 Increase(%)0.56% 3.06% 6.67%9.25%11.63%
Likelihood
Productivity2577.82759.32872.82886.02904.0 Increase(%)0.11% 1.96% 6.46%8.84%11.14% Table2:Comparison among Different Strategies
From Table2,we can easily compare different promotion strategies.The staff num-ber declines under all three strategies,which is consistent with the steadily increasing churn rate.But the staff number under Likelihood strategy is2.61%higher after ten years compared with that under Experience.Besides,both Centrality and Likelihood strategies can increase organizational productivity by more than11%in Year10.So changing pro-motion strategy can contribute to the improvement of human capital management.
6Task6:Extension-Team Science and Multilayers
To ful?ll the HR’s vision,we manage to apply team science and discoveries from multi-player network research into our model.
6.1Incorporating Team Science
Recent studies on team performance have pointed out many possibilities to rigorously model teamwork[7].There are many established?ndings from team science that can be merged into our models.The two most prospective ones are related to"shared cogni-tion"and"team training".Here we point out the possibilities and de?ne some of them informally.Incorporating them rigorously needs more modi?cations.
6.1.1Shared Cognition
Shared cognition has been stressed by many researchers to be one of the crucial factors that shape the team performance[3].In order to work more ef?ciently,team members must predict what other teammates are going to do and what they need to do so,so that they can select actions consistent and coordinated with those of their teammates.[6] Shared cognition can form within different kinds of teams.In our context,a reason-able choice is an of?ce.Staff working within an of?ce share the same goal and shared cognition can positively contribute to its performance.Different measures have been de-veloped to scienti?cally measure shared cognition[3].We can take advantage of these measures to design an extension of our models.Especially when given empirical data (the time and numerical shared cognition),we can estimate the shared cognition growth using established models.We won’t go into details on these measures or empirical meth-ods.Instead,we put forward one possible measure within our context.
Consider an of?ce O with several individuals.We assume team cognition is separable and additive and focus on interpersonal relation in pairs.De?ne t(i,j)as the length i and
j has been working together.We can then use
i∈O w i
j∈O\{i}
t2(i,j)as a measure of shared
cognition.The squared term can re?ect the increasing speed of forming cognition and the weight re?ect the relative importance.
Another possibility is to use network related concepts.We can de?ne that two indi-viduals are connected if their co-working time is larger than a certain threshold.Then we can calculate measures such as the average degree in this network.Similarly,we can attach a weight to each edge based on how long individuals have been working together and build a weighted graph.
6.1.2Team Training
Another concept that we can potentially utilize in our models is team training.Currently, HR manager does not offer any training to the"team"or"of?ce"as a whole.Like offering individual’s training,we can take into account the team training.Being trained as a team can improve team members’understanding of each other’s roles,promote teamwork and enhance team performance[4].We can view team training as an accelerator of team cognition development.So we can multiply the team cognition measure we developed before by a function of team training to re?ect this effect.
Adding this process will not directly affect the dynamic process of staff’s leaving and HR manager’s?lling vacancy.However,it will affect the productivity of the company by promoting the ef?ciency of teamwork.effect may change the decision making of HR manager with the aim to maximize organizational productivity per unit of cost.
6.2Incorporating Multilayer Networks
In this section,we will explore the progress made by recent studies about multilayer networks[5]and apply this concept to our company context to help HR achieve better human resource management.
In previous sections,we have formulated a model based on the of?ce organizing structure.This network describes how the information?ows among all individuals. However,interpersonal relationship is much more complicated.On the one hand,peo-ple are consistently entering and leaving the company,causing a change of structure in the network.On the other hand,people have different types of interaction,such as:?People in the same of?ce work as a team and usually cooperate with each other more frequently.
?People can be close friends with each other regardless of their positions.
?People can have trust in his/her supervisors,and vice versa.
These interaction networks can provide us with more information,while shedding some light on more complex problems and enabling better solutions to problems that we are currently dealing with.Since we lack data on other layers of networks,it is impossible to implement a simulations.Hence,we provide some rules for connecting the information with other layers,as well as some solutions for better churn rate analysis. Network based on position Since the information network is already based on po-sition,such networks can be easily connected by introducing coupling edges.A simple example is the supervision networks,which is a tree structure among different positions. Since supervision in of?ce is a strong relationship,information transmitted through su-pervision is stronger than the average information network.Another example is the teammate relationship,where information is transmitted more frequently.
Network based on people Some relationships-such as friendship and trust-depend on people.Friendship allows for the transmission of more personal information,and trust enables directed transmission of information.Both increase the information intensity, for one tends to accept advice from friends and mentors more often.However,people
can switch positions or leave the company,so maintaining a static network is unfeasible. One approach,however,is to introduce direct cross-layer links with length zero from a person node to a position node.A quali?ed HR manager should track the person-position relationship and modify the structure of the network when necessary.
Now let us assume that we have incorporated teammate,friendship and trust rela-tionship layers to our information network.We provide some improvements over our previous solution to churn rate analysis and productivity estimation.
?Churn information can now also transmit along other layers of networks;
?We reduce our time slice from one month to a week,which allows more frequent information transmission between friends and teammates;
?We increase the impact of turnover decisions made by trusted individuals;
?We take friendship into account when calculating shared cognition,where friends in the same of?ce tend to have increased shared cognition,and hence productivity. 7Sensitivity Analysis
In this section,we implement sensitivity analysis for our model.Speci?cally,we test the sensitivity of parameterα,which we de?ne in calculating productivity.The value ofαis previously0.1.Now we test the effects when its value is changed to0.05,0.06,...,0.10, 0.11,...,0.15.Figure9show the results.
Figure9:Sensitivity Analysis on Parameterα
As we can see,the productivity is insensitive to our parameterα.A50%change inαwill bring no more than5%change in calculated productivity in most cases.
It is quite different if we look into the ratio between indirect effect and direct effect. It is actually sensitive to the change ofα.Whenαchanges from0.05to0.25,the median value of the ratio changes from less than2to nearly4.However,we have valid reasons for this.Consider how this parameter acts in our model.It is actually a re?ection of
the psychological impact intensity for how the dissatisfaction in job affects a person’s productivity.This impact,in reality,can come from shared cognition,the closeness be-tween staff members,company morale and other subtle in?uence.Quite an amount of research has been done for tracking these in?uences.[8].So some empirical research may help when we want to determine the actual level of this parameter.
8Strengths and Weaknesses
8.1Strengths
?Simplicity:Our measures are based on easily understood principles and there are simple ways to compute them.In addition,we make minimum assumptions on individual characteristics:onlyαandβare required for inference.
?Parameters:Most of our strategies are non-parametric,and the parameters of the churn model have nice properties,allowing for simple but effective parameter es-timation,reducing the need of tuning to a minimum.
?Coverage:Our model and measures are capable of simulating various scenarios associated with different churn rates,recruiting and/or promoting strategies.
?Flexibility:Our model can be easily incorporated with other assumptions.For example,if we assume that an individual accumulates dissatisfaction even without external in?uence,our model can be modi?ed to cover this assumption simply by increasing theβ-to-αratio every time period.
?Appealing simulation results:Simulation results of our model are very appealing.
Not only do them effectively re?ect the current situation,but they present insight-ful predictions for the effect of changes on company situation.In the case of Task 4,we discovered that higher churn rates leads to lower productivity-cost ratio.
?Heuristics for HR:HR can gain considerable heuristics from our paper,e.g.how to change recruiting strategies to sustain a required number of positions,and how to reduce churn rates by providing incentives for those who are likely to churn. 8.2Weaknesses
?Simulation volatility:Although our model has nice statistical properties,results generated by different simulations suffer high volatility.One possible remedy is to increase the sampling time,which reduces outcome variance at the cost of compu-tational resources.
?Unrealistic assumptions:Some of our measures are based on unrealistic founda-tions,e.g.productivity increases linearly with training costs,employees have no inclination towards different positions,etc.,which results in imperfect characteri-zation for the corresponding problem.
?Incomplete assumptions:There are also some other perspectives where we fail to consider,such as the positive effects of team cognition on productivity.