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Demand, Throughput rate, Utilization and Overtime are pillars of Capacity planning. Overtime is a function of Demand, Throughput rate and Utilization. A firm’s objective is to determine the distribution of the Overtime risk estimate using the independent marginal distributions and dependency structure between the capacity levers. This additional operating cost can be managed using capacity levers and cross training efforts in order to help the firm manage labor expenses to meet seasonal demand patterns. If not managed efficiently it can lead to higher than expected costs. Unlike a manufacturing firm the capacity planning for a services firm would vary significantly. Throughput rate for a services firm would vary in parallel with the seasonal demand because of the high dependency on human effort, in high volume period associates complete more transactions in the given time while in a lean period the processing time for a transaction would increase leading to decline in throughput rate. Given this dependency among capacity levers the paper describes the application of t-copula as a joint density estimation technique to model the correlation among Demand, Throughput rate and Utilization. Data is hypothecated for illustration purpose and reflects the business case to model the dependency among capacity levers.
Keywords: Capacity planning, Copula, Overtime, Throughput rate, Utilization,Sensitivity analysis is the study of uncertainty in the dependent variable apportioned to uncertainty in the independent variables. Understanding the variability and predictability of Overtime hours becomes important in the Capacity planning domain. Demand, Throughput & Utilization are key pillars of Capacity planning that are used as levers to meet the incoming demand and thus to effectively manage the overtime cost. The following example is the case of a back office operations within a financial services firm where Volume is the demand in terms of transactions or requests to be processed. In this case, Volumes are driven primarily through market conditions and a firm’s investment decisions, which might be difficult to predict and could be random in nature. Throughput is the number of transactions completed in a given unit of time, Utilization relates to the proportion of time spent on core work, and Overtime is the excess of man-hours spent over and above the available hours in a given timeframe. Core work is the time spent on processing the transactions that are billable to the client.
Understanding the distribution of overtime is crucial to manage the staffing strategy. It is important to study how capacity levers vary under different scenarios and how dependency among them impacts the Overtime distribution. If they are independent then each lever can be studied independently and can be modeled to study the distribution of Overtime, while this will not be the case in a scenario where there is some dependency and it becomes important to study the joint density of the Capacity planning levers.
Figure 3 & 4 describes a classic case of Parkinson’s Law that states “work expands so as to fill the time available for its completion”. Demand and the Throughput rate vary in parallel to each other. High peak seasons are managed by an increase in Throughput rate and low Volume period brings the Throughput rate down. Utilization varies in relation to the demand but the relation is not highly correlated.
These key levers will not be independent of each other and there will always be some level of correlation between them, thus modeling the joint density estimation becomes crucial.
The sensitivity analysis method described in this paper thus uses a copula based methodology to model the joint density estimation of these levers and then uses simulation to derive the overtime distribution. Copula is a multivariate probability distribution used to model the dependence between random variables using their marginal distributions
Capacity planning as a domain involves a firm’s strategic goals towards staffing at a tactical level, with a short-term horizon and long-term horizon. At a tactical level (i.e. one day or a week) these metrics would generally vary in a random manner while on a medium to long-term there may be some level of stability and possible trends in the data.
This paper also describes the application of Brownian motion and Monte Carlo simulation to derive staffing strategies in the short-term level (i.e. intraday/daily/weekly) where demand may follow a random pattern. Descriptive analysis of Capacity levers
Utilization=((Time spent on core work))/((Total hours) )
A higher Utilization means that the time spent on core work (excluding internal deliverables or meetings that are not billable to the client) contributes to a larger portion of the total billable hours.
Throughput Rate=((Volume of transactions processed))/((Time spent on core work) )
Throughput rate in isolation as a point estimate will not help in deriving any conclusion. It needs to be compared on relative terms on a time scale or along the service lines with similar type of work.
Overtime hours=[((Demand Volume))/((Target Rate) )-(Staffing supply*Available hours*Utilization*(Throughput rate)/(Target rate))*staffing adjustment ]
If the total available hours in a week stands at 40 (8 working hours/day), then any additional time spent over and above 40 is categorized as overtime (additional cost to the firm). Data for this analysis uses a weekly interval scale to model the Capacity levers.
3.3 Marginal distributions of Independent variables and simulation
A marginal distribution is the probability distribution of individual variables contained in the subset of variables. For Utilization, Throughput Rate and Demand the log normal distribution provides the closest fit to the data. The data being skewed to the right with a lower bound of ‘0’ makes it an ideal candidate for log normal fit. If there is an interest in modeling extreme tail scenarios for independent variables, Extreme value theory can be incorporated along with copulas to model it. A GEV (Generalized extreme value distribution) or POT (‘peak over threshold’ approach) can be used to model the tails of the distributions. Diagnostic plots of the fitted log normal distribution on ‘Demand Volume’ are given below
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