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Deterministic Throughput Capacity Planning Model
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Consider the following pattern of tax return arrivals at an IRS Regional Processing Center (from Exhibit D, Section J):

Week Returns
1 373,000
2 246,000
3 461,000
4 1,019,000
5 2,309,000
6 2,519,000
7 2,594,000
8 2,408,000
9 1,611,000
10 1,344,000
11 1,362,000
12 1,287,000
13 1,208,000
14 1,028,000
15 1,362,000
16 3,704,000
17 578,000
18 888,000
19 920,000
20 635,000
21 512,000
22 375,000
23 89,000
24 216,000
25 442,000
26 210,000

PROBLEM

Find minimum daily processing capacity needed to make sure that every return is processed within 17 days of arriving at the Center.

SOLUTION

The assumptions and the mathematical model for solving this problem are given below:

Assumptions
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This derivation has the following limitations:

1. The weekly input volume (arrival pattern) provided in Exhibit D, Section J is used as is, with no consideration of the fact that the weekly input volumes are random variables in reality, rather than constants as depicted in that section; further, weekly input volumes are assumed to be equally spread during the number of working days in each week.

2. First-come-first-served processing sequence is used; in reality, other proven operations management techniques could be used to optimize throughputs.
Aside: IRS may be required to use first-come-first-served sequencing by external forces.

3. Service rate is assumed to be constant per return; in reality, this would be a random variable dependent upon the type of return, the performance levels of data perfectionists, etc.

Definitions
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Window = Constraint depicting the number of days within which every return must be processed.

Window capacity = The number of returns that can be processed within a window at a given level of daily service rate (processing capacity).

Problem Formulation
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Objective: Find the minimum daily processing capacity necessary to meet window requirements.

Let b[t] = beginning inventory of leftover returns on day t

A[t] = arrivals on day t

S = service rate (control variable) in terms of returns per day

W = number of days (window) within which a return should be processed

Then C = WS = window capacity

b[t] = max { b[t-1] + A[t-1] - S , 0 }

r[t] = max { C - b[t] , 0 } = remaining window capacity for handling A[t]

V[t] = number of returns out of A[t] that will blow the window constraint

= max { A[t] - r[t] , 0 }

Solution
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Find minimum value of S that will ensure the following:

V[t] = 0 for all t = 1,2,3,...,26*n (26 weeks times n working days per week)