Math Guru

Example 1 Service Charge Allocation (Adapted from Finite Mathematics, Models, and Struc- ture, Revised Edition by William J. Adams, copyright 2009.)

The Arkin Company, which makes television sets, has three production departments, which we shall denote by P1, P2, and P3, and four service departments—accounting, maintenance, market-

ing, and purchasing. Each service department’s cost must be distributed to the production depart- ments and to other service departments based on their respective usages of the services provided.

For each service department listed in the leftmost column of Table 1, the fraction of its total cost assigned to the service and production departments of the firm is given. Thus from row 1 we have that 2 percent of the total cost of accounting is assigned to maintenance, 10 percent of the total cost of accounting is assigned to marketing, and so on. Column 1 specifies the fraction of the ser- vice departments’ costs that is assigned to accounting, column 2 specifies the fraction of the ser- vice departments’ costs that is assigned to maintenance, and so on. (Such determinations can be made in various ways; one way is based on the floor space occupied by each department.) The problem is to determine each service department’s total costs (overhead plus charges for services provided by other departments) and to allocate these costs to the production departments.

Let x, y, z, and w denote the total costs in January, measured in dollars, of the accounting, maintenance, marketing, and purchasing departments, respectively. From column 1 we have that the costs of the maintenance, marketing, and purchasing departments that are assigned to accounting are 0.10y, 0z, and 0.10w, respectively. The overhead of the accounting department (costs that are directly assigned to accounting such as salaries of employees in accounting, equipment, supplies, etc.) for January is $20,000. Thus x, the total cost of accounting, must satisfy the following condition:

A similar analysis for the maintenance, marketing, and purchasing departments yields the following conditions:

Table 1:

Service Department

Service Departments Production

Departments January

Overhead

Acc. Main. Mar. Pur. P1 P2 P3

Accounting 0 0.02 0.10 0.10 0.24 0.26 .028 $20,000

Maintenance 0.10 0 0.20 0.10 0.20 0.20 0.20 $18,000

Marketing 0 0 0 0 0.30 0.30 0.40 $80,000

Purchasing 0.10 0.10 0.10 0 0.20 0.20 0.30 $10,000

x 20000 0.1y 0.1w+ +=

Rearranging terms gives us the following system:

Here I have included the 0z terms to make the switch below to matrix notation easier to follow.

If let , , and . We can write the above system as

. If A is invertible, we can find and hence know the values of x, y, z, and w.

Using Excel, with the MINVERSE command, we can easily find the answer:

y 18000 0.02x 0z 0.1+ w+ + z 80000 0.1x 0.2y 0.1w+ + + w 10000 0.1x 0.1y 0z+ + +

= = =

1x 0.1y– 0z 1w–+ 20000 0.02x– 1y 0z 0.1w–+ + 18000 0.1x– 0.2y– 1z 0.1w–+ 80000

0.1x– 0.1y– 0z 1w+ + 10000

= = = =

1 0.1– 0 1–

0.02– 1 0 0.1–

0.1– 0.2– 1 0.1–

0.1– 0.1– 0 1

x

y

z

w

20000

18000

80000

10000

=

A

1 0.1– 0 1–

0.02– 1 0 0.1–

0.1– 0.2– 1 0.1–

0.1– 0.1– 0 1

= u

x

y

z

w

= b

20000

18000

80000

10000

=

Au b= u A 1– b=

b u

1 ‐0.1 0 ‐0.1 1.013514 0.112613 0 0.112613 $20,000 $23,423.42

‐0.02 1 0 ‐0.1 0.030713 1.013514 0 0.104423 $18,000 $19,901.72

‐0.1 ‐0.2 1 ‐0.1 0.117936 0.225225 1 0.134316 $80,000 $87,755.94

‐0.1 ‐0.1 0 1 0.104423 0.112613 0 1.021704 $10,000 $14,332.51

Matrix  A ‐1

Matrix  A

We find and

. To compute the values

of u in Excel we need the MMULT command.

Hence the costs of running the Accounting, Maintenance, Marketing, and Purchasing for the month of January is $23,423.42, $19,901.72, $87,755.04, and $13,332.51, respectively.

From these cost values and the percentages given in columns 5,6, and 7 of Table 1, we obtain the following allocation of service departments costs to each production department. Here we let c1,

c2, and c3, be the cost in dollars of the cost the service departments’ costs assigned to departments,

P1, P2, and P3, respectively.

Here we let C be the cost matrix of the production departments. , where

is the matrix respective cost allocations (i.e, pi,j is the proportion production depart-

ment Pi assigned to service department j.

. We compute C as follows

. So that the service cost for the

month of January of Departments P1, P2, and P3 are approximately, $38,795.25, $39,263.72, and

$49,941.03, respectively. We can compute the costs using Excel, with the MMULT command.

A 1–

1.013514 0.112613 0 0.112613

0.030713 1.013514 0 0.104423

0.117936 0.225225 1 0.134316

0.104423 0.112613 0 1.021704

=

u

x

y

z

w

1.013514 0.112613 0 0.112613

0.030713 1.013514 0 0.104423

0.117936 0.225225 1 0.134316

0.104423 0.112613 0 1.021704

20000

18000

80000

10000

23423.42

19901.72

87755.94

14332.51

= = =

C

c1 c2 c3

Pu= =

P pi j =

P 0.24 0.20 0.30 0.20

0.26 0.20 0.30 0.20

0.28 0.20 0.40 0.30

=

C

c1 c2 c3

0.24 0.20 0.30 0.20

0.26 0.20 0.30 0.20

0.28 0.20 0.40 0.30

23423.42

19901.72

87755.94

14332.51

38795.249

39263.717

49941.031

= = =

Exercise:

The Sonin Company, which makes computers, has two production departments, P1, and P2, and three service departments S1, S2 and S3. Each service department’s total cost must be distributed to the production departments and to the other service departments based on their respective usages of the services provided. For each service department listed in the leftmost column of Table 2, the fraction of its total cost assigned to the service and product departments of the firm is given.

(a) Set up the system of equations that describes the conditions to be satisfied. Be sure to label your variables including units. (b) Solve this system to determine the total costs of the service departments. (c) Determine the allocation of the total costs of the service departments to the production departments.

Table 2:

Service Department

Service Departments Production Department

s

January Overhead

S1 S2. S3. P1 P2

S1 0 0.10 0.05 0.40 0.45 $40,000

S2 0.10 0 0.10 0.40 0.40 $30,000

S3 0.20 0.05 0 0.35 0.40 $20,000

C

0.24 0.2 0.3 0.2 38795.25

0.26 0.2 0.3 0.2 39263.72

0.28 0.2 0.4 0.3 49941.03

Matrix P

Sheet1