Operations Research

American Office Systems

William D. Whisler

California State University, Hayward

1. (a) There are six groups of variables, each of which represents the production, sales, inventory, and advertising of IBM and Macintosh adapters in the four quarters, and the cash balance and line of credit used at the end of each quarter. This gives a total of 40 nonnegative variables.

Pi,j = the number of adapters i, i = I (IBM) or M (Macintosh), produced in quarter j, j = 1, 2, 3, 4.

Si,j = the number of adapters i, i = I (IBM) or M (Macintosh), sold in quarter j, j = 1, 2, 3, 4.

Ii,j = the number of adapters i, i = I (IBM) or M (Macintosh), in inventory at the end of quarter j, j = 1, 2, 3, 4.

Ai,j = the advertising for adapter i, i = I (IBM) or M (Macintosh), in quarter j, j = 1, 2, 3, 4, in dollars

Cj = the cash balance at the end of quarter j, j = 1, 2, 3, 4.

Lj = the amount of the line of credit outstanding at the end of quarter j, j = 1, 2, 3, 4.

The objective is to maximize profit. This is the revenue from the adapters’ sales plus the interest from any outstanding cash balances invested in treasury securities minus the costs associated with production, advertising, inventory, and interest on the amount of money borrowed from the line of credit. The quarterly operating expenses are not included because they are fixed. Using the variables defined above gives,

The coefficients of the sales variables are the revenues from the adapter sales minus the bad debts not collected, 1% for IBM adapters and 2% for Macintosh adapters. From Table 3 of the case 2% of IBM adapter sales and 4% of Macintosh adapter sales are not collected. However, 50¢ on the dollar is received from a collection agency for these bad debts. Thus, American Office Systems receives half of the bad debts and only the other half is lost. The coefficients of the inventory variables represent the holding cost, 1% of the cost of the adapters per month. Thus, for the IBM adapters, 100(0.01)(3) is $3 per adapter per quarter and for the Macintosh adapters, 110(0.01)(3) is $3.30 per adapter per quarter.

There are 40 constraints grouped into six categories: (i) inventory balance; (ii) minimum inventory; (iii) maximum sales; (iv) maximum production; (v) cash flow balance; and (vi) line of credit utilized. The first four categories have eight constraints each, one for each product for each of the four quarters and the last two categories have four constraints, one for each of the quarters. The table in the answer to Question 1 (b), below, summarizes all of the constraints.

(i) Inventory Balance. The eight inventory balance constraints indicate that the inventory at the end of a quarter is equal to the inventory at the beginning of the quarter plus production minus sales during the quarter.

Ii,j = Ii,j-1 + Pi,j – Si,j

for i = I (IBM) or M (Macintosh) and j = quarter 1, 2, 3, or 4. Note that II,0 = 10,000 and IM,0 = 5,000, the initial inventories of the adapters.

(ii) Minimum Inventory. Six minimum inventory restrictions come from the 10% safety cushion.

Pi,j + Ii,j > 1.1Si,j+1

for i = i = I (IBM) or M (Macintosh) and j = quarter 1, 2, or 3. The ending inventory conditions yield the last two restrictions.

PI,4 + II,4 > 24,000

PM,4 + IM,4 > 4,800

(iii) Maximum Sales. The third category of constraints deals with the eight maximum sales conditions.

Si,i < Di,j + 0.1Ai,j

where Di,j are the maximum expected 1996 sales from Table 2 of the case, i represents the type of adaptor, i = I (IBM) or M (Macintosh), and j is quarter 1, 2, 3, or 4. The 0.1 coefficient represents the $10 needed to increase sales by one adapter.

(iv) Maximum Production. Eight maximum production restrictions are the fourth group of constraints.

PI,j < 28,800

PM,j < 9,600

for quarter j = 1, 2, and 3. For the last quarter, capacity is 10% larger,

PI,4 < 31,680

PM,4 < 10,560

(v) Cash Balance. Each quarter has a cash flow balance equation. This results in four equations. For the first quarter,

131.25SI,1 + 160SM,1 + L1 – 100PI,1 –110PM,1 – AI,1 – AM,1 – C1 = 2,626,200

where the coefficient of SI,1 is determined from the percent of the sales collected in the quarter from Table 3 of the case, 175(0.75) = 131.25. Likewise, the coefficient of SM,1 is 200(0.8). The right-hand side comes from the quarterly fixed operating cost in Table 1 of the case. Depreciation is not a cash expense so it is not included. From this expense, the collections from prior months’ sales, $383,800, must be deducted. The table below gives the calculations to determine the collections that are deducted. Note that the collections from the third quarter of 1995 include the money received for the bad debts. For example, since 2% of the IBM sales are sold to the collection agency, American Office Systems receives 50¢ on the dollar, that is, (0.5)(0.02) = 0.01.

First Quarter Cash Flow

Quarter IBM Macintosh

Sales Collections Sales Collections

Units $ % $ Units $ % $

1995-3

1995-4 7,000

8,000 1,225,000

1,400,000 0.04

0.20 49,000

280,000 1,400

1,600 280,000

320,000 0.07

0.11 19,600

35,200