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## Capacity Practice Problems (Updated 02/24/2007 06:42 PM)

1. A grocery store has five regular checkout lanes and one express lane (12 items or less).  Based on a sampling study, it take 11 minutes on average for a customer to go through the regular lines and 4 minutes to go through the express line.  The store is open from 9 am to 9 pm daily.

a) What is the store's capacity?

b) What is the capacity if the regular checkout lines operate according to the schedule in Table 1 (the express lane is always open)?

Table 1: Number of Regular Checkout Lanes Available per Day

 Period MON TUES WED THUR FRI SAT SUN 9 am - noon 1 1 1 1 3 5 2 noon - 4 pm 2 2 2 2 3 5 4 4 pm - 6 pm 3 3 3 3 5 3 2 6 pm - 9 pm 4 4 4 4 5 3 1

Solution:

a) Determine the capacity (or design capacity) for a 12 hour day (9AM-9PM)

We have 5 regular lines and 1 express line.

The regular lines can handle (1 customer/11 min)*60 min/hr*12 hr/day*5 lines=328 customers/day

The express line can handle (1 customer/4 min)*60 min/hr*12 hr/day*1 line=180 customers/day

Capacity=328+180=508 customers/day

b) The solution below shows Monday results:

The regular lines can handle:

 Period Hrs # lines open Capacity (people/day) 9-12 noon 3 1 1/11*60*3*1=16.4 12-4 PM 4 2 1/11*60*4*2=43.6 4-6 PM 2 3 1/11*60*2*3=32.7 6-9 PM 3 4 1/11*60*3*4=65.4

For a total of 16.4+43.6+32.7+65.4=158 customers/day

The express line is always open, and it can handle 180 customers/day (see part a)

Capacity on Monday is 158+180=338 customers

2. A manufacturing firm is considering developing a new product for commercial use.  During the economic analysis phase of the product development process, analysts developed two alternative for making the product:

Alternative A.  Expand current manufacturing facilities and make the product at one of the firm's existing plants.  Fixed cost of the expansion is estimated at \$200,000.  Variable costs are projected to be \$15 per unit.

Alternative B.  Purchase the primary component of the product from a supplier and assemble the product at one of the firm's plants.  Since this alternative would not require facility expansion, its fixed manufacturing, selling, and administrative expenses are estimated at only \$48,000.  The purchased primary component would cost \$13 per unit, and there would be an additional \$5 variable cost per unit to assemble the final product.

a) Conduct a break-even analysis for the new product under both alternatives for a selling price of \$20 per unit.

b) If the projected volume for the new product were 30,000 units, which production alternative, if any, would you recommend?

c) What would the projected volume need to be before the alternative NOT recommended in part b would be the preferred alternative?

Solution:

a) At breakeven, revenue=cost

The expanded version of this equation is (\$unit price)*(units)=fixed cost+[(unit variable cost)*units]

How many units to breakeven?

 Alternative A Alternative B 20Q=200,000+15Q Q=40,000 20Q=48,000+[(13+5)Q] Q=24,000

b) If projected volume were 30,000 units, B is attractive. We would not even be "breaking even" with A.

c) We’d be indifferent if profits from A and B were equal.

A profit=rev-cost=20Q-[200,000+15Q]=5Q-200,000

B profit=rev-cost=20Q-[48,000+18Q]=2Q-48,000

Set equal and solve: 5Q-200,000=2Q-48,000

Q=50,667 units

3.  A producer of electronic games is evaluating a new product that will have a variable cost of \$18 and is expected to sell for \$49.  The fixed cost allocated to production will be \$50,000.  What is the breakeven volume?

Solution:

How many new e-games to breakeven?

At breakeven, rev=cost

49Q=50,000+18Q

Q=1613 units

NOTE:  These problems are adapted from Evans, J.R. (1997). Production/Operations Management, 5th ed. Minneapolis/St. Paul: West.
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