Question

A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20. There are several conditions that the store needs to worry about. One of these is space to hold the inventory. An adult's bike needs two sq-feet, but a child's bike needs only one sq-foot. The store has 500 sq-feet of space. There are 1200 hours of assembly time available. A child's bike need 4 hours of assembly time; the Open Trail needs 5 hours and the Cityscape needs 6 hours. The store would like to place an order for at least 275 bikes.

1. (6 pts) Formulate a model for this problem.

2. (2 pts) How many of each kind of bike should be ordered and what will the profit be?

Corporate Headquarters has approved additional Assembly Time:   +25% (maximum).    However, this will require working additional shifts, i.e., Overtime.    There is an Overtime premium of 50% per hour, commonly known as ‘Time-and-a-Half’.   The standard employee hourly wage, for Bike Assemblers, is $11.75. Because there are multiple approaches to the treatment of Overtime, apply the following simplifying assumptions:

ASSUMPTION #1:   The marginal profits, Rev-Cost, account for the cost of labor, i.e., a standard hour of labor.    In the second part of the problem, you are asked to consider OT hours.    An hour of OT Labor = Standard + Premium.  

ASSUMPTION #2:   Standard component of OT Labor already covered via marginal profit coefficients (i.e., already paid for, regardless of the bike assembly time upper limit: 1200, or higher, or lower); so, the relevant cost you must account for is (only) the Premium component of OT Labor

3. (6 pts) Formulate a model for this problem.

4. (4 pts) How many of each kind of bike should be ordered and what will the profit be?    How does this compare with the original model (i.e., the original Order Plan)? What’s changed?

5. (2 pts) How many OT hours are required?   What is the Total cost of OT?

6. (4 pts) Does it make sense to work additional hours (i.e., work OT)?    Explain.   (Hint:    What is the profit?   How does this compare to the profit from the original model?)

Answer 1

1) LP model is following:

Let T, C, S, B be the number of bicycles of each of the four models, viz. the adult Open Trail (T), the adult Cityscape (C), the girl's Sea Sprite (S), and the boy's Trail Blazer (B) to be ordered.

Max 30T+25C+22S+20B

s.t.

2T+2C+S+B <= 500 (space)

5T+6C+4S+4B <= 1200   (assembly time)

T+C+S+B >= 275

T, C, S, B >= 0

Soution using LINDO follows:

2) Number of bikes to be order

Adult Open Trail (T) = 100

Girl's Sea Sprite (S) = 175

Total profit = 6850