Question

Great Northwoods Outfitters is a retail phone-catalog company that specializes in outdoor clothing and equipment. A phone station at the company will be staffed with either full-time operators or temporary operators 8 hours per day. Full-time operators, because of their experience and training, process more orders and make fewer mistakes than temporary operators. However, temporary operators are cheaper because they receive a lower wage rate and are not paid benefits. A full-time operator can process about 360 orders per week, whereas a temporary operator can process about 270 orders per week. A full-time operator averages 1.1 defective orders per week, and a part-time operator incurs about 2.7 defective orders per week. The company wants to limit defective orders to 200 per week. The cost of staffing a station with full-time operators is $610 per week, and the cost of a station with part-time operators is $450 per week. Using historical data and forecasting techniques, the company has developed estimates of phone orders for an 8-week period, as follows:

Week	Orders	Week	Orders
1	19,500	5	33,400
2	21,000	6	29,800
3	25,600	7	27,000
4	27,200	8	31,000

The company does not want to hire or dismiss full-time employees after the first week (i.e., the company wants a constant group of full-time operators over the 8-week period). The company wants to determine how many full-time operators it needs and how many temporary operators to hire each week to meet weekly demand while minimizing labor costs.
a. Formulate a linear programming model for this problem
b. Solve this model by using the computer (using Excel Solver)

Answer 1

(a)

Let F be the full-time employee level, Hj, Lj, and Wj be the number of temporary workers hired, laid off, and present in the week-j for j=1,2,...,8. Assume W0=0.

Minimize Z = 8*610 F + 450 ∑_j Wj

Subject to,

360 F + 270 Wj = Dj for j=1,2,...,8 Where Dj is number of orders in the j-th week

1.1 F + 2.7 Wj <= 200 for j=1,2,...,8

W_j = W_j-1 + H_j - Lj for j=1,2,...,8

(b)

Solution

Week	Full-timers (F)	Part-timers (Wj)	Hire (Hj)	Lay-off (Lj)	Productions	Orders	Rejections	Wj-1 + Hj - Lj
0		0
1	54	1	1	0	19500	19500	61	1
2	54	6	6	0	21000	21000	76	6
3	54	23	17	0	25600	25600	122	23
4	54	29	6	0	27200	27200	138	29
5	54	52	23	0	33400	33400	200	52
6	54	39	0	13	29800	29800	164	39
7	54	29	0	10	27000	27000	136	29
8	54	43	15	0	31000	31000	176	43
Totals	429	223
Cost/ unit	610	450	Total cost
Cost	261568	100220	361788