solution

Having trouble with Part C of this question. Unable to figure out the Min and developing a minimum cost schedule for the part-time employees. With this, I also need help to figure out the inequality for 7 pm, 8 pm, and 9 pm. I also need help with the optimal solution of Part C. The other parts are all correct.

A linear programming computer package is needed.

Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.

Time	Number of Employees Needed
11:00 a.m.–Noon	9
Noon–1:00 p.m.	9
1:00 p.m.–2:00 p.m.	9
2:00 p.m.–3:00 p.m.	3
3:00 p.m.–4:00 p.m.	3
4:00 p.m.–5:00 p.m.	3
5:00 p.m.–6:00 p.m.	6
6:00 p.m.–7:00 p.m.	12
7:00 p.m.–8:00 p.m.	12
8:00 p.m.–9:00 p.m.	7
9:00 p.m.–10:00 p.m.	7

One full-time employee comes on duty at 11:00 a.m., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 p.m. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern.

(a)

Develop a minimum-cost schedule for part-time employees. (Let

x_i

= the number of part-time employees who start work beginning at hour i where

i = 1 = 11:00 a.m.,

i = 2 = Noon,

etc).Min

30.4×1?+30.4×2?+30.4×3?+30.4×4?+30.4×5?+30.4×6?+30.4×7?+30.4×8?

s.t.11:00 a.m.

x1?=8

Noon

x1?+x2?=8

1:00 p.m.

x1?+x2?+x3?=7

2:00 p.m.

x1?+x2?+x3?+x4?=1

3:00 p.m.

x2?+x3?+x4?+x5?=2

4:00 p.m.

x3?+x4?+x5?+x6?=1

5:00 p.m.

x4?+x5?+x6?+x7?=5

6:00 p.m.

x5?+x6?+x7?+x8?=10

7:00 p.m.

x6?+x7?+x8?=10

8:00 p.m.

x7?+x8?=6

9:00 p.m.

x8?=6

x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈ = 0

Find the optimal solution.

(x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈) =

(8, 0, 0, 0, 2, 0, 4, 6)

(b)

What is the total payroll (in $) for the part-time employees?

$

608

How many part-time shifts are needed?

20 shifts

Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.

There are 7 surplus employees that are scheduled from 2-3 p.m. and 4 surplus employees scheduled from 8-9 p.m. Based on this it would be desirable to rotate employees off sooner making this more beneficial.

(c)

Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. (Let

x_i

be the same variables from part a. Let

y_i

= the number of part-time employees who start work beginning at hour i where

i = 1 = 11:00 a.m.,

i = 2 = Noon, etc).

Min : _________________________

s.t.11:00 a.m.

x1?+y1?=8

Noon

x1?+x2?+y1?+y2?=8

1:00 p.m.

x1?+x2?+x3?+y1?+y2?+y3?=7

2:00 p.m.

x1?+x2?+x3?+x4?+y2?+y3?+y4?=1

3:00 p.m.

x2?+x3?+x4?+x5?+y3?+y4?+y5?=2

4:00 p.m.

x3?+x4?+x5?+x6?+y4?+y5?+y6?=1

5:00 p.m.

x4?+x5?+x6?+x7?+y5?+y6?+y7?=5

6:00 p.m.

x5?+x6?+x7?+x8?+y6?+y7?+y8?=10

7:00 p.m.

_________

8:00 p.m.

_________

9:00 p.m.

_________

x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, y₁, y₂, y₃, y₄, y₅, y₆, y₇, y₈, y₉ = 0

Find the optimal solution.

(x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, y₁, y₂, y₃, y₄, y₅, y₆, y₇, y₈, y₉) =

(_______________________________________________)

How many part-time shifts are needed, and what is the cost savings compared to the previous schedule?

part-time shifts needed

20

cost savings

$106.40