Question

The Air Force will begin operating a 24-hour-a-day shipping distribution center in Dayton Ohio. The distribution center plans to employ workers during six “shifts” throughout a 24-hour period. The minimum required number of employees for each shift is provided in the table below. All employees will work two, and only two, consecutive shifts per 24-hour period; each shift is four hours long. Assume that each employee will work the same two shifts every day. All employees are paid at the same basic hourly rate, i.e., $40 per hour. However, there is a 20% premium for each hour worked in shift #1, #5, and #6.

Formulate a Linear Programming Model that will minimize the total daily labor cost of the center while meeting the employment needs of the shipping distribution center.

Shift #	1	2	3	4	5	6
Period	0200-600	0600-1000	1000-1400	1400-1800	1800-2200	2200-0200
Minimum # of Employees	5	15	12	10	19	7

Answer 1

^Solution

^{As stipulated only formulation is given.}

^{The given data are given below in a tabular form.}

Shift #	1	2	3	4	5	6
Period	0200-600	0600-1000	1000-1400	1400-1800	1800-2200	2200-0200
Minimum # of Employees	5	15	12	10	19	7
Let # of workers employed be	x1	x2	x3	x4	x5	x6
Per employee cost**	48	40	40	40	48	48

^{Note: **All employees will work two, and only two, consecutive shifts and each shift is four hours long => per employee cost per shift = 4 x hourly rate = 4 x 10 = 40.}

^{There is a 20% premium for each hour worked in shift #1, #5, and #6 => for these 3 shifts hourly rate is 12 and hence per employee cost per shift = 4 x 12 = 48.}

^{Then, the formulation of LPP is:}

^{Minimize z = 48x1 + 40x2 + 40x3 + 40x4 + 48x5 + 48x6}

^{Subject to}

^{x1≥ 5}

^{x2≥ 15}

^{x3≥ 12}

^{x4≥ 10}

^{x5≥ 19}

^{x6≥ 7}

^{x1≥ 0,x2≥ 0,x3≥ 0,x4≥ 0,x5≥ 0,x6≥ 0.}

^Answer

^DONE

^Solution

^{As stipulated only formulation is given.}

^{The given data are given below in a tabular form.}

Shift #	1	2	3	4	5	6
Period	0200-600	0600-1000	1000-1400	1400-1800	1800-2200	2200-0200
Minimum # of Employees	5	15	12	10	19	7
Let # of workers employed be	x1	x2	x3	x4	x5	x6
Per employee cost**	48	40	40	40	48	48

^{Note: **All employees will work two, and only two, consecutive shifts and each shift is four hours long => per employee cost per shift = 4 x hourly rate = 4 x 10 = 40.}

^{There is a 20% premium for each hour worked in shift #1, #5, and #6 => for these 3 shifts hourly rate is 12 and hence per employee cost per shift = 4 x 12 = 48.}

^{Then, the formulation of LPP is:}

^{Minimize z = 48x1 + 40x2 + 40x3 + 40x4 + 48x5 + 48x6}

^{Subject to}

^{x1≥ 5}

^{x2≥ 15}

^{x3≥ 12}

^{x4≥ 10}

^{x5≥ 19}

^{x6≥ 7}

^{x1≥ 0,x2≥ 0,x3≥ 0,x4≥ 0,x5≥ 0,x6≥ 0.}

^Answer

^DONE