Question

Mr. Meadows Cookie Company makes a variety of chocolate chip cookies in the plant in Albion,
Michigan. Based on the orders received and forecasts of buying habits, it is estimated that the
demand for the next 4 months is 850, 1260, 510 and 980, expressed in thousands of cookies.
During a 46-day period when there were 120 workers, the company produced 1.7 million
cookies. Assume that the number of workdays over the 4 months are, respectively, 26, 24, 20 and
16. There are currently 100 workers employed, and there is no starting inventory of cookies.

Consider the company describe in the previous problem. Suppose that the cost of hiring workers
each period is $100 for each worker until 20 workers are hired, $400 for each worker when
between 21 and 50 workers are hired, and $700 for each worker hired beyond 50.

a. Write down the complete linear programming formulation of the revised problem.

b. Solve the revised problem for the optimal solution. What difference does the new hiring
cost function make in the solution?

Answer 1

During a 46-day period when there were 120 workers, the company produced 1.7 million
cookies
This means average output per worker per day = 1700/46/120=0.308													Workers hired during the periods
	Demand	Work Days	Average Production Required Per Day	Average Output per worker per day	Workers Required Per Day	Workers Available	Workers Required during the month	Workers Hired	Total Number of workers available	Opening Inventory of cookies	Total Number of Cookies Produced	Closing Inventory of cookies	1 to 20	21 to 50	more than 50	Cost for hiring from 1 to 20 workers	Cost for hiring from 21 to 50 workers	Cost for hiring > 50 workers	Total Cost incurred for hiring additional workers
	1	2	3=1/2	4	5=3/4	6	7=5-6	8	9=6+8	10	11=9*2*4	12=10+11-1	13	14	15	16	17	18	19=13*16+14*17+15*18
Period I	850	26	32.692	0.308	106.14	100.00	6.14	7	107.00	0.00	856.86	6.86	7	0	0	100	400	700	700
Period II	1260	24	52.500	0.308	170.45	100.00	70.45	71	171.00	6.86	1264.032	10.89	0	0	71	100	400	700	49700
Period III	510	20	25.500	0.308	82.79	100.00	-17.21	0	100.00	10.89	616	116.89	0	0	0	100	400	700	0
Period IV	980	16	61.250	0.308	198.86	100.00	75.14	75.144	175.14	116.89	863.11	0.00	0	0	76	100	400	700	53200
Total Cost Incurred A																			103600
Here number of additional workers required for Period are calculated as (980-116.89)/0.308/16-100
The linear programming equation is minimize 19=13*16+14*17+15*18
Here 19 , 13 , 16 , 14,17 , 15, 18 are specified above
The above linear programming equation is subject to
Demand Constraints		1-10 = 850 for period I
		1-10=1260 for period II
		1-10=510 for period III
		1-10=980 for period IV
Here 1 & 10 are specified above in Chart
Production Constraints		Average out put per worker per day = 0.308 units					Calculated above
Work Force Constraints		100+8=9 for period I
		100+8=9 for period II
		100+8=9 for period III
		100+8=9 for period IV
Here we have assumed workers required over and above 100 are hired and fired every month.
8,9 are specified in chart above
I cannot make a comparision between old hiring system and new hiring system as old hiring system is not specified in the question.