Mr. Meadows Cookie Company makes a variety of chocolate chip cookies in the plan

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?

Explanation / Answer

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.

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