Question

A local firm manufactures children's toys. The project demand over the next four months for one particular model of toy robot is

Month	Workdays	Forecasted Demand
July	23	3825
August	16	7245
September	20	2770
October	22	4440

Assume that a normal workday is 8 hours. Hiring cost are $350 per worker and firing cost (including severance) are 850 per worker. Holding cost are $4 per aggergate unit held per month. Assume that it requires an average of 1 hour and 40 minutes for one worker to assemble one toy. Shortages are not permitted. Assime that the ending inventory for June was 600 of these toys and the mangager wishes to have at least 800 units on hand at the end of October. Assume that the current workforce level is 35 workers. Find the optimal plan by formulating as a linear program.  

Answer 1

Since there is 600 units ending inventory in June, the demand for July will be reduced to be 3825– 600 = 3225 units. By the same way, the demand for October will be increased to be 4440 + 800 = 5240 units. This is based on a First In First Out policy.

Month	Demand	Days	Units/workr/mo.	Cumulative Units/workr	Cumulative Demand	Worker Req.
July	3225	23	110.4	110.4	3225	30
August	7245	16	76.8	187.2	10470	56
Sept	2770	20	96	283.2	13240	47
Oct	5240	22	105.6	388.8	18480	48

Note:-

1. To calculate the number of units produced per worker for a particular month (column 4), we first calculate the number of units produced per day, which is equal to (8 hr)/(1 hr 40 min/ unit) = 4.8 units/day. Thus, the number of units produced per month is Days available * 4.8. For example, in July, unit/worker/mo. = 23 days * 4.8 units/day – 110.4 units/worker/mo.

2. To calculate the number of workers required for meeting the demand without any stockouts (column 7), for every month, we divide the cumulative demand up to that month (column 6) by the cumulative number of units produced by a worker (column 5) over the corresponding time interval. Then, we know that for any given month, the cumulative demand up to that month will be met if we employ the number of workers indicated in the corresponding cell, and have them work full-time until that month, possibly building anticipatory inventories in certain months, to be consumed in the subsequent months. Obviously, if we want to experience no stockouts over the entire planning horizon, we must employ the maximum number of workers appearing in the aforementioned column (i.e., 56). These workers will need to work full-time for the months of July and August, but they will be under-utilized in the subsequent months of September and October.

• Table below calculates the production based on 56 workers:

Month	Cumulative Demand	Production	Cumulative Production	Ending Inv.
July	3225	6182	6182	2957
August	10470	4288	10470	0
Sept	13240	2770	13240	0
Oct	18480	5240	18480	800

The total inventory held is 2,957 + 0 + 0 + 800 = 3757 units, which results in an inventory holding cost of (3757)(4) = $15,028.

The cost of hiring is (56-35)(350) = $7,350, giving a total cost of $15,028 + $7,350 = $22,378.

>> Now the plan that corresponds to demand to chase, i.e., the plan that changes the workforce level each month to most closely match the demand, and compute the cost(s) of this plan.

Month	Demand	Days	Unit/worker/mo.	Worker Req./mo.
July	3225	23	110.4	30
August	7245	16	76.8	95
Sept	2770	20	96	29
Oct	5240	22	105.6	50

Month	Cumulative Demand	Production	Cumulative Production	Ending Inv.
July	3225	3225	3312	0
August	10470	7245	10608	0
Sept	13240	2770	13392	0
Oct	18480	5240	18672	800

The total number of workers fired is 5+66 = 71 at a cost of (5+66)(850) = $60,350

The total hired is 65+21 = 86 at a cost of (86)(350) = $30,100.

inventory holding cost is (800)(4) = $3,200.

The total cost of this plan is $60,350 + $30,100 + $3,200 = $93,650.