Question

Oberweis is a dairy company based out of Chicago. Oberweis produces milk at a fixed rate of 6,000 gallons/hour. Oberweis serves clients who, in total, request 100,000 gallons of milk per day. Clients send trucks to pick up their orders, and trucks arrive uniformly from 8am to 6pm. If Oberweis runs out of milk, trucks will wait until enough is produced to satisfy their client’s requests. Per Oberweis policy, the company begins and ends each day with 24,000 gallons in finished goods inventory. The plant begins its daily production at 8am and closes once it has satisfied all client demand and replenished its finished goods inventory. In your analysis, treat trucks/milk as a continuous flow process.

1) If Oberweis were charged $30 per hour per waiting truck, how much would Oberweis be charged per day (in dollars)?

Answer 1

Arrival hours = 8am to 6pm = 10 hours
Demand rate = 100000/10 = 10000 gallons per hour

Company starts the day with 24000 gallons in FG inventory.
The queue starts to form when inventory becomes 0.

Let T be the time elapsed (in hours) from 8am till the time queue starts to form.
24000+6000*T = 10000T
T = 24000/(10000-6000) = 6 hours
So, queue starts to form at = 8am + 6 hours = 2 pm

We determined that queue starts to form at 2pm. Total demand after 2pm = 4*10000 = 40,000 gallons
Hours of production required to satisfy the total demand = 40000/6000 = 6.67 hours
So, truck queue length returns to zero after 6+6.67 = 12.67 hours after 8am , i.e. 8:40 pm

Maximum length of queue forms by 6pm, when the arrival of truck stops. This is 4 hours after the queue starts to form.
Total Arrivals during these four hours = (40000/2000) = 20 trucks
Total demand satisfied during these four hours = 6000*4/2000 = 12 trucks
Maximum queue length = 20-12 = 8 trucks

From 2pm onwards, 2 trucks will add to the queue every hour until 6pm.
After 6pm, queue length decreases at a rate of 3 trucks (6000 gallons) per hour and becomes 0 in 2.67 hours

Total waiting cost per day = (2+4+6+8+5+2+2/3)*30 = $ 830