University of Chicago Booth School of Business
Operations Management, Bus 20500
Birge
Midterm Solutions, OM 20500, Winter 2021
Total: 33 , Mean: 27.6, Median: 28, Standard Deviation: 4.68, 25%ile: 25, 75%ile: 32
Figure 1: Histogram of midterm scores.
1. Here is the inventory build-up graph for this problem:
INV(t) (Lbs*1000)
26
20
16
0
12 midnight
2am
5am 6am
10pm
9:20 am
time, t (CST)
Time t (CST)
10pm-12midnight
12-5am
5am-9:20am
In(t)
8k
8k
0
Out(t)
0
6k
6k
(a) [2] Yes
26, 000 pounds
(b) [1] 4 hours
(This had a typo; so, it is the same as (c) below.)
(c) [1]4 hours (2am to 6am)
∆IN V (t)
+8k
+2k
-6k
2. (a) [1] 5 couples per hour (one per 12 minutes)
1 1
(b) [1] min{ 10
, 5} =
1
10
per minute or 6 couples per hour.
(c) [2] IN V = λ ∗ CT , λ = 3 couples per hour, CT = 1/4 hour (15 minutes)
IN V = 3 ∗ 14 = 0.75 couples.
(d) [1] Only part (c) (inventory) would change (increase) since variation would increase cycle time.
3. The couples have the following arrival, departure, and total times.
Couple
1
2
3
4
Arrival time
1
3
5
6
Begin Service
1
4
7
9
Finish Service
4
7
9
11
System Time
3
4
4
5
(a) [1] 11 minutes
(b) [1] CT =
3+4+4+5
4
= 4 minutes.
(c) [1] IN V = λ ∗ CT =
4
11
∗4=
16
11
couples.
(d) [1] Waiting time = Total system time minus total processing time=16-10=6 minutes.
4. (a) [2] Place your formulation here:
max 5W + 3P
s. t. 0.05W + 0.1P
≤ 400,
0.1P
≤ 300,
0.05W
W, P
≤ 250,
≥ 0.
8.0
7.5
P ≥0
7.0
6.5
0.1P ≤ 300
6.0
5.5
5.0
0.05W ≤ 250
4.5
0.05W + 0.1P ≤ 400
4.0
5W + 3P = 29, 500
3.5
W (1000s)
3.0
2.5
2.0
1.5
1.0
0.5
0
W ≥0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
P (1000s)
This is the graph for this problem. The constraint boundaries and the feasible region are in
blue, and the optimal objective line is in red.
(b) [3] W ∗ = 5000 wedding photos.
P ∗ = 1500 passport photos.
(c) [1]Yes
Photographer and Wedding Setup Area time.
(d) [1] Expand an hour of the wedding area time since
that has the highest shadow price ($70 per hour).
5. [1] (b) - variation can lead to far less output than the maximum throughput.
6. [1](b) - the extra dryer will allow bins to empty faster but cannot change throughput.
7. [1](c) - capacity can impact cycle time whenever there is variability.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
[1](b) - variability impacts cycle time.
[1](c) - where throughput is the constant of proportionality.
[1](a) - the Gantt chart showed each step in the sub process.
[1](b) - poor Herbie was the bottleneck on the hike.
[1](b) - this counts up all the callers time in queue.
[1](a) - as above, capacity has an effect on cycle time with variability.
[1](b) - with lower cycle time, throughput can increase for the same inventory (bed capacity).
[1](a) - only the first answer is true.
[1](a) - these were Jonah’s key metrics for profitability.
[1](c) - the data showed that times with 60% or lower utilization of the CSR’s had almost zero lost
calls, but that lost calls increased quickly for higher utilization levels.