Professor
Dr.-Ing. Bernd Hellingrath
Chair for Information Systems and
Supply Chain Management
Leonardo-Campus 3
48149 Münster
Tel. +49 251 83-38000
Fax +49 251 83-38009
Tutorial Operations Management
Chapter 4 – Process Design
Solutions
Exercise 1: Process flow analysis of the robot arm production
To calculate the number of necessary cuts, we begin best with the last production step. Pro-
duction step 4 “Packing” can be skipped, since it does not yield waste. Production step 3
“Painting” creates 0.5% waste and 99.5% good parts. To have 1200 faultless robot arms to
deliver from “Painting” to “Packing”, ⌈1200/0.995⌉ = 1207 robot arms (rounded up) must
be painted. Production step 2 “Welding” creates 99% good parts. To have 1207 faultless robot
arms, from production step 1 “Cutting” are ⌈1207/0.99⌉ = 1220 robot arms (rounded up)
needed. Since every cut cuts 4 metal sheets at once, ⌈1220/4⌉ = 305 cuts are needed. Under
consideration of a scrap rate of 2.5%, resp. 97.5% good parts, at this productions step
305/0.975 ≈ 312.82 cuts are needed. Rounded to integer cuts 313 times.
, 2
a) Exercise 2: M/M/1-Queues – Packaging Process
8
µ= ∙ 60 = 32[Pcs./Min. ]
15
𝜆𝜆 24 [Pcs./Min.] 3
𝜌𝜌 = = = = 0.75
𝜇𝜇 32 [Pcs./Min.] 4
𝜌𝜌2 36 1
𝐿𝐿𝐿𝐿 = = 16 = 2 4 = 2.25 [Pcs. ] 1
(1−𝜌𝜌)
b)
𝜌𝜌2
𝐿𝐿𝐿𝐿 =
(1 − 𝜌𝜌)
insert LQ = 30 and solve for 𝜌𝜌:
𝜌𝜌2 + 30𝜌𝜌 − 30 = 0
𝑝𝑝 𝑝𝑝 2
Solve using the pq-formula: 𝑥𝑥 2 + 𝑝𝑝𝑝𝑝 + 𝑞𝑞 = 0 => 𝑥𝑥1.2 = − 2 ± ��2� − 𝑞𝑞
𝜌𝜌 = −15 + √ 255 ≈ 0.97 2
and 𝜆𝜆 = 32 [Pcs./Min. ] ∙ 0.97 = 31.04 [Pcs./Min.]
c)
𝜆𝜆 31.04 [Pcs./Min. ]
𝜌𝜌 = = = 0.65
𝜇𝜇 48 [Pcs./Min. ]
𝜌𝜌2 0.652
𝐿𝐿𝐿𝐿 = (1−𝑝𝑝) = ≈ 1.21 [Pcs.]
0.35
1 The measures LQ and LS generally do not have units. The units can be deducted usually based on the con-
text. The units are given here only for comprehension.
2 Under consideration of the pq-formula and the condition that for M/M/1-queues ρ <1 must hold.
Dr.-Ing. Bernd Hellingrath
Chair for Information Systems and
Supply Chain Management
Leonardo-Campus 3
48149 Münster
Tel. +49 251 83-38000
Fax +49 251 83-38009
Tutorial Operations Management
Chapter 4 – Process Design
Solutions
Exercise 1: Process flow analysis of the robot arm production
To calculate the number of necessary cuts, we begin best with the last production step. Pro-
duction step 4 “Packing” can be skipped, since it does not yield waste. Production step 3
“Painting” creates 0.5% waste and 99.5% good parts. To have 1200 faultless robot arms to
deliver from “Painting” to “Packing”, ⌈1200/0.995⌉ = 1207 robot arms (rounded up) must
be painted. Production step 2 “Welding” creates 99% good parts. To have 1207 faultless robot
arms, from production step 1 “Cutting” are ⌈1207/0.99⌉ = 1220 robot arms (rounded up)
needed. Since every cut cuts 4 metal sheets at once, ⌈1220/4⌉ = 305 cuts are needed. Under
consideration of a scrap rate of 2.5%, resp. 97.5% good parts, at this productions step
305/0.975 ≈ 312.82 cuts are needed. Rounded to integer cuts 313 times.
, 2
a) Exercise 2: M/M/1-Queues – Packaging Process
8
µ= ∙ 60 = 32[Pcs./Min. ]
15
𝜆𝜆 24 [Pcs./Min.] 3
𝜌𝜌 = = = = 0.75
𝜇𝜇 32 [Pcs./Min.] 4
𝜌𝜌2 36 1
𝐿𝐿𝐿𝐿 = = 16 = 2 4 = 2.25 [Pcs. ] 1
(1−𝜌𝜌)
b)
𝜌𝜌2
𝐿𝐿𝐿𝐿 =
(1 − 𝜌𝜌)
insert LQ = 30 and solve for 𝜌𝜌:
𝜌𝜌2 + 30𝜌𝜌 − 30 = 0
𝑝𝑝 𝑝𝑝 2
Solve using the pq-formula: 𝑥𝑥 2 + 𝑝𝑝𝑝𝑝 + 𝑞𝑞 = 0 => 𝑥𝑥1.2 = − 2 ± ��2� − 𝑞𝑞
𝜌𝜌 = −15 + √ 255 ≈ 0.97 2
and 𝜆𝜆 = 32 [Pcs./Min. ] ∙ 0.97 = 31.04 [Pcs./Min.]
c)
𝜆𝜆 31.04 [Pcs./Min. ]
𝜌𝜌 = = = 0.65
𝜇𝜇 48 [Pcs./Min. ]
𝜌𝜌2 0.652
𝐿𝐿𝐿𝐿 = (1−𝑝𝑝) = ≈ 1.21 [Pcs.]
0.35
1 The measures LQ and LS generally do not have units. The units can be deducted usually based on the con-
text. The units are given here only for comprehension.
2 Under consideration of the pq-formula and the condition that for M/M/1-queues ρ <1 must hold.
