hw2 solutions
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hw2 solutionsTRANSCRIPT
University of Toronto Department of Mechanical and Industrial Engineering
MIE468: Facility Planning (Winter 2014)
Homework #2 Solutions
Problem 1 All Ik values are rounded up.
𝐼𝑋𝐶 =𝑂𝑋
(1 − 𝑑𝑋𝐶)=
100,000
0.97= 103,093
𝐼𝑋𝐵 =𝑂𝑋
(1 − 𝑑𝑋𝐶)(1 − 𝑑𝑋𝐵)=
100,000
0.97 × 0.96= 107,389
𝐼𝑋𝐴 =𝑂𝑋
(1 − 𝑑𝑋𝐶)(1− 𝑑𝑋𝐵)(1 − 𝑑𝑋𝐴)=
100,000
0.97 × 0.96 × 0.95= 113,041
𝐼𝑌𝐶 =𝑂𝑌
(1 − 𝑑𝑌𝐶)=
200,000
0.97= 206,186
𝐼𝑋𝐴 =𝑂𝑌
(1 − 𝑑𝑌𝐶)(1 − 𝑑𝑌𝐴)=
200,000
0.97 × 0.95= 217,038
𝐼𝑋𝐵 =𝑂𝑌
(1 − 𝑑𝑌𝐶)(1 − 𝑑𝑌𝐴)(1 − 𝑑𝑌𝐵)=
200,000
0.97 × 0.95 × 0.96= 226,081
Setup times are identical for machines A, B and C for a particular product. The setup time for product X on each machine is 20 mins; the setup time for product Y on each machine is 40 mins. Assuming a single setup is needed to produce the annual requirement of a product on a machine, the number of machines required is determined as follows:
𝑭𝑨 =𝟎.𝟏𝟓 × 𝟏𝟏𝟑,𝟎𝟒𝟏
(𝟎.𝟖𝟓)(𝟏𝟔𝟎𝟎)(𝟎.𝟗𝟓)+
𝟎.𝟏𝟎 × 𝟐𝟏𝟕,𝟎𝟑𝟖(𝟎.𝟖𝟓)(𝟏𝟔𝟎𝟎)(𝟎.𝟗𝟓)
+𝟐𝟎 + 𝟒𝟎
(𝟔𝟎)(𝟏𝟔𝟎𝟎)× 𝑭𝑨
𝑭𝑨 = 𝟐𝟗.𝟗𝟒 → 𝟑𝟎 𝐦𝐚𝐜𝐡𝐢𝐧𝐞𝐬
𝑭𝑩 =𝟎.𝟐𝟓 × 𝟏𝟎𝟕,𝟑𝟖𝟗
(𝟎.𝟗𝟎)(𝟏𝟔𝟎𝟎)(𝟎.𝟗𝟎)+
𝟎.𝟏𝟎 × 𝟐𝟐𝟔,𝟎𝟖𝟏(𝟎.𝟗𝟎)(𝟏𝟔𝟎𝟎)(𝟎.𝟗𝟎)
+𝟐𝟎 + 𝟒𝟎
(𝟔𝟎)(𝟏𝟔𝟎𝟎)× 𝑭𝑩
𝑭𝑩 = 𝟑𝟖.𝟏𝟖 → 𝟑𝟗 𝐦𝐚𝐜𝐡𝐢𝐧𝐞𝐬
𝑭𝑪 =𝟎.𝟏𝟎 × 𝟏𝟎𝟑,𝟎𝟗𝟑
(𝟎.𝟗𝟓)(𝟏𝟔𝟎𝟎)(𝟎.𝟖𝟓)+
𝟎.𝟏𝟓 × 𝟐𝟎𝟔,𝟏𝟖𝟔(𝟎.𝟗𝟓)(𝟏𝟔𝟎𝟎)(𝟎.𝟖𝟓)
+𝟐𝟎 + 𝟒𝟎
(𝟔𝟎)(𝟏𝟔𝟎𝟎)× 𝑭𝑪
𝑭𝑪 = 𝟑𝟏.𝟗𝟒 → 𝟑𝟐 𝐦𝐚𝐜𝐡𝐢𝐧𝐞𝐬 If setups occur more frequently, then additional machines might be required due to the lost production cost by setups.
University of Toronto Department of Mechanical and Industrial Engineering
MIE468: Facility Planning (Winter 2014)
Problem 2
I =O
(1 − d) =7500.80
= 937.5 → 938
𝑭 =𝑺𝑸𝑬𝑯𝑹
=(𝟏𝟓𝟔𝟎) × 𝟗𝟑𝟖
�𝟏𝟓𝟐𝟎� × 𝟕 × 𝟏= 𝟒𝟒.𝟔𝟕 → 𝟒𝟓 𝐦𝐚𝐜𝐡𝐢𝐧𝐞𝐬
Alternatively, you could say that the loss of one hour per shift is reduction in the reliability:
𝑭 =𝑺𝑸𝑬𝑯𝑹
=(𝟏𝟓𝟔𝟎) × 𝟗𝟑𝟖
�𝟏𝟓𝟐𝟎� × 𝟖 × (𝟕𝟖)= 𝟒𝟒.𝟔𝟕 → 𝟒𝟓 𝐦𝐚𝐜𝐡𝐢𝐧𝐞𝐬
Problem 3 1
Machine 1 2 3 4 5 6
Part
1 1 1 1 2 1 1 3 1 1 1 4 1 1 5 1 1 6 1 1 7 1 1 1 8 1 1 9 1 1 1
2. DCA a. Tabulate Sums
Machine sum 1 2 3 4 5 6
Part
1 1 1 1 3 2 1 1 2 3 1 1 1 3 4 1 1 2 5 1 1 2 6 1 1 2 7 1 1 1 3 8 1 1 2 9 1 1 1 3
Sum 4 2 4 4 4 4
University of Toronto Department of Mechanical and Industrial Engineering
MIE468: Facility Planning (Winter 2014)
b. Sort Sums
Machine sum 2 6 5 4 3 1 Pa
rt
9 1 1 1 3 7 1 1 1 3 3 1 1 1 3 1 1 1 1 3 8 1 1 2 6 1 1 2 5 1 1 2 4 1 1 2 2 1 1 2
Sum 2 4 4 4 4 4 c. Sort Columns
Machine 6 1 4 5 3 2
Part
9 1 1 1 7 1 1 1 3 1 1 1 1 1 1 1 8 1 1 6 1 1 5 1 1 4 1 1 2 1 1
d. Sort Rows
Machine 6 1 4 5 3 2
Part
9 1 1 1 3 1 1 1 6 1 1 5 1 1 4 1 1 7 1 1 1 1 1 1 1 8 1 1 2 1 1
Group machines (6, 1, 4) and machines (5, 3, 2) 2. SCA Iteration 1 Machine pair SC Combine? {1,2} 0
{1,3} 0 {1,4} 3/5
University of Toronto Department of Mechanical and Industrial Engineering
MIE468: Facility Planning (Winter 2014)
{1,5} 0 {1,6} 3/5 {2,3} 1/2 {2,4} 0 {2,5} 1/2 {2,6} 0 {3,4} 0 {3,5} 1 Yes {3,6} 0 {4,5} 0 {4,6} 3/5 {5,6} 0 => Combine (3,5)
Iteration 2 Machine pair SC Combine?
{1,2} 0 {1,(3,5)} 0 {1,4} 3/5 Yes {1,6} 3/5 Yes {2,(3,5)} 1/2 {2,4} 0 {2,6} 0 {(3,5),4} 0 {(3,5),6} 0 {4,6} 3/5 Yes => Combine (1,4,6)
Iteration 3 Machine pair SC Combine?
{(1,4,6),2} 0 {(1,4,6),(3,5)} 0 {2,(3,5)} 1/2 Yes => Combine (2,3,5)
Group machines (1,4,6) and machines (2,3,5) 3
1.0
0.8
0.5
0.33
0
2 3 5 1 4 6
Part-Machine Relationship 1 2 3 4 5 6 7
1 1 0 0 1 0 1 0
2 0 1 1 0 1 0 0
3 0 0 0 1 0 1 0
4 0 1 1 0 0 0 0
5 0 0 1 0 0 0 1
Calculating the matrix
dij 1 2 3 4 5
1 0 6 1 5 5
2 6 0 5 1 3
3 1 5 0 4 4
4 5 1 4 0 2
5 5 3 4 2 0
Distance Matrix
dij 1 2 3 4 5
1 0 6 1 5 5
2 6 0 5 1 3
3 1 5 0 4 4
4 5 1 4 0 2
5 5 3 4 2 0
Variables:
Part, i 1 2 3 4 5 each part at most selected once
1 0 0 1 0 0 1
2 0 0 0 1 0 1
3 0 0 1 0 0 1
4 0 0 0 1 0 1
5 0 0 0 1 0 1
2
Objective 4.000001
family, j
number of families