Nightingale College Mixed Integer Programing Worksheet

Mixed integer programing:Use math solver program To solve this model a reduced example has been considered because of the
large number of constraints
Jobs
Machine/jobs
1
2
3
4
5
M1
17
15
17
14
12
M2
21
22
13
19
20
M3
14
15
13
19
18
𝑚−1 𝑛
𝑛−1
𝑚𝑖𝑛 ( ∑ ∑ 𝑥𝑗1 𝑃𝑖𝑗 + ∑ 𝐼𝑚𝑗 )
𝑖=1 𝑗=1
𝑗=1
𝑀𝑖𝑛 17𝑋11 + 15𝑋21 + 17𝑋31 + 14𝑋41 + 12𝑋51 + 21𝑋11 + 22𝑋21 + 13𝑋31 + 19𝑋41 + 20𝑋51
+ 14𝑋11 + 15𝑋21 + 13𝑋31 + 19𝑋41 + 18𝑋51
= 𝑀𝑖𝑛 53𝑋11 + 54𝑋21 + 46𝑋31 + 56𝑋41 + 55𝑋51
𝑛
∑ 𝑥𝑗𝑘 = 1 , 𝑘 = 1, … ,5
𝑗=1
𝑥11 +𝑥21 +𝑥31 +𝑥41 + 𝑥51 = 1
𝑥12 +𝑥22 +𝑥32 +𝑥42 + 𝑥52 = 1
𝑥13 +𝑥23 +𝑥33 +𝑥43 + 𝑥53 = 1
𝑥14 +𝑥24 +𝑥34 +𝑥44 + 𝑥54 = 1
𝑥15 +𝑥25 +𝑥35 +𝑥45 + 𝑥55 = 1
𝑛
∑ 𝑥𝑗𝑘 = 1 , 𝑗 = 1, … , 𝑛
𝑘=1
𝑥11 +𝑥12 +𝑥13 +𝑥14 + 𝑥15 = 1
𝑥21 +𝑥22 +𝑥23 +𝑥24 + 𝑥25 = 1
𝑥31 +𝑥32 +𝑥33 +𝑥34 + 𝑥35 = 1
𝑥41 +𝑥42 +𝑥43 +𝑥44 + 𝑥55 = 1
𝑥51 +𝑥52 +𝑥53 +𝑥54 + 𝑥55 = 1
𝑛
𝑛
𝐼𝑖𝑘 + ∑ 𝑥𝑗,𝑘+1 𝑃𝑖𝑗 + 𝑊𝑖,𝑘+1 − 𝑊𝑖𝑘 − ∑ 𝑥𝑗,𝑘 𝑃𝑖+1,𝑗 − 𝐼𝑖+1,𝑘 = 0 𝑓𝑜𝑟𝑘 = 1, . . ,4; 𝑖 = 1, … ,2
𝑗=1
𝑗=1
k= 1, i=1
𝐼11 + 17 ∗ 𝑥12 + 15 ∗ 𝑥22 + 17 ∗ 𝑥32 + 14 ∗ 𝑥42 + 12 ∗ 𝑥52 + 𝑊12 − 𝑊11
− 21 ∗ 𝑥11 −22 ∗ 𝑥21 −13 ∗ 𝑥31 −19 ∗ 𝑥41 −20 ∗ 𝑥51 + −𝐼21 = 0
For k=1, i=2
𝐼21 + 21 ∗ 𝑥12 + 22 ∗ 𝑥22 + 13 ∗ 𝑥32 + 19 ∗ 𝑥42 + 20 ∗ 𝑥52 + 𝑊22 − 𝑊21
− 14 ∗ 𝑥11 −15 ∗ 𝑥21 −13 ∗ 𝑥31 −19 ∗ 𝑥41 −18 ∗ 𝑥51 − 𝐼31 = 0
For k=2 , i=1
𝐼12 + 17 ∗ 𝑥13 + 15 ∗ 𝑥23 + 17 ∗ 𝑥33 + 14 ∗ 𝑥43 + 12 ∗ 𝑥53 + 𝑊13 − 𝑊12 − 21 ∗ 𝑥12 − 22 ∗ 𝑥22 − 13
∗ 𝑥32 − 19 ∗ 𝑥42 − 20 ∗ 𝑥52 − 𝐼22 = 0
For k=2 , i=2
𝐼22 + 21 ∗ 𝑥13 + 22 ∗ 𝑥23 + 13 ∗ 𝑥33 + 19 ∗ 𝑥43 + 20 ∗ 𝑥53 + 𝑊23 − 𝑊22 − 14 ∗ 𝑥12 − 15 ∗ 𝑥22 − 13
∗ 𝑥32 − 19 ∗ 𝑥42 − 18 ∗ 𝑥52 − 𝐼32 = 0
For k=3 , i=1
𝐼13 + 17 ∗ 𝑥14 + 15 ∗ 𝑥24 + 17 ∗ 𝑥34 + 14 ∗ 𝑥44 + 12 ∗ 𝑥54 + 𝑊14 − 𝑊13 − 21 ∗ 𝑥13 − 22 ∗ 𝑥23 − 13
∗ 𝑥33 − 19 ∗ 𝑥43 − 20 ∗ 𝑥53 − 𝐼23 = 0
For k=3 , i=2
𝐼23 + 21 ∗ 𝑥14 + 22 ∗ 𝑥24 + 13 ∗ 𝑥34 + 19 ∗ 𝑥44 + 20 ∗ 𝑥54 + 𝑊24 − 𝑊23 − 14 ∗ 𝑥13 − 15 ∗ 𝑥23 − 13
∗ 𝑥33 − 19 ∗ 𝑥43 − 18 ∗ 𝑥53 − 𝐼33 = 0
For k=4 , i=1
𝐼14 + 17 ∗ 𝑥15 + 15 ∗ 𝑥25 + 17 ∗ 𝑥35 + 14 ∗ 𝑥45 + 12 ∗ 𝑥55 + 𝑊15 − 𝑊14 − 21 ∗ 𝑥14 − 22 ∗ 𝑥24 − 13
∗ 𝑥34 − 19 ∗ 𝑥44 − 20 ∗ 𝑥54 − 𝐼24 = 0
For k=4 , i=2
𝐼24 + 1 ∗ 𝑥15 + 22 ∗ 𝑥25 + 13 ∗ 𝑥33 + 19 ∗ 𝑥43 + 20 ∗ 𝑥56 + 𝑊24 − 𝑊24 − 14 ∗ 𝑥14 − 15 ∗ 𝑥24 − 13
∗ 𝑥34 − 19 ∗ 𝑥44 − 18 ∗ 𝑥54 − 𝐼34 = 0
𝑊11 = 0 , 𝑖 = 1, … , 𝑚 − 1
𝑊21 = 0
𝐼1𝑘 = 0 ; 𝑘 = 1, … , 𝑛 − 1
𝐼11 = 0
𝐼12 = 0
𝐼13 = 0
𝐼14 = 0