                               SIPS 
                            version 1.0
                     
                            Qiang  Li             
                      University of Tsukuba     
                            Yike Guo               
                Dept. of Computing, Imperial College  
                                           
                      Co-operated under AITEC      
                       research projet 1997      


Following are some examples:

Ex1 (job n=1, machine m=2)

RIP:
Min :  5 x 1 + 3 x 2 + 12 x 3 + 16 x 4 .

           x 3 + x 4 = 3 ,
           x 3 - 3 x 1 + x 5 = 0 ,
           x 4 - 3 x 2 + x 6 = 0 ,
           x 1 + x 7 = 1 ,
           x 2 + x 8 = 1 .
           
Complicated constraints:

       4 x 3 + x 1 =< 10 ,
       4 x 4 + x 2 =< 10 .
       
Probabilistic constraints:

       d 1 x 3 + x 1 =< 10 ,
       d 1 x 4 + x 2 =< 10 .
       
probability prob=0.83, 0.9 .
lot splitting M=3
the number of samples sn=100

100 samples (d1):

12 2 8  9  1  4  1  6  21  1  0  20 2  12 30 3  9  10  4  8
10 3 5  10 4  84 9  5  24  12 5  2  4  58 9  9  91  93 9  3
2  1 50 10 11 8  3  2  1  11  10 3  12 10 9  12 0  10  0  0
6  7 6  0  3  12 3  0  10 11  9  7  0  3  8  7  1  1   8  12
3  9 0  77 1  6  42 18 2  77  9  20 50 3  15 2  17 3   6  4 .

Result:

Probability = 0.83 Cost=48
Optimal solution: x1=1, x2=1, x3=2, x4=1, x5=1, x6=2, x7=0, x8=0, 

Probability = 0.9
there is no optimal solution.

------------------------------------------------------------

Ex2 (job n=2,machine m=3)

RIP:
Min: 3 x 1 + 4 x 2 + 3 x 3 + 4 x 4 + 4 x 5 + 3 x 6 + 
     50 x 7 + 10 x 8 + 40 x 9 + 24 x 10 + 8 x 11 + 16 x 12 .

x 7 + x 8 + x 9 = 2 ,
x 10 + x 11 + x 12 = 2 ,
x 7 - 2 x 1 + x 13 = 0 ,
x 8 - 2 x 2 + x 14 = 0 ,
x 9 - 2 x 3 + x 15 = 0 ,
x 10 - 2 x 4 + x 16 = 0 ,
x 11 - 2 x 5 + x 17 = 0 ,
x 12 - 2 x 6 + x 18 = 0 ,
x 1 + x 19 = 1 ,
x 2 + x 20 = 1 ,
x 3 + x 21 = 1 ,
x 4 + x 22 = 1 ,
x 5 + x 23 = 1 ,
x 6 + x 24 = 1 .

Complicated constraints:

10 x 7 + 8 x 10 + x 1 + x 4 =< 25 ,
10 x 8 + 8 x 11 + x 2 + x 5 =< 38 ,
10 x 9 + 8 x 12 + x 3 + x 6 =< 25 .

Probabilistic constraints:
d 1 x 7 + d 2 x 10 + x 1 + x 4 =< 25 ,
d 1 x 8 + d 2 x 11 + x 2 + x 5 =< 38 ,
d 1 x 9 + d 2 x 12 + x 3 + x 6 =< 25 .


probability prob=0.84, 0.86, 0.88
lot splitting M=2
the number of samples sn=100

100 samples (d1,d2):
12 2 8  9  1  4  1  6  3  1  0  4  2  12 3  3  9  10  4  8
10 3 5  10 4  6  9  5  2  12 5  2  4  3  9  9  2  6   9  3
2  1 9  10 11 8  42 18 1  11 10 3  12 10 9  12 50 58  0  0
6  7 6  0  3  12 3  0  10 11 9  7  0  3  8  7  1  1   8  12
3  9 93 77 1  84 3  2  24 77 9  10 0  10 15 91 17 21  93 77 
12 2 8  9  1  4  15 91 3  1  0  4  2  12 3  3  9  10  4  8
10 3 5  10 4  6  9  5  2  12 5  2  4  3  9  9  2  6   17 21
2  1 50 20 11 8  3  2  1  11 10 3  12 10 9  12 0  10  0  0
6  7 6  0  3  12 3  0  10 11 9  7  0  3  8  7  1  1   8  12
3  9 0  4  1  84 42 18 24 77 50 20 50 58 1  6  9  3   0  4 .

Result:

Probability = 0.84 Cost=44
Optimal solution: x2=1, x5=1, x8=2, x11=2, x19=1, x21=1, x22=1, x24=1, 
                  others = 0

Probability = 0.86 Cost=55                  
Optimal solution:  x2=1, x5=1, x6=1, x8=2, x11=1, x12=1, x17=1, x18=1,
                   x19=1, x21=1, x22=1,
                   others = 0
                   
Probability = 0.88 Cost=97
Optimal solution: x2=1, x3=1, x4=1, x5=1, x8=1, x9=1, x10=1, x11=1,
                  x14=1, x15=1, x16=1, x17=1, x19=1, x24=1, 
                  others = 0
 
 
------------------------------------------------------------
Ex3 (job n=4,machine m=3)

RIP:
Min

3 x 1 + 4 x 2 + 3 x 3 + 4 x 4 + 4 x 5 + 3 x 6 + 3 x 7 + 2 x 8 + 2 x 9
+ 2 x 10 + 2 x 11 + 2 x 12 + 50 x 13 + 10 x 14 + 40 x 15 + 
24 x 16 + 8 x 17 + 16 x 18 + 18 x 19 + 24 x 20 + 30 x 21 +
4 x 22 + 20 x 23 + 16 x 24 .


x 13 + x 14 + x 15 = 2 ,
x 16 + x 17 + x 18 = 2 ,
x 19 + x 20 + x 21 = 2 ,
x 22 + x 23 + x 24 = 2 ,

x 13 - 2 x 1 + x 25 = 0 ,
x 14 - 2 x 2 + x 26 = 0 ,
x 15 - 2 x 3 + x 27 = 0 ,
x 16 - 2 x 4 + x 28 = 0 ,
x 17 - 2 x 5 + x 29 = 0 ,
x 18 - 2 x 6 + x 30 = 0 ,
x 19 - 2 x 7 + x 31 = 0 ,
x 20 - 2 x 8 + x 32 = 0 ,
x 21 - 2 x 9 + x 33 = 0 ,
x 22 - 2 x 10 + x 34 = 0 ,
x 23 - 2 x 11 + x 35 = 0 ,
x 24 - 2 x 12 + x 36 = 0 ,

x 1 + x 37 = 1 ,
x 2 + x 38 = 1 ,
x 3 + x 39 = 1 ,
x 4 + x 40 = 1 ,
x 5 + x 41 = 1 ,
x 6 + x 42 = 1 ,
x 7 + x 43 = 1 ,
x 8 + x 44 = 1 ,
x 9 + x 45 = 1 ,
x 10 + x 46 = 1 ,
x 11 + x 47 = 1 ,
x 12 + x 48 = 1 .

Complicated constraints:

10 x 13 + 8 x 16 + 6 x 19 + 4 x 22 + x 1 + x 4 + x 7 + x 10 =< 25 ,
10 x 14 + 8 x 17 + 6 x 20 + 4 x 23 + x 2 + x 5 + x 8 + x 11 =< 38 ,
10 x 15 + 8 x 18 + 6 x 21 + 4 x 24 + x 3 + x 6 + x 9 + x 12 =< 20 .

Probabilistic constraints:
d 1 x 13 + d 2 x 16 + d 3 x 19 + d 4 x 22 + x 1 + x 4 + x 7 + x 10 =< 25 ,
d 1 x 14 + d 2 x 17 + d 3 x 20 + d 4 x 23 + x 2 + x 5 + x 8 + x 11 =< 38 ,
d 1 x 15 + d 2 x 18 + d 3 x 21 + d 4 x 24 + x 3 + x 6 + x 9 + x 12 =< 20 .

probability prob=0.5, 0.62
lot splitting M=2
the number of samples sn=100

100 samples (d1,d2,d3,d4):
12 2  8  9  17 21 93 77 1  4  1  6  3  1  0  4  50 58 15 91
26 21 14 8  2  12 3  3  24 77 50 20 9  10 4  8  25 22 16 6
10 3  5  10 84 42 18 24 24 19 16 6  4  6  9  5  2  12 5  2
20 50 42 18 4  3  9  9  21 93 77 50 25 22 16 6  2  6  9  3
58 15 91 17 2  1  9  10 11 8  3  2  77 50 20 50 1  11 10 3
12 10 9  12 21 93 77 84 23 21 16 6  0  10 0  0  6  7  6  0
84 42 18 24 24 20 16 6  3  12 3  0  10 11 9  7  42 18 24 77
24 19 16 6  0  3  8  7  58 15 91 17 1  1  8  12 26 21 14 8
3  9  0  4  77 50 20 50 1  4  12 2  25 22 16 6  8  9  1  4
23 21 16 6  1  6  3  1  84 42 18 24 0  4  2  12 12 2  8  9
1  4  1  6  20 50 58 21 3  1  0  4  24 20 16 6  2  12 3  3
18 24 77 50 93 77 84 42 9  10 4  8  15 91 17 21 10 3  5  10
24 19 16 6  4  6  9  5  2  12 5  2  50 58 15 91 4  3  9  9
2  6  9  3  50 20 50 58 2  1  9  10 21 93 77 20 11 8  3  2
26 21 14 8  1  11 10 3  12 10 9  12 58 15 91 17 0  10 0  0
77 50 20 50 6  7  6  0  84 42 18 24 3  12 3  0  77 50 20 50
10 11 9  7  84 42 18 24 0  3  8  7  50 58 15 91 1  1  8  12
17 21 93 77 3  9  0  4  24 77 50 20 58 15 91 17 1  4  12 2
84 42 18 24 8  9  1  4  20 50 42 18 1  6  3  1  21 93 77 50
0  4  2  12 77 50 20 50 24 20 16 6  23 21 16 6  77 50 20 50 .


results:
probability = 0.5  cost=93

optimal solution: x2=1,x5=1,x7=1,x10=1,x14=2,x17=2,x19=2,x22=2,
                   x37=1,x39=1,x40=1,x42=1,x44=1,x45=1,x47=1,x48=1
                   others = 0

probability = 0.62 cost=104
optimal solution: x2=1, x5=1, x6=1, x7=1, x10=1, x14=2, x17=1, x18=1,
                  x19=2, x22=2, x29=1, x30=1, x37=1, x39=1, x40=1,
                  x44=1, x45=1, x47=1, x48=1,
                  others = 0

------------------------------------------------------------
Ex4 (job n=6, machine m=3)

RIP:
Min

3 x 1 + 4 x 2 + 3 x 3 + 4 x 4 + 4 x 5 + 3 x 6 + 3 x 7
+ 2 x 8 + 2 x 9 + 2 x 10 + 2 x 11 + 2 x 12 + 3 x 13 +
4 x 14 + 4 x 15 + 3 x 16 + 3 x 17 + 2 x 18 + 
50 x 19 + 10 x 20 + 40 x 21 + 24 x 22 + 8 x 23 + 16 x 24 +
18 x 25 + 24 x 26 + 30 x 27 + 4 x 28 + 20 x 29 + 16 x 30 +
4 x 31 + 12 x 32 + 16 x 33 + 6 x 34 + 2 x 35 + 4 x 36 .

x 19 + x 20 + x 21 = 2 ,
x 22 + x 23 + x 24 = 2 ,
x 25 + x 26 + x 27 = 2 ,
x 28 + x 29 + x 30 = 2 ,
x 31 + x 32 + x 33 = 2 ,
x 34 + x 35 + x 36 = 2 ,

x 19 - 2 x 1 + x 37 = 0 ,
x 20 - 2 x 2 + x 38 = 0 ,
x 21 - 2 x 3 + x 39 = 0 ,
x 22 - 2 x 4 + x 40 = 0 ,
x 23 - 2 x 5 + x 41 = 0 ,
x 24 - 2 x 6 + x 42 = 0 ,
x 25 - 2 x 7 + x 43 = 0 ,
x 26 - 2 x 8 + x 44 = 0 ,
x 27 - 2 x 9 + x 45 = 0 ,
x 28 - 2 x 10 + x 46 = 0 ,
x 29 - 2 x 11 + x 47 = 0 ,
x 30 - 2 x 12 + x 48 = 0 ,
x 31 - 2 x 13 + x 49 = 0 ,
x 32 - 2 x 14 + x 50 = 0 ,
x 33 - 2 x 15 + x 51 = 0 ,
x 34 - 2 x 16 + x 52 = 0 ,
x 35 - 2 x 17 + x 53 = 0 ,
x 36 - 2 x 18 + x 54 = 0 ,


x 1 + x 55 = 1 ,
x 2 + x 56 = 1 ,
x 3 + x 57 = 1 ,
x 4 + x 58 = 1 ,
x 5 + x 59 = 1 ,
x 6 + x 60 = 1 ,
x 7 + x 61 = 1 ,
x 8 + x 62 = 1 ,
x 9 + x 63 = 1 ,
x 10 + x 64 = 1 ,
x 11 + x 65 = 1 ,
x 12 + x 66 = 1 ,
x 13 + x 67 = 1 ,
x 14 + x 68 = 1 ,
x 15 + x 69 = 1 ,
x 16 + x 70 = 1 ,
x 17 + x 71 = 1 ,
x 18 + x 72 = 1 .

Complicated constraints:
10 x 19 + 8 x 22 + 6 x 25 + 4 x 28 + 4 x 31 + 2 x 34 + x 1 + x 4 + x 7 + x 10 + x 13 + x 16 =< 35 ,
10 x 20 + 8 x 23 + 6 x 26 + 4 x 29 + 4 x 32 + 2 x 35 + x 2 + x 5 + x 8 + x 11 + x 14 + x 17 =< 45 ,
10 x 21 + 8 x 24 + 6 x 27 + 4 x 30 + 4 x 33 + 2 x 36 + x 3 + x 6 + x 9 + x 12 + x 15 + x 18 =< 30 .

Probabilistic constraints:
d 1 x 19 + d 2 x 22 + d 3 x 25 + d 4 x 28 + d 5 x 31 + d 6 x 34 + x 1 + x 4 + x 7 + x 10 + x 13 + x 16 =< 35 ,
d 1 x 20 + d 2 x 23 + d 3 x 26 + d 4 x 29 + d 5 x 32 + d 6 x 35 + x 2 + x 5 + x 8 + x 11 + x 14 + x 17 =< 45 ,
d 1 x 21 + d 2 x 24 + d 3 x 27 + d 4 x 30 + d 5 x 32 + d 6 x 36 + x 3 + x 6 + x 9 + x 12 + x 15 + x 18 =< 30 .



probability prob=0.5,0.56
lot splitting M=2
the number of samples sn=100

100 samples (d1,d2,d3,d4,d5,d6):

12 2  8  9  1  4  58 15 91 17 21 93 1  6  3  1  0  4  18 24
77 50 20 50 2  12 3  3  9  10 20 50 58 21 84 42 4  8  10 3
5  10 84 42 18 24 77 50 4  6  9  5  2  12 15 91 17 21 93 77
5  2  4  3  9  9  20 50 50 20 50 58 2  6  9  3  2  1  84 42
18 24 77 50 9  10 11 8  3  2  15 91 17 21 93 77 1  11 10 3
12 10 24 77 50 20 50 58 9  12 0  10 0  0  77 50 20 50 42 18
6  7  6  0  3  12 58 15 91 17 21 93 3  0  10 11 9  7  18 24
77 50 20 50 0  3  8  7  1  1  42 18 24 77 84 42 8  12 3  9
0  4  91 17 21 93 77 84 1  4  12 2  8  9  77 50 20 50 58 15
1  4  1  6  3  1  58 21 84 42 18 24 0  4  2  12 12 2  18 24
77 50 20 50 8  9  1  4  1  6  17 21 93 77 84 42 3  1  0  4
2  12 50 20 50 58 15 91 3  3  9  10 4  8  77 20 50 58 15 91
10 3  5  10 4  6  58 15 91 17 21 93 9  5  2  12 5  2  18 24
77 50 20 50 4  3  9  9  2  6  77 50 20 50 84 42 9  3  2  1
9  10 93 77 84 42 18 24 11 8  3  2  1  11 10 3  12 10 9  12
50 58 15 91 17 21 0  10 0  0  6  7  20 50 24 77 50 20 6  0
3  12 3  0  84 42 18 24 77 50 10 11 9  7  0  3  15 91 17 21
93 77 8  7  1  1  8  12 24 77 50 20 50 58 3  9  0  4  1  4
77 50 20 50 42 18 12 2  8  9  1  4  58 15 91 17 21 93 1  6
3  1  0  4  18 24 77 50 20 50 2  12 2  1  9  10 42 18 24 77
84 42 11 8  3  2  1  11 77 50 20 50 58 15 10 3  12 10 9  12
58 21 84 42 18 24 0  10 0  0  6  7  18 24 77 50 20 50 6  0
3  12 3  0  17 21 93 77 84 42 10 11 9  7  0  3  50 20 50 58
15 91 8  7  1  1  8  12 21 93 77 50 20 50 3  9  0  4  1  4
20 50 58 15 91 17 12 2  8  9  1  4  1  6  3  1  0  4  84 42
18 24 77 50 2  12 12 2  8  9  24 20 16 6  8  6  1  4  1  6
3  1  24 20 16 6  10 8  0  4  2  12 3  3  25 22 16 6  10 6
9  10 4  8  10 3  24 20 16 6  8  6  5  10 4  6  9  5  26 20
17 6  10 7  2  12 5  2  4  3  25 22 16 6  10 6  9  9  2  6
9  3  24 21 16 6  8  6  77 84 42 18 24 77 58 15 91 17 21 93 .


Result:
Probability = 0.5 Cost=111
Optimal solution:
x2=1,x5=1,x7=1,x10=1,x13=1,x17=1,x20=2,x23=2,
x25=2,x28=2,x31=2,x35=2,x55=1,x57=1,x58=1,x60=1,
x62=1,x63=1,x65=1,x66=1,x68=1,x69=1,x70=1,x72=1,
others = 0.

Probability = 0.56 Cost=122
Optimal solution:
x2=1, x5=1, x6=1, x7=1, x10=1, x13=1, x17=1, 20=2, x23=1, x24=1,
x25=2, x28=2, x31=2, x35=2, x41=1, x42=1, x55=1, x57=1, x58=1,
x62=1, x63=1, x65=1, x66=1, x68=1, x69=1, x70=1, x72=1, 
others = 0. 

------------------------------------------------------------

Ex5 (job n=7, machine m=4)

RIP:
Min

3 x 1 + 4 x 2 + 3 x 3 + 3 x 4 + 4 x 5 + 4 x 6 + 3 x 7 + 4 x 8 + 3 x 9
+ 2 x 10 + 2 x 11 + 2 x 12 + 2 x 13 + 2 x 14 + 2 x 15 + 3 x 16 + 3 x
17 + 4 x 18 + 4 x 19 + 3 x 20 + 3 x 21 + 3 x 22 + 2 x 23 + 4 x 24 + 4
x 25 + 2 x 26 + 4 x 27 + 2 x 28 + 50 x 29 + 10 x 30 + 40 x 31 + 50 x
32 + 24 x 33 + 8 x 34 + 16 x 35 + 32 x 36 + 18 x 37 + 24 x 38 + 30 x
39 + 24 x 40 + 4 x 41 + 20 x 42 + 16 x 43 + 16 x 44 + 4 x 45 + 12 x 46 
+ 16 x 47 + 4 x 48 + 6 x 49 + 2 x 50 + 4 x 51 + 10 x 52 + 10 x 53 +
4 x 54 + 8 x 55 + 4 x 56 .

x 29 + x 30 + x 31 + x 32 = 2 ,
x 33 + x 34 + x 35 + x 36 = 2 ,
x 37 + x 38 + x 39 + x 40 = 2 ,
x 41 + x 42 + x 43 + x 44 = 2 ,
x 45 + x 46 + x 47 + x 48 = 2 ,
x 49 + x 50 + x 51 + x 52 = 2 ,
x 53 + x 54 + x 55 + x 56 = 2 ,

x 29 - 2 x 1 + x 57 = 0 ,
x 30 - 2 x 2 + x 58 = 0 ,
x 31 - 2 x 3 + x 59 = 0 ,
x 32 - 2 x 4 + x 60 = 0 ,
x 33 - 2 x 5 + x 61 = 0 ,
x 34 - 2 x 6 + x 62 = 0 ,
x 35 - 2 x 7 + x 63 = 0 ,
x 36 - 2 x 8 + x 64 = 0 ,
x 37 - 2 x 9 + x 65 = 0 ,
x 38 - 2 x 10 + x 66 = 0 ,
x 39 - 2 x 11 + x 67 = 0 ,
x 40 - 2 x 12 + x 68 = 0 ,
x 41 - 2 x 13 + x 69 = 0 ,
x 42 - 2 x 14 + x 70 = 0 ,
x 43 - 2 x 15 + x 71 = 0 ,
x 44 - 2 x 16 + x 72 = 0 ,
x 45 - 2 x 17 + x 73 = 0 ,
x 46 - 2 x 18 + x 74 = 0 ,
x 47 - 2 x 19 + x 75 = 0 ,
x 48 - 2 x 20 + x 76 = 0 ,
x 49 - 2 x 21 + x 77 = 0 ,
x 50 - 2 x 22 + x 78 = 0 ,
x 51 - 2 x 23 + x 79 = 0 ,
x 52 - 2 x 24 + x 80 = 0 ,
x 53 - 2 x 25 + x 81 = 0 ,
x 54 - 2 x 26 + x 82 = 0 ,
x 55 - 2 x 27 + x 83 = 0 ,
x 56 - 2 x 28 + x 84 = 0 ,

x 1 + x 85 = 1 ,
x 2 + x 86 = 1 ,
x 3 + x 87 = 1 ,
x 4 + x 88 = 1 ,
x 5 + x 89 = 1 ,
x 6 + x 90 = 1 ,
x 7 + x 91 = 1 ,
x 8 + x 92 = 1 ,
x 9 + x 93 = 1 ,
x 10 + x 94 = 1 ,
x 11 + x 95 = 1 ,
x 12 + x 96 = 1 ,
x 13 + x 97 = 1 ,
x 14 + x 98 = 1 ,
x 15 + x 99 = 1 ,
x 16 + x 100 = 1 ,
x 17 + x 101 = 1 ,
x 18 + x 102 = 1 ,
x 19 + x 103 = 1 ,
x 20 + x 104 = 1 ,
x 21 + x 105 = 1 ,
x 22 + x 106 = 1 ,
x 23 + x 107 = 1 ,
x 24 + x 108 = 1 ,
x 25 + x 109 = 1 ,
x 26 + x 110 = 1 ,
x 27 + x 111 = 1 ,
x 28 + x 112 = 1 .

Complicated constriants:
10 x 29 + 8 x 33 + 6 x 37 + 4 x 41 + 4 x 45 + 2 x 49 + 2 x 53 +
x 1 + x 5 + x 9 + x 13 + x 17 + x 21 + x 25 =< 25 ,
10 x 30 + 8 x 34 + 6 x 38 + 4 x 42 + 4 x 46 + 2 x 50 + 2 x 54 +
x 2 + x 6 + x 10 + x 14 + x 18 + x 22 + x 26 =< 45 ,
10 x 31 + 8 x 35 + 6 x 39 + 4 x 43 + 4 x 47 + 2 x 51 + 2 x 55 +
x 3 + x 7 + x 11 + x 15 + x 19 + x 23 + x 27 =< 20 ,
10 x 32 + 8 x 36 + 6 x 40 + 4 x 44 + 4 x 48 + 2 x 52 + 2 x 56 +
x 4 + x 8 + x 12 + x 16 + x 20 + x 24 + x 28 =< 25 .

Probabilistic constraints:
d 1 x 29 + d 2 x 33 + d 3 x 37 + d 4 x 41 + d 5 x 45 + d 6 x 49 + d 7 x 53 +
x 1 + x 5 + x 9 + x 13 + x 17 + x 21 + x 25 =< 25 ,
d 1 x 30 + d 2 x 34 + d 3 x 38 + d 4 x 42 + d 5 x 46 + d 6 x 50 + d 7 x 54 +
x 2 + x 6 + x 10 + x 14 + x 18 + x 22 + x 26 =< 45 ,
d 1 x 31 + d 2 x 35 + d 3 x 39 + d 4 x 43 + d 5 x 47 + d 6 x 51 + d 7 x 55 +
x 3 + x 7 + x 11 + x 15 + x 19 + x 23 + x 27 =< 20 ,
d 1 x 32 + d 2 x 36 + d 3 x 40 + d 4 x 44 + d 5 x 48 + d 6 x 52 + d 7 x 56 +
x 4 + x 8 + x 12 + x 16 + x 20 + x 24 + x 28 =< 25 .

probability prob=0.42, 0.47
lot splitting M=2
the number of samples sn=100

100 samples (d1,d2,d3,d4,d5,d6,d7):

12 2  8  9  1  4  1  42 18 24 77 50 20 50 6  3  1  0  4  2
12 58 21 84 42 18 24 77 3  3  9  10 4  8  10 50 20 50 58 15
91 17 3  5  10 4  6  9  5  21 93 77 84 42 18 24 2  12 5  2
4  3  9  77 84 42 18 24 77 50 9  2  6  9  3  2  1  20 50 58
15 91 17 21 9  10 11 8  3  2  1  93 77 50 20 50 42 18 11 10
3  12 10 9  12 24 77 50 20 50 58 15 0  10 0  0  6  7  6  91
17 21 93 77 84 42 0  3  12 3  0  10 11 18 24 77 50 20 50 24
9  7  0  3  8  7  1  77 50 20 50 58 15 91 1  8  12 3  9  0
4  17 21 93 77 84 42 18 1  4  12 2  8  9  1  24 77 50 20 50
84 42 4  1  6  3  1  0  4  18 24 77 50 20 50 58 2  12 12 2
8  9  1  15 91 17 21 93 77 20 4  1  6  3  1  0  4  50 58 15
91 50 20 50 2  12 3  3  9  10 4  58 15 91 17 21 93 77 8  10
3  5  10 4  6  84 42 18 24 77 50 20 9  5  2  12 5  2  4  50
58 21 84 42 18 24 3  9  9  2  6  9  3  77 50 20 50 58 15 91
2  1  9  10 11 8  3  17 21 93 77 84 42 18 2  1  11 10 3  12
10 24 77 84 42 18 24 77 9  12 0  10 0  0  6  50 20 50 58 15
91 17 7  6  0  3  12 3  0  21 93 77 50 20 50 42 10 11 9  7
0  3  8  18 24 77 50 20 50 58 7  1  1  8  12 3  9  15 91 17
21 93 77 84 0  4  1  4  12 2  8  42 18 24 77 50 20 50 9  1
4  1  6  3  1  50 20 50 58 15 91 17 0  4  2  12 2  1  9  21
93 77 84 42 18 24 10 11 8  3  2  1  11 77 50 20 50 58 21 84
10 3  12 10 9  12 0  42 18 24 77 50 20 50 10 0  0  6  7  6
0  58 15 91 17 21 93 77 3  12 3  0  10 11 9  84 42 18 24 77
84 42 7  0  3  8  7  1  1  18 24 77 50 20 50 58 8  12 3  9
0  4  1  15 91 17 21 93 77 20 4  12 2  8  9  1  4  50 58 15
91 50 20 50 1  6  3  1  0  4  2  58 15 91 17 21 93 77 12 12
2  8  9  1  4  84 42 18 24 77 50 20 1  6  3  1  0  4  2  50
58 21 84 42 18 24 12 3  3  9  10 4  8  77 50 20 50 58 15 91
10 3  5  10 4  6  9  17 21 93 77 84 42 18 5  2  12 5  2  4
3  24 77 84 42 18 24 77 9  9  2  6  9  3  24 50 20 50 58 15
91 17 21 16 6  8  6  8  25 21 93 77 50 20 50 42 9  9  2  6
9  3  24 18 24 77 50 20 50 58 22 16 6  10 6  5  26 20 17 6
10 7  8  24 9  9  2  6  9  3  24 15 91 17 21 93 77 84 22 16
6  10 6  5  24 9  9  2  6  9  3  24 21 16 7  10 4  9  84 42
18 24 50 20 50 58 9  9  2  6  9  3  24 42 18 24 77 50 20 50 . 


Result:

Probability = 0.42  Cost=121

Optimal solution:
x2=1,x6=1,x9=1,x13=1,x17=1,x24=1,x28=1,x30=2,x34=2,x37=2,
x41=2,x45=2,x52=2,x56=2,x85=1,x87=1,x88=1,x89=1,x91=1,x92=1,
x94=1,x95=1,x96=1,x98=1,x99=1,x100=1,x102=1,x103=1,x104=1,
x105=1,x106=1,x107=1,x109=1,x110=1,x111=1,
others = 0.

Probability = 0.47 Cost=124
Optimal solution:
x2=1,x6=1,x9=1,x13=1,x20=1,x23=1,x26=1,x30=2,x34=2,x37=2,x41=2,x48=2,
x51=2,x54=2,x85=1,x87=1,x88=1,x89=1,x91=1,x92=1,x94=1,x95=1,x96=1,x98=1,
x99=1,x100=1,x101=1,x102=1,x103=1,x105=1,x106=1,x108=1,x109=1,x111=1,x112=1,
others = 0.
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