1. Group members: Ibrahim jan Qesar Habib Najeebullah
Integer programming
Group members:
Ibrahim jan
Qesar Habib
Najeebullah

2. Linear programming
Linear programming (LP, or linear optimization) is a mathematical method.
for determining a way to achieve the best outcome (such as maximum profit or lowest cost).
mathematical model for some list of requirements represented as linear relationships.
Linear programming is a specific case of mathematical programming (mathematical optimization).

3. More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

4. Integer linear programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.
Integer programming problem are a special class of linear programming where all are some of the variable in the optimal solution are restricted to non-negative integer value.

5. Linear programming problem are based on assumption that the decision variable are continuous with the result that they may take frictional value in the optimal solution e.g. 143/5 unit of product.
However many situation in real life may not allow for this and warrant for the decision variable to take only integer value.

6. For example in production, manufacturing is frequently scheduled in term of lots, batches or production runs.
In allocation of goods, a shipment of goods involve discrete number of trucks, wagon or aircraft and fractional values are meaningless.

7. Types of integers Pure and mixed integer programming.
(1) An integer programming in which all variable are required to be integer is called pure integer.
(2) An integer programming in which some variable are required to be integer is called mixed integer programming.

8. Mixed integer programming
Maximize 20x1 + 8x2 Constraint 5x1 + 7x2+  63 2x1 + 3x2 +  42 x1 + x2 ≥ 0 x1 ,integer

9. Pure integer Maximize 20x1 + 8x2 Constraint 5x1 + 7x2+  63
x1 + x2 ≥ 0 x1 x2 ,integer

10. Zero-one Model Integer programming
A special type of integer programming problem is a case where the value of decision variable are limited to logical variable like Yes or No, Matched or No-Matched which are symbolized by the value 0 and 1

11. Maximize 20x1 + 8x2 Constraint 5x1 + 7x2+  63 2x1 + 3x2 +  42 x1 + x2 ≥ 0 x1=0 (if product is not start) =1(A product is start)

12. Applications of integer programming
Integer program is used in following.
Traveling sales man problem.
Capital budgeting.
Facility location.
Fixed charge problem.

13. Knapsack
The knapsack problem is a particularly simple integer program: it has only one constraint. Furthermore, the coefficients of this constraint and the objective are all non-negative. For example, the following is a knapsack problem:
Maximize 8x1 + 11x2 + 6x3 + 4x4
subject to 5x1 + 7x2 + 4x3 + 3x4 ≤14
xj 2 f0;1g:

14. Capital Budgeting
Suppose we wish to invest $14,000. We have identified four investment opportunities.
Investment 1 requires an investment of $5,000 and has a present value (a time-discounted value) of $8,000; investment 2 requires $7,000 and has a value of $11,000; investment 3 requires $4,000 and has a value of $6,000; and investment 4 requires $3,000 and has a value of $4,000. Into which we place our money so as to maximize our total present value?

15. As in linear programming, our first step is to decide on our variables
As in linear programming, our first step is to decide on our variables. we will use a 0-1 variable xj for each investment. If xj is 1 then we will make investment j. If it is 0, we will not make the investment. This leads to the 0-1 programming problem:
Maximize 8x1 + 11x2 + 6x3 + 4x
subject to 5x1 + 7x2 + 4x3 + 3x4 ≤ 14
xj 2 f0;1g j = 1; : : :4:

16. Salesman travelling problem
A sales man must travel city to city to maintain his account. This week he has to leave his home base and visit four other cities. The home city is city A. use the assignment method to determine the tour that will minimize the total distance of visiting all cities. The total distance of visiting of all cities and returning is given in the table.

17. From city A B C D E −
375
600
150
190
300
350
175
500
160

18. Solution Put M to prohibit travelling from particular city to itself.
From city A B C D E M
375
600
150
190
300
350
175
500
160

19. NOW PREPARE REDUCE COST TABLE
Minimum value subtract from all element in the rows.
From city A B C D E M
375
600
150
190
300
350
175
500
160

20. We have obtained this table
From city A B C D E M
225
450 40
200
125
175
300 50
190
140 15
325

21. All column have zero. From city A B C D E M 225 450 40 200 175 300 5040
200
175
300 50
190
140 15
325
125

22. . Now draw line covering all zero.
From city A B C D E M
225
450 40
200
175
300 50
190
140 15
325
125

23. From city A B C D E M
240
325 40
200
175
285 35
185
205 65
140
110

24. Number of line covering zero is smaller than n. it need for adjustment
Number of line covering zero is smaller than n. it need for adjustment. From the uncovered value we select smallest value and subtract it from uncovered values and add with intersection values. we obtained the following table
From city A B C D E M
240
325 40
200
175
285 35
185
205 65
140
110

25. From city A B C D E M
275
325 40
235
175
285
150
205 30
105 75

26. Again not equal. Repeat the procedure and obtained the table.
From city A B C D E M
275
325 40
235
175
285
150
205 30
105 75

27. From city A B C D E M
275
295 10
265
205
285
120 75
150

28. Again not equal. Repeat procedure. We have obtained the following table.
From city A B C D E M
275
295 10
265
205
285
120 75
150

29. From city A B C D E M
275
295 10
265
205
285
120
105 75

30. = A-D,B-E,C-B,D-C,E-A =A-D-C-B-E-A PUT THE VALUES
=1165KM
From city A B C D E −
375
600
150
190
300
350
175
500
160

31. Example of integer programming
The owner of ready made garments store sell two type of premium shirt known as zee shirt and star shirts. He make a profit of Rs 200 and Rs 300 per shirt of zee and star shirt. He has two tailor A and B. A tailor can devote a total of 17 hours per day. while tailor B can devote 15 hours a day. Both type of shirts are stitched by both the tailor. the time needed for stitching zee shirt is 2 hour by tailor A and 3 hour by tailor B. a star shirt require 4 hour of tailor A and 3 hour by tailor B. how many shirts of each should be stitched to maximize daily profit. Solve it as integer problem.

32. Solution
maximize 2oox x subject to 2x1 + 4x2+  17 Available hours by A 3x1 + 3x2 +  15 Available hours by B x1 + x2 ≥ 0, x1 integer

33. Z =2oox x+0S1+0S2
subject to
2x1 + 4x2+ S1+0S2= 17
3x1 + 3x2 + 0S1+S2= 15

34. Basic x1 x2 s1 s2 bi S1 2 4 1 17 S2 3 15 Cj
200
300
sloutio 17 15 J

35. J Basic x1 x2 s1 s2 bi S1 2 4* 1 17 4.25 S2 3 15 5 Cj 200 300 sloutio S2 3 Cj
200
300
sloutio 17 15 J

36. Basic x1 x2 s1 s2 bi
X2 300 .5 1
.25 S2
1.5*
-.75 Cj
200
300
sloutio
4.25
2.25 J 50
-75

37. OPTIMAL SOLOUTION J Basic x1 x2 s1 s2 bi X2 300 1 .5 -.333 3.5 S2 2001 .5
-.333
3.5
S2 200
-.5
.666
1.5 Cj
200
300
sloutio
1.5
3.5 J O
-50
-33.33

38. IN THE PREVIOUS TABLE NO INTEGER VALUE IN THE SOLUTION
Now we apply cutting plane algorithm( gormay cut),
Select first row of the table.
0x1 + x2+.5S1+(-.333)S2= 3.5
0x1 + x2+.5S1+(-1+2/3)S2= 3.5
5S1+2/3S2= .5 (3 +0x1 + 1x2 - 1S2)
-5S1-.666S2+S3= -.5

39. Basic x1 x2 s1
S S3 bi
X2 300 1 .5
3.5
S2 200 S3
-.5
-.666* 1
1.5 Cj
200
300
sloutio
1.5
3.5 J O
-50

40. Basic x1 x2 s1
S S3 bi
X2 300 1
.75
3.5
S2 200 S3 -1
1.5
-.5 Cj
200
300
sloutio 1
3.75 J
-25

41. TAHNKS