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USING INTEGER CUT ON GENERAL INTEGER LINEAR PROGRAMMING PROBLEMS TO EXPLORES ALTERNATIVE

SOLUTIONS

Victor Hariadi1, Yudhi Purwananto1, Ade Vidician S.P.1 Informatics Department, Faculty of Information technology, Sepuluh Nopember Institute of Technology

Gedung Teknik Informatika, kampus ITS Sukolilo Surabaya East Java Indonesia 60111 email : victor@its-sby.edu1, yudhi@its-sby.edu1, ade_vidician_sp@yahoo.co.id1

ABSTRACT

The many problems occurred in the allocation of limited resources among a number of competing activities, such as the production facility allocation problems, problems of allocation of national resources for domestic needs, production scheduling, the solution game, and the selection pattern of delivery (shipping), must be solved by the best (optimal) possible to do. One way to solve these problems is used linear programming (LP). Numerous research have been discovered to solve the problems of linear programming. For certain problems, such as if an entire variable of a problem is an integer should be used integer linear programming (ILP) to solve it is to obtain optimal solutions and objective values. Having obtained the optimal solution can be searched so that other optimal solutions or alternative solutions beyond the initial optimal solution, one way is to use a general integer cut model. The general integer cut is obtained by adding new constraint function formulation containing the optimal solution over previously obtained to produce a new alternative solution by eliminating the possibility of the existence of optimal solution that already exist with the same objective value.

Keywords: linear programming, integer linear programming, general integer cut model, alternative solution.

1. INTRODUCTION Linear Programming (LP) is a way to solve the problem of allocating resources among a limited number of competing activities, the best way possible done. This allocation problem will appear when someone must choose the level of these activities. Some examples of the above situations include problems of allocation of production facilities, problems of allocation of national resources for domestic needs, production scheduling, the game solution, and the selection pattern of delivery (shipping). One of the things

that characterize the above situation is the necessity to allocate resources for the activity. This linear program using a mathematical model to explain the problems he faced. The characteristic of "Linear" here gives the sense that all the mathematical functions in this model is a linear function, whereas the word "program" is a synonym for planning. Thus, Linear Program is planning activities to obtain optimum results, ie an outcome that best goal among all feasible alternatives. Linear programs related to continuous variables and discrete variables, in this case using Integer Linear Programming (ILP) which is pure where all variables are integer. General integer cut model is developed so the alternative solution of the ILP problem can be found if more than one solution can meet the same optimal value of objective function, because they allow the decision maker to choose from many solutions without experiencing any deterioration in the objective function.

2. LINEAR PROGRAMS Linear program, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function of the convex polyhedron defined by linear and non-negativity constraints. In particular, the linear program problem is a problem to determine the value of each variable (decision variables) such that the value of objective function is linear optimum (maximum or minimum) with respect to the restrictions (constraints) that there is this limitation should be expressed by linear inequalities.

2.1. Formulation of the Linear Program Model Problems

Basically, linear program problems can be formulated in a basic model / standard model / mathematical model below. Determine the value of the 푋 ,푋 ,푋 ,⋯ ,푋 so :

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푍 = 퐶 푋 + 퐶 푋 + ⋯+ 퐶 푋 + ⋯+ 퐶 푋 = ∑ 퐶 푋 (optimum [max/min])

Who then called the objective function with restrictions (Constraints): 푎 푋 + 푎 푋 + ⋯+ 푎 푋 ≤ or ≥ 푏 푎 푋 + 푎 푋 + ⋯+ 푎 푋 ≤ or ≥ 푏 ⋯ ⋯ ⋯ 푎 푋 + 푎 푋 + ⋯+ 푎 푋 ≤ or ≥ 푏 or ∑ 푎 푋 ≤ or ≥ 푏 for 푖 = 1,2,3, … ,푚. and 푋 ≥ 0,푋 ≥ 0,⋯ ,푋 ≥ 0 or 푋 ≥ 0, where j = 1, 2, 3,...., n (non-negative term).

Description: Z is the value of the objective function. There are n kinds of goods to be produced

each of 푋 ,푋 ,푋 ,⋯ ,푋 unit. Cj are parameters that are used as

optimization criterion or coefficient decision variable in objective function (eg price per unit of goods to-j).

Xj are the decision variable or activity that you want to search (eg the number of goods production to-j, where j=1,2,…,n).

푏 is a finite resource, which restrict the activities or business in question is also a constant, or "the right value” from the constraint to i (ie size of raw materials to i, i = 1, 2, .., m) . There are m types oh raw materials, each of which is available 푏 , 푏 , , 푏 .

푎 is the coefficient of decision variable (activity concerned) within the constraint of the i (ie size of the raw material i used to produce 1 unit of goods to-j).

Some problem is called the problem a linear program if it meets the following: 1. Objective

What is the purpose of the problems faced by the wish to solve and find solutions. These goals must be clearly and unequivocally called objective function. The objective function can be a positive impact, benefits, or negative impacts, the losses, risks, costs, distance, time to be minimized.

2. Alternative comparison. There must be something or alternatives to be compared, for example between the combination of the fastest and highest costs in time and delayed the lowest cost, or capital-

intensive alternative to labor-intensive, high demand with projected low, and so on.

3. Resources Resources must be analyzed in limited circumstances. For example the limited manpower, limited raw materials, limited capital, limited space to store goods, and others. Restricted to the linear inequalities. Limitations in resources is called as a constraints.

4. Quantitative formulation Objective function and constraints must be formulated quantitatively in mathematical models.

5. Members of the set attachment Members of the set forming the objective function constraint function must have the attachment relationship or functional relationships.

3. INTEGER LINEAR PROGRAMMING

Integer linear programming (ILP) is part of the linear programming where necessary decisions must be made in the form of integers (not fractions is often the case when using the simplex model). Mathematical model of the ILP is equal to the linear programming model, with the additional restriction that the variables must be integers. There are 3 kinds of problems in ILP, namely:

1. Pure ILP, ie cases where all decision variables must be integers.

2. Mixed ILP (MILP), that is the case where some, but not all, the decision variables must be integers.

3. Binary ILP, cases with special problems where all decision variables must be valued 0 and 1.

One of the ways used to solve the ILP problems is using branch and bound model.

4. BRANCH AND BOUND MODEL Branch and bound model has become the standard computer code for integer programming, and applications in practice seems to suggest that this model is more effective than the model of rounding. This technique can be applied both to pure integer programming and mixed integer programming problems. Step-by-step branch and bound model for maximizing problem can be done as follows:: 1. Solve LP with regular simplex model.

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2. Thorough optimal solution. If the base variable is expected round, the round optimal solution has been reached.

3. Value of a proper solution branching into sub-sub-problems. The goal is to eliminate the continuous solution that does not meet the round requirements of the problem.

4. For each sub-problem, the value of continuous optimal solution set as the objective function upper bounds. The best round solution to the lower limit (in the beginning, this is a continuous solution is rounded down). Sub-problems that have upper limits less than the lower limit of the existing, not included in further analysis. A feasible solution is the same round good or better than the upper limit for each sub-problem is sought. If such a solution occurs, a sub problem with the best the upper limit chosen for the branching. Back to step 3.

5. GENERAL INTEGER CUT MODEL

General integer cut model is used to find all alternative solutions of the ILP problems containing binary variables and non-binary, by way of a general integer cut is added to the original model to make the previous model to be infeasible to produce new models with the same objective value.

5.1. Model 1

Given an ILP problem with an optimal solution (푥 , 푥 , … , 푥 ) and the expected objective value of Q, developed the following model to produce the optimal solution: Model 1: Minimize / Maximize : 푓(푋) = 푄 Subject to: ∑ |푥 − 푥 | ≥ 1 (1). Where :

푓(푋) : objective function 푄 : objective function value z : number of variable, z = 1, 2, 3, ..., n. 푥 : desired variables 푥 : Variables that have been obtained

previously. Based Model (1) above, the absolute value must be linearized so that Model 1 can be transferred into the problem Mixed Integer Linear Programs (MILP). Consider the following propositions for linearized Model (1):

Proposition 1 : Let 훼 ∈ {0,1}, 푊 ≥ 0, 0 ≤ 휃 ≤ 1, and 푀(푖푠 푎 푙푎푟푔푒 푐표푛푠푡푎푛푡)

|푥 − 푥 | ≥ 1 ↔

⎩⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎧

(푖) 0 ≤ 푊 − 푥 + 푥 ≤ 푀(1− 훼 ), 푧 = 푚 + 1, 푚 + 2,⋯ ,푛,(푖푖) 0 ≤ 푊 − 푥 + 푥 ≤ 푀훼 , 푧 = 푚 + 1, 푚 + 2, … , 푛,

(푖푖푖) 푊 + 휃 ≥ 1,

(푖푣) 푥∈

− 푥∈

≤ |퐵| − 휃,

퐵 = {푖|푥 = 1, 1 ≤ 푖 ≤ 푚}, 푁 = {푖|푥 = 0, 1 ≤ 푖 ≤ 푚}.

The proof of Proposition 1 above: a. If 푥 − 푥 > 0 for some z and m + 1 ≤ 푧 ≤

푛 then 훼 = 1 based on (i) and (ii) which results in: |푥 − 푥 | = 푊

b. If 푥 − 푥 < 0 for some z and m + 1 ≤ 푧 ≤푛 then 훼 = 0 based on (i) and (ii) which results in: |푥 − 푥 | = 푊

c. Based on (a) and (b) then: ∑ |푥 − 푥 | =∑ 푊 .

d. If ∑ |푥 − 푥 | = 0 then 휃 =0 based on (iv) which results in∑ 푊 =∑ |푥 − 푥 | ≥ 1 by proof (a) and (b) without violating (iii).

e. If ∑ |푥 − 푥 | = 0 then ∑ 푊 = 0 by (a) and (b), and 휃 = 1 based on (iii)

f. which results in∑ |푥 − 푥 | ≥ 1 without violating (iv).

From the proof above it ∑ |푥 − 푥 | ≥ 1 can be replaced by formula (i), (ii), (iii) and (iv). With a notation of proposition (1) above used to find an alternative solution of an ILP problem containing binary variable (0 and 1) for the objective function (푥 ∊ Z for all i).

5.2. Model 2

By Proposition (1) the model (1) can be converted into other models such as the following :

Model 2-1: Minimize / Maximize : 푓(푋) = 푄 Subject to : 0 ≤ 푊 − 푥 + 푥 ≤ 푀(1 −훼 ), 푧 = 푚 + 1,푚 + 2,⋯ , 푛, (2) 0 ≤ 푊 − 푥 + 푥 ≤ 푀훼 , 푧 = 푚+ 1,푚 + 2, … ,푛, (3)

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∑ 푊 + 휃 ≥ 1, (4)

푥∈

− 푥∈

≤ |퐵| − 휃,

퐵 = {푖|푥 = 1, 1 ≤ 푖 ≤ 푚}, 푁 = {푖|푥 = 0, 1 ≤ 푖 ≤ 푚} (5).

Where:

훼 : is a binary variable (0 and 1), 푊 : is a continuous variable

provided 푊 ≥ 0, 휃 : is a continuous variable

provided 0 ≤ 휃 ≤ 1, M : is a large constant, B : set index variable of value 1, N : set index variable of value 0.

After re-formulating the function of absolute constraints (1) with the proposed method, initial problem into another MILP problems can be solved using integer programming using conventional techniques (conventional IP techniques). So for linearized inequality constraints with a total value of the absolute ∑ |푥 − 푥 | ≥ 1where 푥 ∊ {0,1} for 1 ≤ i ≤ m and 푥 ∊ Z for m + 1 ≤ i ≤ n , formula in proposition (1) need n – m additional variable 0–1, n – m + 1 additional continuous variable and 4 ( n – m ) + 2 additional constraints. Whereas for finding an alternative solution of an ILP problem which contains only non-binary variables in the objective function ( ∊ Z for all i), proposition (1) can be simplified to the proposition (2) as follows: Proposition 2 : 푳풆풕 휶풛흐{ퟎ, ퟏ},푾풛 ≥ ퟎ, 풂풏풅 푴 (풊풔 풂 풍풂풓품풆 풄풐풏풔풕풂풏풕)

|풙풛 − 풙풛ퟎ| ≥ ퟏ ↔

⎩⎪⎪⎨

⎪⎪⎧

(풊)ퟎ ≤ 푾풛 − 풙풛 + 풙풛ퟎ ≤ 푴(ퟏ −∝풛),풛 = 풎 + ퟏ,풎 + ퟐ, … ,풏,

(풊풊)ퟎ ≤ 푾풛 − 풙풛ퟎ + 풙풛 ≤ 푴 ∝풛, 풛 = 풎 + ퟏ,풎 + ퟐ, … ,풏,

(풊풊풊) 푾풛 ≥ ퟏ.풏

풛 ퟏ

풏

풛 ퟏ

Based on proposition (2) then model (2) can be simplified to solve ILP problems that contains only non-binary variables in the objective function becomes as follows:

Model 2-2: Minimize / Maximize : Subject to :

(6)

(7)

(8).

6. GENERAL INTEGER CUT OPTIMIZATION ALGORITHM

Step 1 Same as ILP problems to determine solutions in general, step 1 is to determine an optimal solution of ILP problems, let j = 0, solve ILP to find optimal solutions and the objective value of Q.

Step 2 In step 2 this is the step to find the optimum solutions. Let j = j + 1, the existing constraints to add new constraints to find alternative solutions .

Minimize / Maximize :

Subject to: for all (9).

Constraint (9) should be changed to 2 so that the model can be solved using integer programming with conventional techniques. Repeat steps 2 and hold for the condition f (X) = Q (a new objective values generated are equal to the initial objective value), and if the condition f (x) ≠ Q (new objective value is not equal to the initial objective value) then do j = j - 1 is the optimal solution is used to the previous optimal solution and go directly to step 3.

Step 3 In this step 3 is to write all the optimal solutions :

, where p=0,1,2,...,j.

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6.1. System Flow

Figure 1 General integer cut algorithm system flow

7. IMPLEMENTATION, TEST AND ANALYSIS

Test performed on a PC with Intel (R) Core (TM) 2 Duo CPU T5550@1.83GHz (2 CPUs) with the memory of 2024 MB of RAM. Operating system is Windows Vista Home Premium and the language used computational method for implementation is the Lingo v8.0. Test alternative solutions to general integer cut will be done in 2 scenarios, ie testing at two examples of problems with adding two general models and also integer cut by combining the model (2) with a magnitude M different which is a large constant. Combining the model (2) with M different intended to determine the function of M is and what caused it.

Test Data 1 Test data problems 1 from the 1st issue Zionts (1974) about the fixed costs problems where the data contains a combination of integer and binary variables. To simplify the calculation, the problem is converted first into a mathematical model like this below:

Maximize : Subject to :

.

Test Data 2 Test data problems 2 derived from the mixing of materials engineering applications where this data contains only non-binary variables. A chemical company is producing two types of substances (A and B) consisting of three types of raw materials (I, II and III). The two substances have different profits and requirements on the compositions of the three materials. The company must also consider the available amount and treatment costs of each material (I, II and III). The problem is formulated below: Maximize :

Subject to :

, , ,

7.1. Implementation This section will be given a description of the implementation of the Model (1) and Model (2) that will be used to obtain optimal solutions and objective values by example using test data 1. Here is the implementation of a general integer cut with the absolute technique: 1 @ABS (x1 - 23) + @ABS (x2 - 53) +

@ABS (x3 - 0) + @ABS (y1 - 1) + @ABS (y2 - 1) >= 1;

Figure 2 Implementation code of general integer cut with the absolute technique, example using data test problem 1

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For an explanation of the program code in the figure 2 is as follows:

Line 1 is the implementation of Model 1 formula 1

@ABS: a lingo function for absoluting a number.

The value of 23, 53, 0, 1 and 1 is the optimal solution that had been produced before.

Here is the implementation of the general integer cut with conventional IP techniques: 1 M = 100000; 2 0 <= W1 - x1 + 23; 3 W1 - x1 + 23 <= M - M*A1; 4 0 <= W2 - x2 + 53; 5 W2 - x2 + 53 <= M - M*A2; 6 0 <=W3-x3; 7 W3 - x3 <= M - M*A3; 8 0 <= W1 – 23 + x1; 9 W1 – 23 + x1 <= M*A1; 10 0 <= W2 – 53 + x2; 11 W2 - 53 + x2 <= M*A2; 12 0 <= W3 + x3; 13 W3 + x3 <= M*A3; 14 W1 + W2 + W3 + T >= 1; 15 y1 + y2 <= 2 - T ; 16 @GIN(A1); 17 @GIN(A2); 18 @GIN(A3); Figure 3 Implementation code of general integer cut with the conventional IP technique, example using data test problem 1

For an explanation of the program code in the figure 3 is as follows:

M: is a large constant. Line 2-7 is implementation Model 2-1

formula 2 Line 8-13 is implementation Model 2-1

formula 3 Line 14 is implementation Model 2-1

formula 4 Line 15 is implementation Model 2-1

formula 5 Line 16-18 is a binary variable conditions

that will be integer Value of 23, 53, 0, 1 and 1 is the optimal

solution that had been produced before. W1, W2, W3 dan T : is a continuous

variable, the model is denoted by , A1, A2, A3: is a binary variable (0 and 1)

the model is denoted by.

7.2. Scenario 1 and Analysis For the scenario 1 performed three times using test data of problems 1, ie the first trial to add a general integer cut with a constraint with absolute technique, the second trial to add a general integer cut with a constraint with conventional technique with M=100, and the third trial to add a general integer cut with a constraint with conventional technique with M=100000. Lists of alternative solutions (including the initial optimal solution) of the three trials are presented in Table 1, Table 2 and Table 3. From the results of optimization scenario 1 can be drawn some conclusions that: 1. By using the general integer cut model can be

generated an alternative optimal solutions outcome results in an initial optimal ILP problems containing integer variables and binary variables.

2. The results of trials 2 and 3 trials showed that the addition done by the general integer cut can generate equal number of alternative solutions but different sequence optimal solution depends on the size of M is filled.

7.3. Scenario 2 and Analysis For the scenario 2 performed three times using test data of problems 2, ie the first trial to add a general integer cut with a constraint with absolute technique, the second trial to add a general integer cut with a constraint with conventional technique with M=100, and the third trial to add a general integer cut with a constraint with conventional technique with M=100000. Lists of alternative solutions (including the initial optimal solution) of the three trials are presented in Table 4, Table 5 and Table 6. From the results of optimization scenario 1 can be drawn some conclusions that: 1. By using the general integer cut model can be

generated an alternative optimal solutions outcome results in an initial optimal ILP problems only containing non-binary variables.

2. The difference size of M is filled which are not seen in trials 2 and 3 because of data problems 2 if done using a general integer cut produce only two optimal solutions.

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Solution X1 X2 X3 Y1 Y2 Constraints Variable Integer Continuous

Variable Variable 1 23 53 0 1 1 4 5 5 -

2 22 54 0 1 1 20 20 5 -

3 24 52 0 1 1 40 40 10 -

4 22 53 1 1 1 60 60 15 -

5 23 52 1 1 1 80 80 20 -

6 22 52 2 1 1 100 100 25 - Table 1 List of Alternative Solution from Test Data 1 Scenario 1

Solution X1 X2 X3 Y1 Y2 Constraints Variable Integer Continuous

Variable Variable

1 23 53 0 1 1 4 5 5 -

2 24 52 0 1 1 18 12 8 4

3 22 54 0 1 1 32 19 11 8

4 22 52 2 1 1 46 26 14 12

5 22 53 1 1 1 60 33 17 16

6 23 52 1 1 1 74 40 20 20 Table 2 List of Alternative Solution from Test Data 2 Scenario 1

Solution X1 X2 X3 Y1 Y2 Constraints Variable Integer Continuous

Variable Variable

1 23 53 0 1 1 4 5 5 -

2 22 53 1 1 1 18 12 8 4

3 22 54 0 1 1 32 19 11 8

4 24 52 0 1 1 46 26 14 12

5 23 52 1 1 1 60 33 17 16

6 22 52 2 1 1 74 40 20 20 Table 3 List of Alternative Solution from Test Data 3 Scenario 1

Solution X1 X2 X3 Y1 Y2 Y3 Constraints Variable Integer Continuous

Variable Variable

1 85 41 300 306 459 0 8 6 6 -

2 85 42 299 306 458 1 24 24 6 - Table 4 List of Alternative Solution from Test Data 1 Scenario 2

Solution X1 X2 X3 Y1 Y2 Y3 Constraints Variable Integer Continuous

Variable Variable

1 85 41 300 306 459 0 8 6 6 -

2 85 42 299 306 458 1 33 18 6 12 Table 5 List of Alternative Solution from Test Data 2 Scenario 2

Solution X1 X2 X3 Y1 Y2 Y3 Constraints Variable Integer Continuous

Variable Variable

1 85 41 300 306 459 0 8 6 6 -

2 85 42 299 306 458 1 33 18 6 12 Table 6 List of Alternative Solution from Test Data 3 Scenario 2

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8. CONCLUSIONS The conclusion that can be taken from a series of tests and analysis done on the general integer cut model is as follows: 1. General integer cut model can be used to search

for alternative solutions outside the initial optimal solution, where the general integer cut is obtained by re-formulating the optimal solution previously to produce a new alternative solution by eliminating the possibility of the emergence of solutions pre-existing.

2. Alternative solution of an ILP problem is obtained using the general integer cut model should be an integer optimal solution with objective value equal to the initial optimal solution is at the moment before adding a new constraint in the form of a general integer cut.

3. By using the general integer cut model can be generated an alternative optimal outcome results in an initial optimal ILP problem that contains only non-binary variables as well as ILP problems containing binary variables.

4. In general integer cut models, differences in size M is filled will be visible if an ILP problem can produce more than 1 alternative solutions. Because the amount of M will result in differences sequences in the optimal alternative solution that found.

5. General integer cut model can help decision makers in an agency or company to combine the decision variables of different composition that

are considered in accordance with actual condition of the company but still produce the same optimal profit, this is because the ability of general integer cut to give a lot of optional alternative solutions.

9. REFERENCES [1] Asror, Mustaghfiri (2010) Linier Programming

[Online]. Available at: http://musafirundip.wordpress.com/2009/08/15/linear-programming/

[2] Dimyati, Tjutju Tarliah, Dimyati, Ahmad (2003). Operation Research Model-Model Pengambilan Keputusan. Sinar Baru Algensindo.

[3] Hartanto, Eko (2010) Integer Linier Programming [Online]. Available at: http://www.google.co.id/url?sa=t&source=web&ct=res&cd=1&ved=0CAkQFjAA&url=http%3A%2F%2Fpeni.staff.gunadarma.ac.id%2FDownloads%2Ffiles%2F4159%2Flinier%2Bprogramming.ppt&rct=j&q=linier+programming.ppt%3B+eko+hartanto&ei=qcxOS9mEFo3e7APagtHCCA&usg=AFQjCNGwpbqRBLaEU4JiADBiUlwIeGak9Q>

[4] Wahyujati, Ajie (2010) Operation Research 2: Integer Programming [Online]. Available at: http://ajiew.staff.gunadarma.ac.id/Downloads/files/8625/Integer+Programming.pdf