Integer Linear Programming Problem in Messebo Cement Factory

To solve integer linear programming problem is very difficult than to solve linear programming problem. In this paper we are going to see the formulation of integer linear programming problem and one of its solution techniques called Branch and Bound method. This paper also contains a real world problem of integer linear programming problem on Messebo Cement Factory (Mekelle, Tigray, Ethiopia) solved using the Branch and Bound method.


Introduction
Many linear programming problems require certain variables to have integer values. Such requirement arises naturally when the variables represent entities like packages or peoples that cannot be fractionally divided. These and other requirements give birth to integer linear programming problem (ILPP).
Integer Linear programming problems are linear programming problems with all the decision variables are restricted to integer value.
In Integer Linear Programming Problem fractional values are meaningless. For example: when we asked how many chairs and tables should the profit maximizing carpenter make, it did not make sense to come up with an answer "three and one half''. May be the carpenter know more than enough to make half a chair (using half the resource needed to make the entire chair), but probably he would not be able to sell half a chair for half the price of a whole chair. So, sometimes it makes sense to add to a problem the additional constraint that all of the variable must take an integer value.
Thus, the general form of Integer linear programming problem is, So, since Integer linear programming problem forced the values of the decision variables to come from the integer part, the optimal solution of Integer linear programming problem is an integer. The feasible set of an Integer linear programming problem is the set of integer points in the feasible region (polyhedral) given by Ax ≤ b Figure 1: Feasible points of ILPP

Method of solving Integer Linear Programming Problem
There are two general methods for solving integer linear programming problem. These are (i) Branch and Bound method (BBM) (ii) Cutting plane method (Gomory's method ) But in this paper we only focus with the Branch and bound method.

Branch -and -Bound -method (BBM)
Branch and bound technique is one method of solving ILPP which is based on the concept of divide and conquer. Since the original "large" problem is too difficult to be solved directly, it is divided into smaller sub problems until these sub problems can be conquered. The dividing (branching) is done by partitioning the entire set of feasible solutions into smaller and smaller subsets. The conquering (fathoming) is done partially by bounding how good the best solution in the subset if it's bound indicates that it cannot possibly contain an optimal solution for the original problem. In general, the -branchand bound approach is based on the principal that the total set of feasible region can be partitioned in to smaller subsets of feasible region. This smaller subset can then be evaluated systematically until the best solution is found.
When the branch and bound approach is applied to an integer linear programming problem, it is used in conjunction with the normal non integer solution approach.
The general formulation of integer linear programming problem is,

Basic steps in branch and bound method (BBM)
There are three steps in branch and bound method.
Step1) Initialization: -Relax the integer programming problem (RILPP) and solve the relaxed problem. Relaxed the integer linear programming means the integer linear programming problem without the integrality restriction. In other word it means the corresponding linear programming problem.
Then if the optimal solution to the LPP happens to be integral, then this solution is also the optimal solution to the original ILPP. Otherwise by identifying the upper and lower bounds as follows and go to the next step.
The optimal solution of the Relaxed Integer Linear Programming Problem = The optimal solution obtained by rounding off down = Step 2) Branching: If the current optimal solution of the Relaxed Integer Linear Programming Problem has fractions, then choose the one with the highest fractional value, and it is the first branching point.
Step3) Fathoming: a sub problem will be fathomed (that is ignore from further branching) for one of the following reasons.
1. If the current optimal solution is in agreement with the integrality Condition.

If the current optimal solution is infeasible
3. If the current optimal value is less than Lb

FORMULATION
Now let's us solve the real Integer Linear Programming Problem of Messebo Cement Factory using the Branch and Bound Method.
 Messebo Cement Factory is mainly engaged in producing two types of products, Ordinary Portland Cement (OPC) and Portland Pozolana Cement (PPC). To produce one unit OPC, 90% clinker, 5% pozolana and 5% gypsum is needed and similarly to produce one unit of PPC, 70% clinker, 25 % pozolana and 5% gypsum is used and the total available quantities are 106.5 tone/hr clinker, 37 tone/hr pozolana and 6.5 tone/hr gypsum. The prices are 190 birr and 150 birr per unit of OPC and PPC respectively.
Here what our objective is to maximize the profit. In other word to maximize our revenue so that our objective function will become = 190 1 + 150 2 , where the decision variables 1 2 represents the number of unit products OPC and PPC to be produced respectively. And 1 unit represents 100kg. 0.05 1 + 0.05 2 ≤ 6.5 1 , 2 ≥ 0 Now let's first solve the RILPP (relaxed integer linear programming problem). And since the RILPP is its corresponding LPP (linear programming problem) that is obtained by omitting the integrality constraint, the RILPP of the above ILPP will become = 190 1 + 150 2 .
As you can examine it from the graph below (Figure1) the feasible region of the ILPP are the dotted or the integer points inside the feasible region of the LPP (RLPP). Then since the optimal solution of the LPP is found at a corner point(s), we have four candidate points where the optimal solution of the RILPP or LPP can be attained

Vertex
Value of the Objective function From this we can see that the optimal solution of the RILPP is at 1 * , 2 * = (77.5,52.5) and the optimal value will be 22,600. But since the optimal solution 1 * , 2 * = (77.5,52.5) does not satisfy the integrality constraint that is 1 * ~∈ 2 * ~∈ the solution of the RILPP cannot be the solution of the ILPP so we go to the next step.  And since SP4 is infeasible it will be fathomed.  Now since the entire sub problems are Fathomed, the solution of a sup problem which results with highest Z-value among all the sup problems will be the solution of the ILPP. In this case the solution of SP14 which is( 1 , 2 ) = (81,48) has the highest Z-value with 22,590 among all the sub problems thus the optimal solution of the ILPP will be( * 1 , * 2 ) = (81,48)and the optimal value will become * = 22,590. this means, MCF has to produce 8100 kg of OPC and 4800 kg of PPC to maximize its profit.

CONCLUSION
In BBM, we add a constraint and the added constraint are used to divide the feasible region in to two sub regions. In integer linear programming problem rounding off is meaningless. For instance, take the ILPP max = 5 1 + 4 2 . . 1 + 2 ≤ 5 10 1 + 6 2 ≤ 45 1 , 2 ≥ 0 Then the optimal value of the RILPP is * = 23.75 at the optimal solution 1 , 2 = 3.75,1.25 and if we rounding off down, we get 1 , 2 = 3,1 and at 3,1 we get the value = 19 but, at the feasible point 1 , 2 = 3,2 we get a better optimal value Z= 23.so in ILPP rounding off is meaningless.