Integer programming, MA Operational Research1 Integer Programming Operational Research -Level 4 Prepared by T.M.J.A.Cooray Department of Mathematics

Integer programming, MA Operational Research2 Introduction  In LP problems,decision variables are non negative values, i.e. they are restricted to be zero or more than zero.  It demonstrates one of the properties of LP namely, continuity, which means that fractional values of the decision variables are possible in the solution of a LP model.  For some problems like: product mix, balanced diet (nutrition) etc. the assumption of continuity may be valid.

Integer programming, MA Operational Research3  Ctd..  Further,in some problems such as, production of different fertilizers (in kilograms or tonnages ),usage of different amounts of food items (in grams) etc. may satisfy the continuity assumption.  But some items cannot be produced in fractions. Say ship,cranes,tables,chairs etc.

Integer programming, MA Operational Research4  Ctd..  If we round off the production volume of such products:  Corresponding solution may be different from the optimal solution.  There would be a significant difference in the total profit, if the profit per unit of each of the products is very high.

Integer programming, MA Operational Research5  There are problems where it is also necessary to assign people, machines, vehicles to activities in integer quantities.  That means these variables should take integer values.  Hence there is a need for Integer programming methods.  The mathematical model for IP or more precisely ILP is the LP model with one additional restriction/constraint: “variables must have integer values”.

Integer programming, MA Operational Research6  Ctd..  If only some of the variables of the problem should be integers then they are called “Mixed IP problems”  If all the variables should take integer values then we have “Pure IP problems”.  There is another area of application,namely problems involving a number of interrelated “yes no decisions”. In such decisions,the only two possible choices are yes or no.

Integer programming, MA Operational Research7  For example, should we undertake a particular project? Should we make a particular fixed investment? Should we locate a facility in a particular site etc.  with just two choices,we can represent such decisions by decision variables that are restricted to just two values.say 0 or 1.  Thus x j =1 if decision j is yes =0 if decision j is no =0 if decision j is no

Integer programming, MA Operational Research8  such variables are binary variables or 0- 1 variables  IP problems that contain only binary variables are called binary integer programming problems. (BIP problems) (BIP problems)

Integer programming, MA Operational Research9 Consider the following ILP problem Maximize Z=7x 1 +10x 2 Subject to : - x 1 +3x 2 ≤6 7x 1 +x 2 ≤35 7x 1 +x 2 ≤35 x 1,x 2 ≥ 0 and integer x 1,x 2 ≥ 0 and integer

Integer programming, MA Operational Research10 Cutting plane algorithm: The general procedure for Gomory’s cutting plane method is as follows: Step1:find the original LP solution (using the simplex method) ignoring the integer constraint. Step 2:if the solution is integer stop. Otherwise, construct a “cut” derived from the row that has a non integer variable with the largest fractional value and add to the current final tableau. If there is a tie select any row arbitrarily. Step 3:solve the augmented LP problem,and return to step 2.

Integer programming, MA Operational Research11  Consider the problem shown in slide 15. slide 15.slide 15.  The optimum tableau is given as Basicx1x2s1s2 solutio n z0063/2231/22 66 ½ x2017/221/22 3 ½ x110-1/223/22 4 ½

Integer programming, MA Operational Research12  The information in the optimum tableau can be written explicitly as  Z+63/22 S1+31/22 S2 =66 ½  X2+ 7/22 S1+1/22 S2 =3 ½  X1 -1/22 S1 + 3/22 S2=4 ½  First,factor out all the non integer coefficients into an integer value and a strictly positive fractional component.  A constraint row with a non integer value, can be used as a source row for generating a cut.

Integer programming, MA Operational Research13  Factoring the x2 row, since s1 and s2 are non negative and the corresponding coefficients are positive fractions middle term is  1/2. since the LHS is all integer values we can say it is  0

Integer programming, MA Operational Research14 Any cut can be used in the first iteration of the cutting plane algorithm. This is the constraint, that is added as an additional constraint which is also known as the cut, to the LP optimum tableau. Resulting tableau is: 

Integer programming, MA Operational Research15 Basicx1x2s1s2 Sg1Sg1Sg1Sg1 solutio n z0063/2231/ ½ x2017/221/220 3 ½ x110-1/223/220 4 ½ Sg1Sg1Sg1Sg100-7/22-1/221-1/2  We get the following optimal,but infeasible tableau. Ratio Smallest absolute value

Integer programming, MA Operational Research16  Applying the dual simplex method to recover feasibility, yields, Basi c x1x2s1s2 Sg1Sg1Sg1Sg1solution z x x11001/7-1/7 4 4/7 S10011/7-22/7 1 4/7 This solution is still non integer in x1 and s1.selecting x1 arbitrarily as the next source row the associated cut is 

Integer programming, MA Operational Research17 Basi c x1x2s1s2 Sg1Sg1Sg1Sg1 Sg2Sg2Sg2Sg2solution z x x11001/7-1/70 4 4/7 S10011/7-22/70 1 4/7 Sg2Sg2Sg2Sg2000-1/7-6/71-4/7 Ratio /2 Smallest absolute value

Integer programming, MA Operational Research18 The dual simplex method yields the following tableau which is optimal, feasible and integer. Basi c x1x2s1s2 Sg1Sg1Sg1Sg1 Sg2Sg2Sg2Sg2solution z x x S S

Integer programming, MA Operational Research19  It is important to note that the fractional cut assumes that all the variables including the slack and the surplus variables are integer.  This means that the cut deals with pure integer problems only.

Integer programming, MA Operational Research20 Consider the following problem.  Maximize Z=2x1+x2  subject to: x1+x2  5 -x1+x2  0 -x1+x2  0 6x1+2x2  21 6x1+2x2  21 x1, x2 are non negative integers. x1, x2 are non negative integers.