(B) SIMPLEX METHODIn the last discussion, we used the graphical method to solve linear programming problems, but the graphical approach will not work for problems that have more than two variables. In real life situations, linear programming problems consist of literally thousands of variables and are solved by computers. We can solve these problems algebraically, but that will not be very efficient. So we need a method that has a systematic algorithm and can be programmed for a computer. The method has to be efficient enough so we wouldn't have to evaluate the objective function at each corner point. We have just such a method, and it is called the simplex method.The simplex method for linear programming developed by Dantzig (1914-2005) in the mid-twentieth century (1963) is the first practically and computationally viable method for solving systems of linear inequalities. This has led to the developments of linear programming (LP), a branch of mathematics developed in the twentieth century as an extension of linear algebra to solve systems of linear inequalities. The development of LP is a landmark event in the history of mathematics and its applications that brought our ability to solve general systems of linear constraints (including linear equations, inequalities) to a state of completion. Since more realistic applications of linear programming involve numerous variables and many constraints, there is a need for a method other than the graphical method. A frequently used method is the simplex method.The ‘simplex algorithm’ is an iterative procedure for finding, in a systematic manner, the optimal solution to a linear programming problem. The working of the simplex method proceeds by preparing a series of tables called simplex tables, the last of which indicates the optimal solution of the given problem. Graphically, we found the procedure as searching different corner points of the feasible region. The solution found at each iteration of the simplex method represents such corner points. However, not all corner points are examined. The search chooses only a subset of these corner points, selecting new one if and only if the objective function is at least as good as the current corner point. The method is quite simple and the first step requires the determination of basic feasible solution. Then, with the help of a limited number of steps the optimum solution can be determined. Suppose there is no objective function to optimize, and only a feasible solution of a system of linear constraints is to be found. When there are inequality constraints in the system, the only practical method to even finding a feasible solution is to solve a linear programming formulation of it as a Phase I linear programming problem. Dantzig developed this Phase I formulation as part of the simplex method for LPs that he developed in the mid-twentieth century.

Remark. The set of corner points of the feasible region corresponds to the set of basic feasible solutions. In other words, corner points are basic feasible solutions, and vice versa.

Note. The feasible solutions are infinite in number (because there is infinite number of points in the feasible region, OABC). So, it is rather impossible to search for the optimum solution

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amongst all the feasible solutions. But fortunately, the numbers of basic feasible solutions are finite in numbers (which are corresponding to extreme points O, A, B, C, respectively). Even then, a great labour is required in finding all the BFSs and to select that one which optimizes the objective function.The simplex method provides a systematic algorithm which consists of moving from one BFS (one vertex) to another in a prescribed manner so that the value of the objective function is improved. This procedure of jumping from vertex to vertex is repeated. If the objective function is improved at each jump, then no basis can ever repeat and there is no need to go back to vertex already covered. Since the number of vertices is finite, the process must lead to the optimal vertex in a finite number of steps.

C B

O A

BINDING AND NON-BINDING CONSTRAINTSOnce the optimal solution to an LPP is obtained, we may classify the constraints as being binding or non-binding. A constraint is termed as binding if the L.H.S and R.H.S of it are equal when optimal values of the decision variables are substituted into the constraint. On the other hand, if the substitution of the decision variables does not lead to equality between the left and right hand sides of the constraint, it is said to be non-binding. For example, consider the LPP as Max Z = 40x + 35ySubject to 2x + 3y 60 4x + 3y 96 And, x, y 0The optimal values of decision variables are: x = 18 and y = 8. Substituting these values in the two constraints we get 2x + 3y or 2 x 18 + 3 x 8 = 60 = R.H.S, and4x + 3y or 4 x 18 + 3 x 8 = 96 = R.H.S.Thus, both the constraints are binding in nature.

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On the other hand, if we consider the following example asMin Z = 40x + 24ySubject to 20x + 50y 4800 80x + 50y 7200 And, x, y 0The optimal values of decision variables are: x = 0 and y = 144. Substituting these values in the two constraints we get20x + 50y or 20 x 0 + 50 x 144 = 7200 4800 (R.H.S), and80x + 50y or 80 x 0 + 50 x 144 = 7200 = R.H.S.Accordingly, the first of the constraints is non-binding and the second one is a binding one.

REDUNDANT CONSTRAINT(S)As we have observed, plotting of each of the constraints on the graph serves to determine the feasible region of a given problem. If and when a constraint, when plotted, does not form part of the boundary marking the feasible region of the problem, it is said to be redundant. The inclusion or exclusion of a redundant constraint obviously does not affect the optimal solution to the problem. In other words, a constraint, which does not affect the feasible region, is called a redundant constraint. Such a constraint is not necessary for the solution of the problem. It can therefore be omitted while formulating the problem. This will save the computation time.

SOME IMPORTANT DEFINITIONSFollowing are defined a few important terms for standard LPP which are necessary to understand further discussion.

1. Basic Solution (BS): For a system of m simultaneous linear equations in n unknowns (n > m), a solution obtained by setting any n-m variables equal to zero and solving for remaining m variables, provided the determinant of the coefficients of these m variables is non-zero is called a basic solution. Such m variables (of course, some of them may be zero) are called basic variables and remaining n-m zero valued variables whose values did not appear in the solution are called non-basic variables. The number of basic solutions thus obtained will be at the most nCm = n! / ((n-m)! m!) , which is the number of combinations of n things taken m at a time.

2. Basis: The set of basic variables which are not restricted to zero in the basic solution and are listed in solution column.

3. Basic Feasible Solution (BFS): A basic feasible solution is a basic solution which also satisfies the non-negativity restrictions, that is, all basic variables are non-negative.Basic feasible solutions are of two types:(a) Non-degenerate BFS: A non-degenerate basic feasible solution is the basic feasible

solution which has exactly m positive xi (i = 1, 2, … , m). In other words, all m basic variables are positive, and the remaining n variables will be all zero.

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(b) Degenerate BFS: A basic feasible solution is called degenerate, if one or more basic variables are zero-valued.

The significance of BFS stems from the fact that for any LPP, there is a unique extreme point of the feasible region corresponding to each BFS and also there is at least one BFS corresponding to each extreme point of the feasible region.

4. Optimum Basic Feasible Solution: A basic feasible solution is said to be optimum, if it also optimizes (maximizes or minimizes) the objective function.

5. Unbounded Solution: If the value of the objective function z can be increased or decreased indefinitely, such solutions are called unbounded solutions. If the feasible region (shaded area) is open-ended, i.e., it has no confined boundary. Then, this may be possible that the LPP has no finite solution, and hence the solution is unbounded.

6. Iteration: It is a sequence of steps taken in moving from one basic feasible solution to another basic feasible solution.

7. Cj row: It is a row in the simplex tableau which contains the coefficients of the variables in the objective function.

8. Zj row: It is a row in the simplex tableau whose elements represent the increase (decrease) of the value of the objective function if one unit of the jth variable is brought into the solution.

9. Zj - Cj row: It is a row whose elements represent net per unit contribution of the jth variable in the objective function, if the variable is brought into the new basic solution.

10. Key column: It is the column with the most negative (positive) Zj - Cj and it indicates which variable will enter the next solution in a maximization (minimization) case.

11. Key row: It is the row with the smallest positive value of the, replacement ratio of the constraint rows. The replacement ratio is obtained by dividing elements in the solution column by the corresponding elements in the key column. The key row indicates the variable that will leave the basis to make room for new entering variable.

12. Key (pivot) element: It is the element at the intersection of key row and key column.Note. Unless otherwise stated, solution means a feasible solution. However, an optimum solution to a linear programming problem implies that z has a finite maximum or finite minimum value.

In addition to these terms in a simplex tableau we have the following terms which are necessary to make a linear programming problem fit to be solved by simplex method. Slack variable: A variable used to convert a less than or equal to ( ) constraint into equality

constraint. It is added to the left hand side of the constraint. In economic terminology, slack variable represents the unused capacity.

Surplus variable: A variable used to convert a greater than or equal to ( ) constraint into equality constraint. It is subtracted from the left hand side of the constraint.

Artificial variable: It is a variable added to equal to (=) and greater than or equal to ( ) type constraint. These variables are used to obtain an initial feasible solution in simplex

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method. These variables are reduced to zero at optimality. Artificial variables are temporary slack variables which are used for purposes of calculation, and are removed later.

The above variables are used to convert the inequalities into equality equations.

STANDARD FORM OF A LPPThe use of the simplex method to solve an LP problem requires that the problem should be converted into its standard form. The standard form of the LP problem should have the following characteristics:

1. All the constraints should be converted to equations except for the non-negativity restrictions which remain as inequalities (≥ 0). The ‘≤’ type constraints can be made into equality type, by adding variables which are known as slack variables. The ‘≥’ type constraints can be made into equality by subtracting variables which are called surplus variables.Consider the equations,Example: a11x1 + a12x2 ≤ b1

a21x1 + a22x2 ≥ b2

On converting, these equations into equality, we get: a11x1 + a12x2 + xs1 = b1

a21x1 + a22x2 xs2 = b2

Here xs2 would be slack variable and xs2 would be surplus variable.In each equation either only one slack variable would be added or only one surplus variable would be subtracted.We assign zero cost to each of the slack and surplus variables.

2. The right hand side element of each constraint should be made non-negative (if not). The right side can always be made positive on multiplying both sides of the resulting equation by 1 whenever it is necessary.For example, consider the constraint as 3x – 4y ≥ 4 which can be written in the form of equation 3x – 4y – w = 4 by introducing the surplus variable w ≥ 0. Again, multiplying both sides by 1, we get 3x +4y + w = 4 which is the constraint equation in standard form.

3. All variables must have non-negative values. Sometimes, a problem might have a variable which is ‘unrestricted in sign’ or ‘free’, so that it can assume negative values as well as non-negative. This situation is handled by treating such a variable as the difference of two variables which are both non-negative, because such a difference may be positive, negative or zero. Consider the following example:

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Maximize Z = 5x +7y 2wSubject to 2x + 5y + 3w ≤ 805x + 2y – 2w ≤ 30y + 6w ≤ 42 And, x, y ≥ 0, w is unrestricted in signSince, w is unrestricted in sign, we can represent it by a difference of two non-negative variables, say and . Substituting w= – in the above and simplifying, we getMaximize Z = 5x +7y –2 + 2Subject to 2x + 5y + 3 – 3 ≤ 805x + 2y – 2 + 2 ≤30y + 6 – 6 ≤ 42And, x, y, , ≥ 0After the solution is obtained, we shall substitute the difference of the values of and

as the value of w.

CONDITIONS FOR APPLICATIONS OF SIMPLEX METHOD1. The R.H.S. of each of the constraints bi should be non-negative. (Refer Point 2 above)2. Each of the decision variables of the problem should be non-negative. (Refer point 3

above)

PROCEDURE OF SOLVING A PROBLEM THROUGH SIMPLEX METHODThe method of solving a problem through simplex method involves the following steps:-Step1: Formulation of LPP.Step2: Convert constraints into equality form by using the rules given in standard form of a LPP.Step3: Construct the starting simplex table.Step4. To set up the Initial Basic Feasible Solution. The fourth step is to find the basic feasible solution. For this we can take the m slack variables or artificial variables s1, s2, …, sm or a1, a2 , … ,am or mixture of them (some slack and some artificial variables) as the basic variables. We can determine their values directly, because the remaining variables x1, x2, …, xn are non basic and therefore each has the value zero. Thus s1 =b1, s2 =b2, …, sm =bm or (a1=b1, a2=b2, …., am=bm) or (s1 = b1, a1 = b2, …, am = bm or any other combination of this type). Each of the slack (s1, s2, …, sm) or surplus (sl1, sl2, …, slm) variable will have a coefficient zero in the objective function, whereas each of the artificial (a1, a2, …, am) variables will have a coefficient +M in case of maximization or –M in case of minimization in the objective function, M being a very large number. This is just to keep them out of the final solution.The simplex method then proceeds to solve the above problem by designing and redesigning successively better basic feasible solutions until an optimal solution is obtained.

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Step 5. To set up the Initial Simplex Table. Having set up the simplex table, the next step is to locate the identity matrix and the variables involved in it. The identity contains all zeros except the diagonal elements as 1. The identity must have this square form with all zeros and a diagonal of (plus) ones. The size of this square matrix would be determined by the number of constraints in the system. Locating the identity is of prime importance as the solution is identified in reference to this. The variables corresponding to the identity matrix are called basic variables and the remaining ones are called non-basic variables. For computational efficiency and simplicity, the simplex algorithm is usually performed in a table called a simplex tableau. The initial basic variables are the slack or artificial variables associated with each constraint (all decision variables are zero). The initial simplex tableau is shown in Tableau 1.The interpretation of the data in Table 1 is given after the table. Other simplex tables will have similar interpretations.

Simplex Table 1Table 1 c1 c2 … cn 0 0 … 0 M or

M0 or M or

M… 0 or M

or MxB cB x1 x2 … xn sl1 sl3 … slj a1 s2 or a2 … sm or am bi

a1 0 or M or M

a11 a12 … a1n -1 0 … 0 1 0 … 0 b1

s2 or a2

0 or M or M

a21 a22 … a2n 0 0 … 0 0 1 … 0 b2

a3 0 or M or M

a31 a32 … a3n 0 -1 … 0 0 0 … 0 b

aj 0 or M

or Maj1 aj2 … ajn 0 0 … -1 0 0 … 0

sm or am

0 or M or M

am1 am2 … amn 0 0 … 0 0 0 … 1 bm

Zj Cj -c1 -c2 … -cn 0 0 … 0 M or M

0 or M or M

… 0 or M or M

a. In the first row, we write the coefficients of the variables in the objective function. These values will remain the same in subsequent table.

b. The second row shows the major column headings. These headings remain the same in the subsequent table and apply to the values listed in the first m rows.

c. In the first column labelled “xB” (also called product mix Column) are listed the basic variables. Thus in the initial simplex table, these variables are the slack variables or artificial variables.

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d. In the second column labelled “cB”, we write the coefficients (in the objective function) of the basic variables included in the specific table. Thus the coefficients of slack or artificial variables which are included in the first table are written in the cB column.

e. The Body Matrix (under non-basic variables) in the initial simplex tableau consists of the coefficients of the decision variables in the constraint set.

f. The identity matrix (under slack or artificial variables) in the initial simplex tableau represents the coefficients of the slack or artificial variables in the constraints set. Each simplex table contains an identity matrix under basic variables.

g. In the column labelled “bi” (also called Quantity column), we write the solution values of the basic variables. Since the values of our initial basic feasible solution are represented by the right-hand sides of the constraints, these values are listed in the initial simplex table.

h. To get an entry for the Zj, we multiply the entries in the respective columns by the corresponding entries of Cj column and add the products. The Zj entries will all be equal to zero in the initial simplex table. The other Zj entries represent the decrease in the objective function that would result if one of the variables not included in the solution were brought into the solution.

i. The last row labeled “Zj – Cj “, called the index row or net evaluation row, and is used to determine whether or not the current solution is optimal. The calculation of Z j – Cj row simply involves subtracting each Cj value from the corresponding Zj value for that column. We observe that Zj – Cj values are meaningful for the non basic variables only. This is because for a basic variable Zj = (value of the jth slack or artificial variable i.e. 1) x Cj = Cj so that Zj – Cj = Cj – Cj = 0.

The numbers in the Zj – Cj row represent the net contribution to the objective function that results by introducing one unit of each of the respective column variables. A plus value indicates that a greater contribution can be made by bringing the variable for that column into the solution. A negative value indicates the amount by which contribution would decrease if one unit of the variable for that column were brought into the solution.Step6. Test the solution for optimality. Examine the entries Zj – Cj row. In case of maximization problem, if all the entries in Zj – Cj row are positive or zero, the optimal solution has been obtained, otherwise the presence of any negative entry indicates that the solution can further be improved. And, in case of minimization problem, if all the entries in Z j – Cj row are negative or zero, the optimal solution has been obtained, otherwise the presence of any positive entry indicates that the solution can further be improved.Step7. Revision of the current simplex table. At each iteration, the simplex method moves from the current BFS to a better BFS. This involves replacing one current basic variable (called the departing variable) by a new non basic variable (called the entering variable).

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Determining the entering variable. In case of maximization problem, the column with the most

negative entry in the Zj – Cj row is called the key (or pivot) column (indicated by ). This

locates the variable to be introduced into the basis. Thus, the non basic variable which will replace a basic variable is the one lying in the key column. And, in case of minimization problem, the column with the most positive entry in the Z j – Cj row

is called the key (or pivot) column (indicated by ). This locates the variable to be introduced

into the basis. Thus, the non basic variable which will replace a basic variable is the one lying in the key column.Determining the departing variable. The decision of which basic variable to replace is made by focusing upon the key column and the solution column. Divide each entry of the solution column by the corresponding positive entry in the key column. These quotients are written in the last column labeled “Ratio” of the simplex table to be replaced. The row that corresponds to the smallest nonnegative quotient is called the key (or pivot) row (indicated by ). The departing variable is the corresponding variable in this row. The number that lies at the intersection of the key column and the key row of a given table is called the key (or pivot) number. We place a circle around this number.Derivation of new table. After identifying the entering and departing variable, all that remain is to find the new BFS by constructing a new simplex table from the current one. This is accomplished by using elementary row operations so that the key number is reduced to 1 and other entries in the key column are transformed to zero. To change the key row so that the key number is 1. We can use the following formula:New key row = old key row / key number (pivot element)To change the non-key rows, we can use the formula:New non key rows = old non-key row – (key column entry) x new key row, where key column entry is the entry in this row i.e. in the key column.Step8. Test for optimality. Again we apply the test for optimality to determine whether the new solution is optimal. In case of maximization problem, if there are no negative Z j – Cj entries, we have an optimal solution and have finished the problem. Otherwise, go back to step7. Similarly, in case of minimization problem, if there are no positive Z j – Cj entries, we have an optimal solution and have finished the problem. Otherwise, go back to step7.

Remark The vector which is removed from the basis at one iteration in the simplex method cannot

immediately re-enter the basis at the next iteration. An existence of a slack variable in the final simplex table indicates that the resource has

not been fully utilized. Non-existence of slack variables indicates the full utilization of the existing resources. Thus if the company plans to enlarge resources, it can look for only those resources which are fully utilized. To assess the value of additional resources, we can consider what difference it would make if we could provide an extra unit of each

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of the resources fully utilized. The increase in profit as a result of the additional unit of resource is seen to be the marginal value of the resource and is referred to as the shadow price for that resource. The shadow price for each resource is shown in the Z j – Cj row of the final simplex table under the corresponding slack variable.

If there are no x1, x2 variables in the final iteration table, the values of x1 and x2 are zero.

Q1. Solve the following problem by using simplex method. Maximize Z = 40x + 35y (Profit) Subject to 2x + 3y ≤ 60 (Raw Material Constraint) 4x + 3y ≤ 96 (Labour Hours Constraint) And x, y ≥ 0 (Ans. x = 18, y = 8 and Z = 1000)

CHARNES’ M-TECHNIQUE FOR PROBLEMS WITH SURPLUS AND ARTIFICIAL VARIABLESSo far, we have seen the linear programming constraints with less than type. We come across problems with ‘greater than’ and ‘equal to’ type also. Each of these types must be converted as equations. In case of ‘greater than’ type, the constraints are rewritten with a negative surplus variable and by adding an artificial variable. Artificial variables are simply used for finding the initial basic solutions and are thereafter eliminated. In case of an ‘equal to’ constraint, just add the artificial variable to the constraint.The co-efficient of artificial variables a1, a2,….. are represented by a very high value –M or M, and hence the method is known as BIG-M Method.Artificial variable is added to any equality constraint in order to bring in a column for the standard basic matrix. Artificial variable is introduced only when there is a need. If the corresponding column of the standard basis is already available by any original variable, we need not introduce the corresponding artificial variable. The value of the artificial variable is zero, if the given LPP is consistent. Therefore, if the LPP is consistent, the artificial variable must either be non-basic or appear at a zero level in the final optimal table. As it has to become non-basic at the final optimal table of the problem solved by simplex method, once an artificial variable leaves the basis we do not consider it for the remaining iterations to get rid of them, which will eventually happen, if the problem is feasible.In Charnes’ big M-method, we give ‘-M’ cost to the artificial variable in a maximization problem and ‘+M’ cost in a minimization problem, where M is very large.The large cost M prevents the artificial variable to re-enter the basis, once it has left. For if it enters there will be an increase in the objective function value in a minimization problem. But simplex method takes care to reduce the value of objective function at each iteration in a minimization problem. Whereas in a maximization problem ‘-M’ being the cost corresponding artificial variable, if it enters at any stage after being removed, there will be heavy drop in the

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objective function value, contrary to the simplex method procedure, as at each iteration it increases the value of the objective function.If all artificial variables are equal to zero in the optimal solution, we have found the optimal solution to the original problem. In any LPP, if the artificial variable remains at the positive level in the final optimal table, then the problem becomes inconsistent and we declare the problem to be infeasible.

Question1 Why do we choose the most negative entry in the bottom row?Answer The most negative (positive) entry in the bottom row represents the largest (smallest) coefficient in the objective function – the coefficient whose entry will increase (decrease) the value of the objective function very quickly.

Question2 Why do we identify the pivot element? Answer As we have mentioned earlier, the simplex method begins with a corner point and then moves to the next corner point always improving the value of the objective function. The value of the objective function is improved by changing the number of units of the variables. We may add the number of units of one variable, while throwing away the units of another. Pivoting allows us to do just that.

Problem with mixed constraintsQ2. (i) Maximize Z = 2x1 + 4x2 (ii) Maximize Z = 2x1 + x2 +3x3 Subject to 2x1 + x2 ≤ 18 Subject to x1 + x2 + 2x3 ≤ 5 3x1 +2 x2 ≥ 30 2x1 + 3x2 + 4x3 = 12 x1 +2 x2 = 26 And, x1, x2, x3 ≥ 0 And, x1, x2 ≥ 0 (iii) Minimize Z = 5x + 8y (iv) Maximize Z = 8x1 – 4x2

Subject to x + y = 5 Subject to 4x1 + 5x2 20 x ≤ 4 –x1 + 3x2 –23 y ≥ 2 And, x1 0, x2: unrestricted in sign. And x, y ≥ 0Ans. (i) The optimal solution is (2, 12) and value of objective function is 52.

(ii) The optimal solution is (3, 2, 0) and value of objective function is 8.(iii) The optimal solution is (3, 2) and value of objective function is 31.(iv) The optimal solution is (175/17, –72/17) and value of objective function is 1688/17.

TIE FOR THE ENTERING VARIABLEIn a simplex table, suppose that two or more non-basic variables are tied for having the most negative (positive) Zj - Cj entry. In such a case, we choose any one of these to find the entering variable. The optimal solution will be reached eventually, regardless of the tied variable chosen.

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MULTIPLE OPTIMAL SOLUTIONSIn every simplex table, the basic variables have all Z j – Cj ≥ 0, but not the non-basic ones. When a solution is indicated to be optimal and the Zj – Cj value for none of the non-basic variables is zero, the solution is unique in the sense that no other solution to the given problem exists, which would yield the identical value of the objective function. When a non-basic variable in an optimal solution has a zero value for Zj – Cj, then the solution is not unique, and multiple optimal solutions exist. The other optimal solutions are usually obtained by performing additional iterations of the simplex method, each time choosing a non-basic variable with a zero Z j – Cj

entry as the entering variable.Q3. (i) Maximize Z = 8x1 + 16x2 (ii) Maximize Z = x1 + 2x2 +3x3 Subject to x1 + x2 ≤ 200 Subject to x1 + 2x2 + 3x3 ≤ 10 x2 ≤ 125 x1 + x2 ≤ 5 3x1 +6 x2 ≤ 900 x1 ≤ 1 and, x1, x2 ≥ 0 and, x1, x2, x3≥ 0 Ans. (i) The optimal basic solutions are (50, 125) and (100, 100) having optimal objective

function value 2400.

(ii) The optimal basic solutions are (0, 0, ), (1, 0, 3), (0, 5, 0) and (1, 4, ) having

optimal objective function value 10.

INFEASIBILITYWhen in the final solution, an artificial variable is in the basis at a positive value then there is no feasible solution to the problem.Q4. (i) Maximize Z = 20x1 + 30x2 (ii) Maximize Z = x1 + 4x2 +3x3 Subject to 2x1 + 3x2 ≤ 40 Subject to 2x1 – x2 + 5x3 = 40 4x1 – x2 ≤ 20 x1 + 2x2 – 3x3 ≥ 22 x1 ≥ 30 3x1 + x2 + 2x3 = 30 and, x1, x2 ≥ 0 and, x1, x2, x3 ≥ 0Ans. (i) & (ii) the problem is infeasible.

UNBOUNDED SOLUTIONAn LPP is said to have an unbounded solution if its objective function value can be increased (in case it is a maximization problem) or decreased (if it is a minimization problem) without limit.If in a certain situation, there are no non-negative replacement ratios or when all the values of key column are negative i.e. all of them are negative or they are equal to (for the reason of the denominator equal to zero), then the algorithm terminates. This indicates that the problem under consideration has unbounded solution.If we come across unboundedness in solving real problems, then the problem is not correctly formulated. Since, no real situation permits any management to have infinite production of goods

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and infinite profits, unbounded solution results if in a maximization problem all constraints are of greater than or equal to type. In such a situation there is no upper limit on feasible region. Similarly, an unbounded solution occurs in a minimization problem if all constraints are of less than or equal to type.Q5. (i) Maximize Z = 10x1 + 20x2 (ii) Maximize Z = 5x1 + 2x2 +2x3 Subject to 2x1 + 4x2 ≥ 16 Subject to 3x1 – 2x2 – 2x3 = –8 x1 + 5x2 ≥ 15 3x1 – 4x2 – x3 = –7 and, x1, x2 ≥ 0 and, x1, x2, x3 ≥ 0Ans. (i) & (ii) the solution is unbounded.

INFEASIBILITY VS UNBOUNDEDNESSBoth infeasibility and unboundedness have a similarity in that there is no optimal solution in either case. But there is a striking difference between the two: while in infeasibility there is not a single feasible solution, in unboundedness there are infinite feasible solutions but none of them can be termed as the optimal.

TIE FOR THE DEPARTING VARIABLE - DEGENERACYDegeneracy in LPP occurs when one or more of the basic variables assume a value of zero. As stated already, for a n-variable, m-constraint problem, there would be m basic and n-m non-basic variables, and the basic variables would assume positive values. However, in case a basic variable assumes a value equal to zero, then the variable and the solution are labeled degenerate. Thus, in condition of degeneracy, the solution would contain a smaller number of non-zero variables than the number of constraints, m. Whenever there is a tie in the replacement ratios for determining outgoing variable, the next table would give a degenerate solution. Also, we may choose any one of these quotients to find departing variable. Q6. Maximize Z = 28x1 + 30x2 (Profit) Subject to 6x1 + 3x2 ≤ 18 (Raw Material Constraint) 3x1 + x2 ≤ 8 (Labour Hours Constraint) 4x1 +5 x2 ≤ 30 and, x1, x2 ≥ 0Ans. The optimal solution is (0, 6) and value of objective function is 180.

Note We know that successive simplex tables represent improvements in the value of the

objective function – increases are observed for the maximization and decreases for the minimum problems. However, when the outgoing variable happens to be a degenerate variable, the objective function value in the next table does not change. The Z value for a ‘non-optimal’ table is not different from the Z value obtained from the next table.

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If there is no improvement in the solution in terms of the objective function, then why should we not stop at the iteration where a degenerate solution appears? The answer to this is that one cannot be sure that the first appearance of a degenerate solution will coincide with the optimal solution. In other words, the problem may be temporarily degenerate. We may consider the next question where a degenerate solution is encountered but later on degeneracy disappears.

Q7. Maximize Z = 5x + 2y Subject to 4x + 2 y ≤ 16 3x + y ≤ 9 3x – y ≤ 9 and, x, y ≥ 0

Ans. The optimal solution is (1, 6) and value of objective function is 17.

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