Consider the following problem.Minimize Z = x₁ + 2 x₂,subject to -x₁ + x₂ $\le$ 15 2 x₁+ x₂ $\le$ 90 x₂ $\ge$ 30 and x₁ $\ge$ 0,x₂ $\ge$ 0.
(a)Solve this problem graphically
(b)Develop a table giving each of the CPF solutions and the corresponding defining equations,BF solution,and nonbasic variables.

Question

Consider the following problem.Minimize Z = x₁ + 2 x₂,subject to -x₁ + x₂

\le

15 2 x₁+ x₂

\le

90 x₂

\ge

30 and x₁

\ge

0,x₂

\ge

0.
(a)Solve this problem graphically
(b)Develop a table giving each of the CPF solutions and the corresponding defining equations,BF solution,and nonbasic variables.

Quizplus · Accepted Answer

The Answer is (a)Thus,the optimal solution is (x₁,x₂)= (15,30)with Z = 75. $(a)Thus,the optimal solution is (x1,x2)= (15,30)with Z = 75. (b)The above graphical solution reveals that the problem has three CPF solutions,as listed in the first column of the table below.The corresponding defining equations are the equations of the constraint boundaries (the solid lines in the above graph)that pass through the respective CPF solutions,as listed in the second column of the table below.To identify the corresponding BF solutions and nonbasic variables,we need to use the augmented form of the problem.Introducing slack variables,x3 and x4,and surplus variable x5,we have Minimize Z = x1 + 2 x2,subject to - x1 + 2 x2 + x3 = 15 2 x1 + x2 + x4 = 90 x2 - x5 = 30 and x1 \ge 0,x2 \ge 0,x3 \ge 0,x4 \ge 0,x5 \ge 0.Also introducing an artificial variable x6 into the last constraint is optional,since this variable would be needed only to initiate the simplex method,which we are not concerned with doing here.If x6 is introduced,it would become an additional nonbasic variable (so its value would be 0)in the last two columns of the table below.For each CPF solution,the corresponding BF solution listed in the third column is obtained by calculating x3,x4 and x5 in the augmented form of the problem.The nonbasic variables listed in the last column are the variables whose values are 0 in the BF solution. $ (b)The above graphical solution reveals that the problem has three CPF solutions,as listed in the first column of the table below.The corresponding defining equations are the equations of the constraint boundaries (the solid lines in the above graph)that pass through the respective CPF solutions,as listed in the second column of the table below.To identify the corresponding BF solutions and nonbasic variables,we need to use the augmented form of the problem.Introducing slack variables,x₃ and x₄,and surplus variable x₅,we have Minimize Z = x₁ + 2 x₂,subject to - x₁ + 2 x₂ + x₃ = 15 2 x₁+ x₂ + x₄ = 90 x₂-x₅ = 30 and x₁

\ge

0,x₂

\ge

0,x₃

\ge

0,x₄

\ge

0,x₅

\ge

0.Also introducing an artificial variable x₆ into the last constraint is optional,since this variable would be needed only to initiate the simplex method,which we are not concerned with doing here.If x₆ is introduced,it would become an additional nonbasic variable (so its value would be 0)in the last two columns of the table below.For each CPF solution,the corresponding BF solution listed in the third column is obtained by calculating x₃,x₄ and x₅ in the augmented form of the problem.The nonbasic variables listed in the last column are the variables whose values are 0 in the BF solution. $(a)Thus,the optimal solution is (x1,x2)= (15,30)with Z = 75. (b)The above graphical solution reveals that the problem has three CPF solutions,as listed in the first column of the table below.The corresponding defining equations are the equations of the constraint boundaries (the solid lines in the above graph)that pass through the respective CPF solutions,as listed in the second column of the table below.To identify the corresponding BF solutions and nonbasic variables,we need to use the augmented form of the problem.Introducing slack variables,x3 and x4,and surplus variable x5,we have Minimize Z = x1 + 2 x2,subject to - x1 + 2 x2 + x3 = 15 2 x1 + x2 + x4 = 90 x2 - x5 = 30 and x1 \ge 0,x2 \ge 0,x3 \ge 0,x4 \ge 0,x5 \ge 0.Also introducing an artificial variable x6 into the last constraint is optional,since this variable would be needed only to initiate the simplex method,which we are not concerned with doing here.If x6 is introduced,it would become an additional nonbasic variable (so its value would be 0)in the last two columns of the table below.For each CPF solution,the corresponding BF solution listed in the third column is obtained by calculating x3,x4 and x5 in the augmented form of the problem.The nonbasic variables listed in the last column are the variables whose values are 0 in the BF solution. $

Consider the Following Problem ≤\le≤ 15 2 X1 + X2 ≤\le≤ 90 X2

Consider the Following Problem $\le$ 15 2 X₁+ X₂ $\le$ 90 X₂