Duality, Complementary Slackness, Dual Simplex Algorithm and Feasibility

Consider the linear program:
Min -2x-y
St x-y<2
x+y<6
x, y> 0
a) By inspection, argue that this problem cannot have an unbounded optimal solution.
b) Convert this problem to simplex standard form, enumerate all the basic solutions, identify which ones are feasible, and compute their objective values.
c) What is the optimal solution to this problem? How do you know?
d) What is the dual of this problem?
e) Use complementary slackness to prove your answer in part c.
f) For each basis in part b, use complementary slackness to find the corresponding basic solution to the dual problem, identify whether it's feasible, and compute its objective value.
g) Select a basis from part b which is primal infeasible but dual feasible. Using this as your initial solution, compute ONE pivot of the dual simplex algorithm. Is your new solution optimal? Why or why not?

Please see the attached file for the fully formatted problems.

Linear Programming, Duality, Complementary Slackness, Dual Simplex Algorithm and Feasibility are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Consider the following linear programming problem:
Maximize 2x1 + 3x2 + 5x3
Subject to
x1 + 2x2 + 3x3 ≤ 8
x1 - 2x2 + 2x3 ≤ 6
x1, x2, x3 ≥ 0
a. Write the dual problem
b. Solve the foregoing proble

Exercise 4.25 This exercise shows that if we bring the dual problem into standard form and then apply the primal simplex method, the resulting algorithm is not identical to the dualsimplex method.
Consider the following standard form problem and its dual.
minimize x1 + x2 maximize p

a) Describe the purpose of using the 2-phase method or the M-technique and highlight the pluses and minuses of these two approaches.
b) Let the linear programming problem (S) be defined as follows:
Minimise Z = 6x1 + 3x2 + 4x3
Such that:
x1 + 6x2 + x3 = 10
2x1 + 3x2 + x3 = 15
x1,x2,x3 >=0
i) Use the M-technique

Consider the following algorithm for solving a linear program in standard form without having to use the Big-M method:
Choose any basis.
Check to see if this basis is primal feasible.
If so, use this as your initial BFS and solve the problem with simplex.
If the basis is primal infeasible, solve the problem using dual si

Exercise 4.26 Let A be a given matrix. Show that exactly one of the following alternatives must hold.
(a) There exists some x does not equal 0 such that Ax = 0, x > 0.
(b) There exists some p such that p'A> 0'.
Exercise 4.27 Let A be a given matrix. Show that the following two statements are equivalent.
(a) Every vector such

1. Consider the LP Max z=c1x1+2x2+c3x3
Subject to x1 + 5x2+a1x3 ≤ b1
X1-5x2+a2x3≤ b2
X1, x2, x3 ≥ 0
The optimal tableau for this LP is
1 d1 2 1 0 30
0 d2 -8 -1 1 10
------------------------------------
0 d3 -7 d4 0 z - 150
Without u

Consider the following LP
Min a + b + c + d
St a + d = 3
b + d = 2
c + d = 0
a, b, c, d > 0
a) Write the dual of this problem.
b) Given the primal basis {a, b, c}, construct the corresponding primal anddual solutions.
c) What can you say about the optimality of this basis and its corresponding primal and d

See the attached file.
Sugarco can manufacture three types of candy bar. Each candy bar consists totally of sugar and chocolate. The compositions of each type of candy bar and the profit earned for each candy bar are shown in the following table.
Bar Amount of Sugar (Ounces) Amount of Chocolate (Ounces) Price (Cents)
1