Let $(LP)$ and its dual $(DLP)$ be defined as

\begin{array}{l l} & \text{maximize} \quad c^{T}x \\ (LP) & \text{subject to} \quad Ax \leq b, \quad \text{and} \quad (DLP) \\ & \quad x \geq 0, \end{array} \quad \begin{array}{l l} & \text{minimize} \quad b^{T}y \\ & \text{subject to} \quad A^{T}y \geq c, \\ & \quad y \geq 0, \end{array}

where

A = \left( \begin{array}{ccc} 8 & 6 & 1 \\ 4 & 2 & 1.5 \\ 2 & 1.5 & 0.5 \end{array} \right), \quad b = \left( \begin{array}{c} 48 \\ 20 \\ 8 \end{array} \right), \quad \text{and} c = \left( \begin{array}{ccc} 6 & 3 & 2 \end{array} \right)^{T}.

(a)

Final exam March 11 2021

SF2812

SOLVE SUMMARY

MODEL example

TYPE LP

SOLVER CPLEX

OBJECTIVE cost

DIRECTION MAXIMIZE

FROM LINE 22

Optimal solution found

Objective: 28.000000

	LOWER	LEVEL	UPPER	MARGINAL
--- EQU of	.	.	.	1.0000
--- EQU r1	-INF	24.0000	48.0000	.
--- EQU r2	-INF	20.0000	20.0000	1.0000
--- EQU r3	-INF	8.0000	8.0000	1.0000
of objective function
r1 constraint number 1
r2 constraint number 2
r3 constraint number 3
	LOWER	LEVEL	UPPER	MARGINAL
--- VAR x1	.	2.0000	+INF	.
--- VAR x2	.	.	+INF	-0.5000
--- VAR x3	.	8.0000	+INF	.
--- VAR cost	-INF	28.0000	+INF	.

The only catch is that GC does not know how to extract the optimal solutions from the GAMS output. Help GC obtain the optimal solutions to $(LP)$ and $(DLP)$ from the GAMS output file.

Hint: Use this as the optimal basis inverted matrix:

B ^ {- 1} = \left( \begin{array}{c c c} 0 & - 1 / 2 & 3 / 2 \\ 0 & 2 & - 4 \\ 1 & 2 & - 8 \end{array} \right),