$1.$ Consider the following standard production problem

Maximize c$$x

s$.$t$.$ Ax $<=$ b

xi $>=0$

Where A is an m by n matrix $($m constraints and n variables$),$ x is an n by $1$ vector, and b

is an m by $1$ vector. A$$ is the transpose matrix of A$.$

Its dual problem is

Minimize b$$y

s$.$t$.$ A$$y $>=$ c

yi $>=0$

For the given primal problem below, write out the dual problem in the table. $(10$ points$)$

The solution to the primal problem is generated by LINDO and reported in the table

below.

LP OPTIMUM FOUND AT STEP $2$

OBJECTIVE FUNCTION VALUE

$1)36\mathrm{.}00000$

VARIABLE VALUE REDUCED COST

X$12\mathrm{.}0000000\mathrm{.}000000$

X$26\mathrm{.}0000000\mathrm{.}000000$

ROW SLACK OR SURPLUS DUAL PRICES

$2)2\mathrm{.}0000000\mathrm{.}000000$

$3)0\mathrm{.}0000001\mathrm{.}500000$

$4)0\mathrm{.}0000001\mathrm{.}000000$

NO$.$ ITERATIONS$=2$

PRIMAL DUAL

Maximize $3$x$1+5$x$2$

Subjected to x$1<=4($resource $1)$

$2$x$2<=12($resource $2)$

$3$x$1+2$x$2<=18($resource $3)$

x$1,$ x$2>=0$

$!$

$1$

$($Due: Sunday $5/52024)$

Answer the following questions:

a$)$ Which constraints are binding? Which resources are fully utilized? $(10$ points$)$

b$)$ How much increase of the objective value do you expect if the manufacturer is

given an additional unit of resource $2?$ How about an additional one unit of

resource $3?$ Solve the problem with one changed constraint $($change the right side

limit of constraint $2$ from $12$ to $13)$ in an LP solver, and compare the objective

value with what you expect. $(10$ points$)$

c$)$ Solve the dual problem in an LP solver, and compare the optimal objectives in the

primal and dual problems. Does your finding agree with the duality theorem $1?$

$(10$ points$)$

d$)$ Does duality theorem $2$ hold in this example? $(10$ points$)$

e$)$ Use this example to show: In production problem, a positive opportunity cost

$($shadow price$)$ of a resource is always to be associated with the full utilization of

the resource in the optimal solution. If the total opportunity cost of producing a

unit of product j is greater than the gross profit from that product, then product j

should not be produced according to the optimal production strategy. $(10$ points$)$