max $5$x$1+4$x$2$The purpose of this question is to illustrate how duality theory can be used in deriving solutions

to practical problems. Consider the following problem

min,${\displaystyle \underset{t=1}{\overset{T}{}}}(c_t{x}_t+h_t{I}_t)$

$s.t.,I_{t-1}+x_t-I_t=d_t,t=1,$dots,$T$

$,x_t,I_{t}0,t=1,$dots,$T.$

The above formulation can be interpreted as a model for finding the optimal production quan$-$

tities $(x_t)$ and inventory levels $(I_t)$ that will minimize the total production $({\displaystyle \underset{t=1}{\overset{T}{}}}{c}_t{x}_t)$ and

inventory holding $({\displaystyle \underset{t=1}{\overset{T}{}}}{h}_t{I}_t)$ costs to meet demand $d_t$ for each period $t=1,$dots,$T.$

$($a$)[10$ points

s$.$t$.2$x$1+3$x$2+$ x$3=8$

$2$x$1+$ x$2+$ x$4=14$

x$1,...,$ x$4>=0$

$($a$)$ Apply the primal affine scaling algorithm to solve the same problem starting

from x$0=(0\mathrm{.}1,0\mathrm{.}1,7\mathrm{.}9,13\mathrm{.}7)$ and taking $\backslash$alpha $=0$