this is the question

Example $2\mathrm{.}4-4($Multiperiod Production Smoothing Model$)$

A company is planning the manufacture of a roduot for March, April, May, and June of next year. The demand quantities are $520,720,520,$ and $620$ units, respectively. The company has a steady workforce of $10$ employees but can meet fluctuating production needs by hiring and firing temporary workers. The extra costs of hiring and firing a temp in any month are $$200$ and $$400,$ respectively. A permanent worker produces $12$ units per month, and a temporary worker, lacking equal experience, produces $10$ units per month. The company can produce more than needed in any month and carry the surplus over to a succeeding month at a holding cost of $$50$ per unit per month. Develop an optimal hiring$/$firing policy over the $4-$month planning horizon.

Problen $2\mathrm{.}4\mathrm{.}4$

Remating Demand March $=520-1012=400$

Remainay Demend Apri$1=720-1012=6000$

Remainitg Demand May $=520-120=400$

Remaining Demeard June $=620-120=500$

operational moder:$-$

DV:

$x_i=$ Net number of Teapocaty at hes star of the molthi fftere wiring or firieg.

si$=$number of Teeppocary ar the stert of cocch matini hiring.

AAiin$\{1,2,3,4\}$

St$=$ number of Tempocing art the stoct of each monthi ficing.

$I_i=$ units of intentory at the ens of eoan month ${?}_i.$

of:$-$

Min $z=50(I_1+I_2+I_3+I_4)+200(S_1^{-}+S_2^{-}+S_3^{-}+s_4^{-})+400(S_1^{+}+S_2^{+}+s_3^{+}+s_4^{+})$

constcuints.

$x_1^{}-I_1=400($inventory constrant$)$

$I_1+x_2-I_2=600($inventory constants$)$

$I_2+x_3-I_3=400($inventory constrant$)$

$I_3+x_4=500($inventory constaint$)$

$x_1=S_1^{-}-S_1^{+}($no$.$ of woiness in $1$ st month$)$

$x_2=x_1+S_2^{-}-S_2^{+}($no$.$ of wolvers in $2^{nd\text{m}\text{o}\text{n}\text{t}\text{h}}$

$x_3=x_2+S_3^{-}-S_3+($no$.$ ot workes in $3^{rd}$ month$)$

$x_4=x_3+S_4^{-}+S_4^{+}($no$.$ of wovers in $4$ th morth$)$

$10$