5. Consider an inventory problem described by the following date: k : cost per item stocked, ky: : revenue per item sold, kz: scrap value per item stocked and not sold. Define q: amount stocked, d: demand; assume that v;(q,d)=-kq+k, min{q,d}+kz max{q d,0}. Show first that a. max{q-d,0}=q- min{q,d} and then substituting (a) into the expression for v,(q,d) b. v,(q,d)=(-k, +k3q +(k k)min{q,d} Show further that c. = dsq , (q) = Ev, (q,)= (-k, + ks )q+(k k;)E"q)+qPl >q)] where E" (q) = {dP = d) d. v,(q+1)- v(a)=(-k, +k; )+(k- kg)Pl 2q+1) [Hint: E (q+1)= E" (q)+(q +1)P(7 =q+1).] e. v, (9+1)> v,() Pl 2q +1)> ", k k - k = >