Question

For a newsvendor product the probability distribution of demand X (in units) is as follows:

xi 0 1 2 3 4 5 6
pi 0.05 0.1 0.2 0.3 0.2 0.1 0.05

The newsvendor orders Q = 4 units.

a) Derive the probability distributions and the cumulative distribution functions of lost sales as well as leftover inventory.

b) Knowing that the expected total cost function is convex in the order quantity Q, demonstrate that Q = 4 gives the minimal expected total cost.

Answer 1

Here newspaperd orders Q = 4 units

(a) So first we take the case of lost sales.

Here lost sales will occur only when Demand is greater than 4 units

so here if lost sales = LS

LS = Q - 4 when Q > = 4

= 0 when Q < 4

then for demand equal or less than 4, Lost sales = 0

for Q = 5, LS = 1

for Q = 6, LS = 2

p(LS) = (0.05 + 0.1 + 0.2 + 0.3 + 0.2) = 0.85 ; LS = 0

= 0.1 ; LS = 1

= 0.05 ; LS = 2

CDF of Lost sales

P(LS) = 0 ; LS < 0

= 0.85 ; 0 LS < 1

= 0.95 ;  1 LS < 2

= 1; LS 2

Now for Leftover inventory

if demand is HIgher than or equal to 4 than leftover inventory would be 0

and if lower than 4

then LI = 4 - Q

f(LI) = (0.2 + 0.01 + 0.05) = 0.35 ; LI = 0

= 0.3 ; LI = 1

= 0.2 ; LI = 2

= 0.1 ; LI = 3

= 0.05 ; LI = 4

CDF of LI

F(LI) = 0 ; LI < 0

= 0.35 ; 0 LI < 1

= 0.65 ; 1 LI < 2

= 0.85 ; 2 LI < 3

= 0.95 ; 3 LI < 4

= 1 ; LI >= 4

(b) here expected total cost function is convex in order quantity Q, that means a quadratic model.

For Q = 4, we can see that total cost including the leftover inventory cost and lost sales cost would be minimal.