Question

10. A bakery sells rolls in units of a dozen. The demand X (in 1000 units) for rolls has a gamma distribution with parameters a = 3,0 = 0.5, where 0 is in units of days per 1000 units of rolls. It costs $2 to make a unit that sells for $5 on the first day when the rolls are fresh. Any leftover units are sold on the second day for $1. If w denotes the number of units to be made, then the above information suggests that the net profit u(x) as a function of the demand x (in units of days per 1000 units of rolls) can be modeled by 5000 u(x) = ? F / W 1000 + - - where we conveniently ignored the case 1000/x > w for simplic- ity. Compute E[u(X)] and determine the number of units that should be made to maximize the expected value of the net profit.

Answer 1

Griver that :- 2=3 0 =0.5 The random varioble & follows gamma dist with parameter 2-3 0-0.5 $2 to make a cenit that sells for 85 on the first day when the rolle are fresh. The profft as the 1st day at $1 Now , find Lp (X (1)) and P(x+1) P.(ncl) - Gamma (x,x, 0) - Gamma (1,31095). (from .f distr) - 012332 P(X>7) = 1- Gamma (x,2,0) - 1-0.32332 0: 67668 The prob distintable Dayl $3 as shown below Day 2 profit probability 0.67668 0.32332

The expected value of the profil e(x) - ($3* 0.67668) + (-81X0.32.392) = $1.70672 There fore, the expected value of the profit days. $ 1.70 672