Question

please I need the answer ( step by step).

H15, Jill, the office manager of a desktop publishing outfit, stocks replacement toner cartridges for aser printers. Demand for cartridges is approximately 30 per year and is quite variable (i.e., can be represented by the Poisson distribution). Cartridges cost $100 each and require 3 weeks to obtain from the vendor. Jill uses a (Q, r) approach to control stock levels. (a) If Jill wants to restrict replenishment orders to twice per year on average, what batch size Q should she use? Using this batch size, what reorder point r should she use to ensure a service level (i.e., probability of having the cartridge in stock when needed) of at least 98 percent? (b) If Jill is willing to increase the number of replenishment orders per year to six, how do Q and r change? Explain the difference in r. (c) If the supplier of toner cartridges offers a quantity discount of $10 per cartridge for orders of 50 or more, how does this affect the relative attractiveness of ordering twice per year versus six times per year? Try to frame your answer in definite economic terms. 16. Moonbeam-Mussel (MA

Answer 1

(a)

Number of orders per annum = 2

So, average batch size, Q = 30 / 2 = 15 units

Average demand during lead time, θ = 30 * (3/50) = 1.8 units (assuming one year=50 weeks)

The desired Fill rate = 98%

We know,

Expected shortage per cycle = Q x (1 - Fill rate)
or, Expected shortage per cycle = 15*(1 - 0.98) = 0.30

Now, approximate Poisson to Normal as μ_LTD = 1.8 units and σ_LTD = √1.8 = 1.34

So,

L(z) . σ_LTD = 0.30
or, L(z) = 0.30 / 1.34 = 0.224

Look up a normal distribution loss function table and the corresponding z will be z = 0.42

So, appropriate reorder point, r = μ_LTD + z.σ_LTD = 1.8 + 0.42*1.34 = 2.36 units i.e. approximately 3 units.

(b)

Number of orders per annum = 6

So, average batch size, Q = 30 / 6 = 5 units

Average demand during lead time, θ = 30 * (3/50) = 1.8 units (assuming one year=50 weeks)

The desired Fill rate = 98%

We know,

Expected shortage per cycle = Q x (1 - Fill rate)
or, Expected shortage per cycle = 5*(1 - 0.98) = 0.10

So,

L(z) . σ_LTD = 0.10
or, L(z) = 0.10 / 1.34 = 0.075

Look up a normal distribution loss function table and the corresponding z will be z = 1.05

So, appropriate reorder point, r = μ_LTD + z.σ_LTD = 1.8 + 1.05*1.34 = 3.2 units i.e. approximately 4 units.

(c)

It is infeasible to attain this discount with 2 or 6 orders in a year because the overall annual demand is 30 units and the minimum order size for this discount is 50 units.