Question

Weekly demand for private label washing machines at Karstadt, a German department store chain, is normally distributed with a mean of 500 and a standard deviation of 300. Karstadt currently has a supply source in China that delivers machines at a cost of 200 euro. The lead time required by the supplier is normally distributed with a mean of nine weeks and a standard deviation of six weeks. A European supplier has offered to deliver washing machines with a guaranteed lead time of one week at a cost of 210 euro. Karstadt has a holding cost of 25 percent and targets a cycle service level of 99 percent. Should Karstadt accept the local supplier’s offer?

Answer 1

d = average weekly demand = 500, So, annual demand (D) = 500*52 = 26000
s = std. dev. of weekly demand = 300

For the Chinese supplier (1)
c1 = cost of a machine = 200 euro, so, h1=holding cost per annum = 200 x 25% = 50 euro
t1 = average lead time = 9 weeks
s_t1 = std. dev. of lead time = 6 weeks

For the European supplier (2)
c2 = cost of a machine = 210 euro, so, h2=holding cost per annum = 210 x 25% = 52.5 euro
t2 = average lead time = 1 week
s_t2 = std. dev. of lead time = 0

Assume an ordering cost = S which is same for both the suppliers

EOQ
Q*₁ = SQRT(2*26000*S/50) = SQRT(1040*S)
Q*₂ = SQRT(2*26000*S/52.5) = SQRT(990.5*S)

Cycle Stock Inventory Cost
(Q*₁/2) x h1 = 25*SQRT(1040*S)
(Q*₂/2) x h2 = 26.25*SQRT(990.5*S)

Standard deviation of lead time demand
s_ltd1 = SQRT(s².t1 + (d.s_t1)²) = SQRT(300*300*9 + ((500*6)^2)) = 3132.09
s_ltd2 = SQRT(s².t2 + (d.s_t2)²) = SQRT(300*300*1 + 0) = 300

Safety stock inventory cost
Z*s_ltd1*h1 = NORMSINV(0.99)*3132.09*50 = 364317 euro
Z*s_ltd2*h2 = NORMSINV(0.99)*300*52.5 = 36640 euro

Ordering cost
(D/Q*₁)*S = 26000*S/SQRT(1040*S)
(D/Q*₂)*S = 26000*S/SQRT(990.5*S)

Cost of purchase
D*c1 = 26000*200 = 5200000 euro
D*c2 = 26000*210 = 5460000 euro

We sum up all the costs (note that 'S' is still unknown) for these two suppliers in the following format -

We can now take a fairly large range of 'S' to check the output i.e. the $C$9 cell using the sensitivity analysis. The output is shown below.

Value of 'S'	Min. cost supplier
1	European
200	European
400	European
600	European
800	European
1000	European
1200	European
1400	European
1600	European
1800	European
2000	European
2200	European
2400	European
2600	European
2800	European
3000	European

So, for almost any value of 'S', the European supplier is the best option. So, the orrder may be accepted.