Question

Please show all steps. Consider the following data pertaining to the three warehouses operated by a company:

Warehouse #	Std. deviation of lead time demand
1	845
2	410
3	225

The company considers consolidating any two of its warehouses (i.e., risk pooling). Based on

the information provided above and given that lead time demands between any two

warehouses are correlated at -0.3 (i.e.,ρ12=ρ23=ρ13= -0.3). In which case will the reduction

in safety stocks resulting from warehouse centralization likely be the greatest if we want to

achieve a 97% service level? (Please provide detailed calculation processes. Typed also)

Answer 1

Answer:

Let us consider the three security stock of 3 warehouses be ss1,ss2, and ss3, three warehouses be w1, w2, w3 respectively and two warehouses after consolidation be w12, w23 and w31 individually.

As we know,

Variance (X1+ X2) = Var (X1)+Var (X2) + Covariance of (X1,X2)

Where, X1 and X2 are two warehouses being consolidated and

corerelation = (covariance) / σ 1 * σ 2

Therefore, after union, lets consider the covariance of consolidated warehouse 1 and 2 = covariance12, .... also, covariance23, covariance 31, for others individually.

covariancel2 = cor-relation(σ l * σ 2) = (-0.3) *845*410 = -103935

covariance23 = co-relation(σ2 * σ 3) = (-0.3) *410*225 = -27675

covariance31 = cor-relation(σ 1 * σ 3) = (-0.3)*845*225 = -57037.5

Variance (wl,w2 ) = σ 12 + σ 22 + 2.covariancel2

= 8452 + 4102+ 2*(-103935) = 674255

Along these std dev for warehouse w12 = square root (674255) = 821.13

Along these safety stock for ss12 = 821.13 *1.88 = 1543.72

Similarly,

Variance (w2,w3) = σ 22 + σ 32 + 2.covariance 23

= 4102 + 2252+ 2*(- 27675) = 170125

along these std dev for warehouse w23 = square root (170125) = 412

Along these safety stock for ss23 = 412 *1.88 = 774.56

Additionally, Variance ( w1,w3) = σ 12 + σ 32 + 2.covariance 23

= 8452 + 2252+ 2*(- 57037.5)

= 650575

along these std dev for warehouse w13 = square root (650575) = 806

Therefore safety stock for ss13 = 806 *1.88 = 1515.28

As we definitely realize that the safety stock of present warehouse are :

ss1 = 845*1.88 =1588.6

ss2 = 410* 1.88 =770.8

ss3 = 225*1.88 = 423

Therefore decrease in safety stock if wi and w2 are combined

= 1588.6+770.8-1543.72 = 815.68

Therefore decrease in safety stock if w2 and w3 are consolidated

= 770.8+ 423 - 774.56 = 419.24

Therefore decrease in safety stock if w3 and w21 are consolidated= 496.38

Therefore biggest decrease in safety stock is seen when w1 and w2 are consolidated.