ANSWERfor Exercise 3We wish to find the optimal values of the reorder point R and the lot size Q. In order to get the calculationstarted we need to find the EOQ. However, this requires knowledge of the annual rate of demand, which doesnot seem to be specified. But notice that if the order lead time is six months and the mean lead time demand is100, that implies that the mean yearly demand is 200,giving a value ofR=200. It follows thatEOQ=√2SR/hC=√2∗50∗200/(0.2∗10) =100.The next step is to findROPfrom EquationCSL=F(ROP) =P(Demand≤ROP) =1−QhCbR.SubstitutingQ=100, we obtainP(Demand≤ROP) =1−QhCbR=1−100∗0.2∗1025∗200=0.96.Since the demand is normally distributed,ROP=Norminv(0.96, 100, 25) =144.As a result, we can run place orders of sizeQ=100 whenever inventory level is belowROP=144.ANSWERfor Exercise 4a) Because of independence, the sum of the demands over 4 2-hour blocks remain to be normally distributedand its mean and variance are 24=6+6+6+6, 4=1+1+1+1. The daily demand hasN(24, 22)distribution.b) With 3 coffee bags everyday, 30 cups are prepared.The stockout happens when these 30 cups are notenough. That probability is given by P(N(24, 22)≥30) =1−P(N(24, 22)≤30) =1−normdist(30, 24, 2, 1) =1−0.99865.c) With 3 coffee bags, the safety stock is 30-24=6. We now need to compute the expected number of stockoutsper dayESC=E(max{0,N(24, 22)−30})=−ss[1−normdist(ss/σ, 0, 1, 1)] +σnormdist(ss/σ, 0, 1, 0)=−6[1−normdist(6/2, 0, 1, 1)|{z}0.99865] +2normdist(6/2, 0, 1, 0)|{z}0.004432=0.00076416

The expected shortage per day is low as consequence of high safety stock of 6.If we reduce the safety stock down to 4, we can see what happens to the corresponding ESC:ESC=−4[1−normdist(4/2, 0, 1, 1)|{z}0.97725] +2normdist(4/2, 0, 1, 0)|{z}0.053991=0.016981With a safety stock of 2, ESC grows further to 0.166631.ANSWERfor Exercise 5a) The demand takes values from{21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}with equal probability (of 1/12) soit has discrete uniform distribution.b) The overstocking and understocking costs areco=10 dollars andcu=30 dollars. We need to solve forP(Demand≤Q) =cu/(cu+co) =0.75=9/12. This equality yieldsQ=29.c) The marginal cost of ordering the 30th book iscoP(Demand≤29) =10(9/12) =15/2. If you also wantto compute the marginal benefit of ordering the 30th book, thencuP(Demand≥30) =30(3/12) =15/2.In this case of discrete demand, the equivalence of marginal cost and benefit is a coincidence. In the case ofcontinuous demand distribution, this equivalence is actually the optimality condition.d) The total cost of overstocking and understocking ise) The manager can consider decreasing the cost overstocking (with a higher buyback price), decreasing thecost of understocking (with transshipments from other bookstores) and fast response from Prentice-Hall sothat two or more orders can be placed in a single semester.TC(Q=29)=coE(max{0, 29−Demand}) +cuE(max{0, Demand−29})=10[(29−28)/12+ (29−27)/12+· · ·+ (29−21)/12]+30[(32−29)/12+ (31−29)/12+ (30−29)/12]=10[3] +30[0.5] =45.