Value of Stock and Profit Calculation

(10)1. At the beginning of December, XYZ Company employed a contractor to cut enough
trees to meet the expected demand for Christmas trees. They sell these trees to a local
wholesaler in batches of 100. Over the past few years, the demand has been as follows,
Batches 1 2 3 4 5 6 7 8 9
Probability 0.03 0.09 0.11 0.17 0.22 0.16 0.13 0.05 0.04
If it costs $14 to cut and trim a tree that sells for $30, how many trees should the company cut
down? What is the expected profit?
 Given,
Unit Cost (UC) = $12
Sales Price (SP) = $30
No scrap (SV) = $0
UC
SP = 12
30 = 0.4
Q P (D ≥ Q) P (D)
1 1 0.03
2 0.97 0.09
3 0.88 0.11
4 0.77 0.17
5 0.60 0.22
6 0.38 0.16
7 0.22 0.13
8 0.09 0.05
9 0.04 0.04
Select Qo value such that it satisfies the below equation,
P (D ≥ Qo) > UC
SP > P (D ≥ Qo + 1)
From the above table the two adjacent values of Q that enclose 0.4 are 5 and 6. Therefore Q
= 5.
P (D ≥ 5) > UC
SP > P (D ≥ 6)
0.60 > 0.4 > 0.38
Hence, it is optimal to cut 5 batches. Then the expected profit will be,
EP(Q) = SP x [∑
D=0
Q
D x P ( D ) +Q x ∑
D =Q+1
∞
P(D)¿ ¿ - Q x UC

EP (5) = 30 x [2.32 + 5(0.38)]- 5 x 12
= 30 x [4.22] – 60
= 66.60
For batch of 100 trees the profit is 100 * 66.60 = $6,660.00
(15)2. The demand for an item follows a Poisson distribution with mean 9.5 units a month.
The lead time is one month, shortage cost is $140 a unit a month, reorder cost is $50, and
holding cost is $6 a unit a month. Calculate optimal values for the order quantity and reorder
level.
 Demand = 9.5 units, Lead time = 1 month, Reorder Cost = $50, Shortage Cost = $140,
Holding Cost = $6 a unit a month.
Now,
Qo = √ 2 × RC × D
HC = √ 2 ×50 × 9.50
6 = 12.583 units
HC × Q
SC × D = 6 ×12.583
140× 9.5 = ∑
D=ROL
∞
Prob(D) = 0.0567
We want the probability that demand is greater than the reorder level to be 0.0567. As
demand follows a Poisson distribution with mean 9.5, we can find the cumulative
probabilities as follows:
VALUES PROPABILITY PROB(D≥VALUE)
0 0.00007485 0.99992515
1 0.00071109 0.99921406
2 0.00337769 0.99583637
3 0.01069601 0.98514035
4 0.02540303 0.95973732
5 0.04826577 0.91147155
6 0.07642080 0.83505076
7 0.10371394 0.73133682
8 0.12316030 0.60817652
9 0.13000254 0.47817398
10 0.12350241 0.35467157
11 0.10666117 0.24801039
12 0.08444010 0.16357029
13 0.06170622 0.10186407
14 0.04187208 0.05999199
15 0.02651898 0.03347300
16 0.01574565 0.01772736
17 0.00879904 0.00892832

18 0.00464394 0.00428438
19 0.00232197 0.00196241
20 0.00110293 0.00085948
21 0.00049895 0.00036053
22 0.00021545 0.00014508
23 0.00008899 0.00005609
24 0.00003523 0.00002086
25 0.00001339 0.00000747
26 0.00000489 0.00000258
27 0.00000172 0.00000086
28 0.00000058 0.00000028
29 0.00000019 0.00000009
30 0.00000006 0.00000003
The poisons distribution is discrete and we want to find the cumulative probability nearest to
0.050. We can see that this corresponds to a value of 14. This is our initial estimate of reorder
level.
Now we substitute this value into equation of Q to find the revised value of Q.
Q= √ 2 × D
HC × ¿¿
Summation is done as follows: -
D PROB(D) D-ROL PROB(D) x (D-ROL)
14 0.04187208 0 0.00000000
15 0.02651898 1 0.02651898
16 0.01574565 2 0.03149129
17 0.00879904 3 0.02639711
18 0.00464394 4 0.01857575
19 0.00232197 5 0.01160984
20 0.00110293 6 0.00661761
21 0.00049895 7 0.00349263
22 0.00021545 8 0.00172363
23 0.00008899 9 0.00080093
24 0.00003523 10 0.00035226
25 0.00001339 11 0.00014724
26 0.00000489 12 0.00005869
27 0.00000172 13 0.00002237
28 0.00000058 14 0.00000817
29 0.00000019 15 0.00000287
30 0.00000006 16 0.00000097
TOTAL 0.12782036
Hence,
Q = √2× 9.5
6 ×[50+(140 ×0.12782036)] = 14.662

Now we find the new value of ROL by substituting the value for Q in following equation: -
HC × Q
SC × D = 6 ×14.662
140 × 9.5 = 0.0661
Hence, this corresponds to the reorder level of 14, so we accept it.
Reorder quantity = 15 (rounded)
ROL= 14
Using this value in variable cost per unit time equation tells us: -
VC = RC × D
Q + HC x [ROL-LT x D + Q
2 ] + SC × D
Q x ∑
D−ROL
∞
( D−ROL ) × PROB( D)
VC = 50× 9.5
15 + 6 x [14-1 x 9.5 + 15
2 ] + 140× 9.5 ×0.12782036
15
VC = $115 a month.
(15)3. ABC Company supplies an item that has a mean demand of 220 units a week and
standard deviation of 25 units. Stock is reviewed every 4 weeks and lead time is constant at 2
weeks. Each unit cost $7.50 a week to store. At 95% service level, what should be the
company’s practice for this particular item and what is the annual cost of this item? What will
change if the service level is set at 98%? (Unit cost = $100, Reorder Cost = $400/order)
 Demand = 220 units a week = 220 * 52 =11,440 units a year
σ = 25
Holding cost = $7.50 per unit per month = $90 per unit per year
T = 4 weeks = 0.07 years
Lead time = 2 weeks = 0.038 years
For 95% safety stock Z = 1.64
Hence,
safety stock = Z × σ × √T +¿
= 1.64 × 25 ×√4 +2
= 100.43 units per week
Target stock level = D × (T+LT) + safety stock
= 220 × (4+2) + 100.43
= 1420.43 units week
Each month, when it’s time to order, the company examines the stock in hand and places an
order of