Chapter 13 Exercise Answers

Chapter 13: Determining Optimal Level of Product Availability
Exercise Solutions

1.

0.2941
Optimal lot-size == NORMINV(0.2941,100,40) = 78.34

Given that p = $200, s = $30, c = $150:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $2,657
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 7.41
Expected understock =
( – O)[1 – NORMDIST((O – )/, 0, 1, 1)] +  NORMDIST((O – )/, 0, 1, 0) = 29.07
EXCEL worksheet 13-1 illustrates these computations

2.

With revised forecasting:

0.2941
Optimal lot-size == NORMINV(0.2941,100,15) = 91.88

Given that p = $200, s = $30, c = $150:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $4,121
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 2.78
Expected understock =
( – O)[1 – NORMDIST((O – )/, 0, 1, 1)] +  NORMDIST((O – )/, 0, 1, 0) = 10.9
EXCEL worksheet 13-2 illustrates these computations
3.
Mean demand during lead time =DL= (2000)(2) = 4000
Standard deviation of demand during lead time = L = = 500= 707
Safety inventory = ROP – DL = 6000 – 4000 = 2000
CSL = NORMDIST (6000, 4000, 707, 1) = 0.9977
Cost of overstocking = (0.25)(40) = $10
Justifying cost of understocking: =
Optimal CSL =
Optimal safety stock = (NORMSINV (0.8889)) (707) = 863 units

EXCEL worksheet 13-3 illustrates these computations

4.

Using the current policy:

0.75
Optimal lot-size == NORMINV(0.75,20000,10000) = 26,745

Given that p = $60, s = $20, c = $30:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $472,889
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 8,236
Using South America option:
0.857
Optimal lot-size == NORMINV(0.857,20000,10000)
= 30,676

Given that p = $60, s = $25, c = $30:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $521,024
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 11,407
So, it is evident that using South America option results in increased expected profits, but also increases the production capacity requirements needed at Champion.
EXCEL worksheet 13-4 illustrates these computations

5.

Current sourcing (one line):

Reguplo:

0.8333
Optimal lot-size == NORMINV(0.8333,10000,1000) =
= 10,967

Given that p = $200, s = $80, c = $100:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $970,018
Each of the other models:

0.7857
Optimal lot-size == NORMINV(0.7857,1000,700) =
= 1,554

Given that p = $220, s = $80, c = $110:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $81,421
Total expected profits = $970,018 + 3($81,421) = $1,214,280
Tailored sourcing policy:
The computations are exactly the same with revised data for Reguplo (c = $90) and for each of the other three models ( c= $120)

Total expected profits = $1,281,670

Thus, it is benefical to utilize the tailored sourcing option due to increased expected profits. This option increases the optimal production lot size for Reguplo and decreases the lot sizes for each of the other three options. EXCEL worksheet 13-5 illustrates these computations

6.
IBM:

0.7447
Optimal lot-size == NORMINV(0.7447,5000,2000) = 6,316

Given that p = $50, s = $3, c = $15:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $144,796
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 1,622
Similarly, the other three are evaluated and the results are summarized below:

Outputs

AT&T
HP
Cisco
Optimal cycle service level 0.7447
0.7447
0.7447
Optimal production size

8,645
5,316
5,447

Expected profits

$207,245
$109,796
$106,776

Expected overstock 2,028
1,622
1,785

Total production lot size = 6316 + 8,645 + 5,316 + 5,447 = 25,723
Total expected profits = $144,796 + $207,245 + $109,796 + $106,776 = $568,612
Total expected overstock = 1,622 + 2,028 + 1,622 + 1,785 = 7,057 (= amount donated to charity on average)

EXCEL worksheet 13-6 illustrates these computations

7.

With aggregation:

Anticipated demand = 5,000 + 7,000 + 4,000 + 4,000 = 20,000
Standard deviation =

0.8889
Optimal lot-size == NORMINV(0.8889,20000,4369)
= 25,333

Given that p = $50, s = $14, c = $18:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $610,210
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 5,568
As can be seen from the results above, postponement increases the expected profit and decreases the amount of overstock.

EXCEL worksheet 13-7 illustrates these computations
8.
(a)

Cost of overstocking, CO = $ 0.50
Cost of understocking, CU = $ 1.00
Mean demand 50,000
Standard deviation of demand = 15,000

Optimal CSL = 0.67

Optimal order quantity = (NORMSINV (0.67))(15,000) + 50,000 = 56,461

(b)

Cost of overstocking, CO = $ 0.50
Cost of understocking, CU = $ 5.00
Mean demand 50,000
Standard deviation of demand = 15,000

Optimal CSL = 0.91

Optimal order quantity = (NORMSINV (0.91))(15,000) + 50,000 = 70,028

EXCEL worksheet 13-8 illustrates these computations

9.

(a)

Mean demand = 5,000
Standard deviation of demand = 2,000
Cost of overstocking, CO $ 40.00
Order size = 6,000

CSL (implied by the order size) = NORMDIST (6000-5000/2000) = 0.691

Implied cost of understocking, CU = (CO)(CSL)/(1-CSL) = (40)(0.691)/(1-0.691) = $89.64

(b)

Mean demand = 5,000
Standard deviation of demand = 2,000
Cost of overstocking, CO $ 40.00
Order size = 8,000

CSL (implied by the order size) = NORMDIST (8000-5000/2000) = 0.933

Implied cost of understocking, CU = (CO)(CSL)/(1-CSL) = (40)(0.933)/(1-0.933) = $558.74

EXCEL worksheet 13-9 illustrates these computations

10.

Current policy:

0.6923
Optimal lot-size == NORMINV(0.6923,4000,1750) = 4879

Given that p = $125, s = $60, c = $80:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $140,001
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 1,224
Southern Hemisphere option:

0.90
Optimal lot-size == NORMINV(0.9,4000,1750) = 6243

Given that p = $125, s = $75, c = $80:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $164,644
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 2,326
EXCEL worksheet 13-10 illustrates these computations
11.
(a)
Mean demand during lead time =DL= (40)(1) = 40
Standard deviation of demand during lead time = L = = 5= 5
Safety inventory = ROP – DL = 45 – 40 = 5
CSL = NORMDIST (45, 40, 5, 1) = 0.8413
Cost of holding one unit for one year = (0.25)(4) = $1
Justifying cost of understocking: =

(b)
Justifying cost of understocking: =
(c)
Desired CSL = = = 0.9909
Desired safety stock = (NORMSINV(0.9909))(5) = 11.8
Desired reorder point = 40 + 11.8 = 51.8
EXCEL worksheet 13-11 illustrates these computations
12.
Without postponement:
For each box:
0.7692
Optimal lot-size == NORMINV(0.7692,20000,8000)
= 25,891

Given that p = $20, s = $7, c = $10:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $168,362
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 6,965
Total expected profits = 4(168,362) = $673,446
Total expected overstock = 4(6,965) = 27,860
Total production quantity = 4(25,891) = 103,564

With postponement:
Anticipated demand = 20,000 + 20,000 + 20,000 + 20,000 = 80,000
Standard deviation =

0.6154
Optimal lot-size == NORMINV(0.6154,80000,16000)
= 84,694

Given that p = $20, s = $7, c = $12:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $560,515
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 9,003
Indifferent:
At a unit cost of $10.7 the two options, i.e., postponement and no postponement would be indifferent. This unit cost is obtained by using the solver option in EXCEL by considering cell 21 as the changing cell while cell 35 is utilized as the target cell with a value of $673,446.

EXCEL worksheet 13-12 illustrates these computations
13.
The with and without postponement calculations are similar to problem 12 (EXCEL worksheet 13-13 illustrates these computations), but what is new in this problem is the tailored postponement which is discussed below:

Tailored postponement:

Popular style without postponement:
0.6818
Optimal lot-size == NORMINV(0.6818,30000,5000)
= 32,364

Given that p = $35, s = $13, c = $20:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $410,757
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 3,396
Other three styles with postponement:
Aggregated expected demand = 8,000 + 8,000 + 8,000 = 24,000
Standard deviation =

0.6182
Optimal lot-size == NORMINV(0.6182,24000,6928)
= 26,083

Given that p = $35, s = $13, c = $21.4:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $268,281
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 18,083
Total expected profit = $410,757 + $268,281 = $679,038
Total expected overstock = 3,396 + 18,083 = 21,479
EXCEL worksheet 13-13 illustrates these computations
14.
Without discount:
0.6842
Optimal lot-size == NORMINV(0.6842,20000,8000)
= 23,836

Given that p = $95, s = $0, c = $30:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $1,029,731
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 5,470
With discount:
Optimal lot-size =25,000

Given that p = $95, s = $0, c = $28:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $1,076,941
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 6,295
Expected profits increase with discount.
EXCEL worksheet 13-14 illustrates these computations
15.
Without discount:
0.7
Optimal lot-size == NORMINV(0.7,70000,25000)
= 83,110

Given that p = $10, s = $0, c = $3:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $403,077
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 17,869
With discount:
Optimal lot-size =100,000

Given that p = $10, s = $0, c = $2.75:

Expected profits = (p – s) NORMDIST((O – )/, 0, 1, 1)
– (p – s) NORMDIST((O – )/, 0, 1, 0) – O (c – s) NORMDIST(O, , , 1)
+ O (p – c) [1 – NORMDIST(O, , , 1)] = $410,974
Expected overstock = (O – )NORMDIST((O – )/, 0, 1, 1) +  NORMDIST((O – )/, 0, 1, 0) = 31,403
Expected profits increase with discount.
EXCEL worksheet 13-15 illustrates these computations

16.
a. the manufacturer should order :
40-Gb
20-Gb
6-Gb
26,772
47,419
84,054 b. The expected profits for the units are:
40-Gb
20-Gb
6-Gb
$1,664,888
$2,048,931
$2,080,846

c. If the available capacity is limited to 140,000 units the manufacturer should order:
40-Gb
20-Gb
6-Gb
26,772
41,300
72,028

expected profits would be:
40-Gb
20-Gb
6-Gb
$1,790,125 $2,072,482 $2,002,170

EXCEL worksheet 13-16 illustrates these computations.