数据、模型与决策(运筹学)课后习题和案例答案018.doc

数据、模型与决策(运筹学)课后习题和案例答案018

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CD CHAPTER 18
INVENTORY MANAGEMENT WITH KNOWN DEMAND
Review Questions

18.1-1 Administrative costs, such as labor charges to place an order， can cause the cost to exceed the purchase price。
18.1-2 Capital tied up in inventory will not be regained until the goods are sold。 If the capital were free to be invested it would be able to earn a return from other opportunities. This cost is called an opportunity cost because it reflects the lost return because alternate opportunities must be foregone to carry inventory。
18.1-3 Other costs associated with holding inventory include the cost of leasing warehouse space, the cost of insurance for inventory, labor costs for warehouse personnel, and taxes based on the value of the inventory。
18。1—4 Customer dissatisfaction and the loss of future sales， price decreases to compensate for delays， delayed revenue, and increased record keeping are all cost consequences for inventory shortages.
18.2—1 The four cost components that may be included in an inventory model are acquisition cost， setup cost， holding cost， and shortage cost。
18.2-2 Retailers and wholesalers replenish their inventory and incur a direct cost by purchasing products。 Manufacturers replenish their inventory and incur a direct cost by manufacturing more of the product involved。
18。2—3 When the replenishment is done by purchasing the product， the setup cost consists of the various administrative costs. When a manufacturer is replenishing its inventory, the setup cost consists of the cost of setting up the manufacturing process for another production run。
18。2-4 An inventory policy prescribes both when inventory should be replenished and by how much。
18.2—5 The total inventory cost per unit time needs to be minimized to determine an optimal inventory policy.
18。2-6 A fixed cost is a cost that remains the same regardless of the decisions made while a variable cost is a cost that is affected by the decisions made。 The variable costs are the only relevant costs since they are the only costs that can be decreased by improving the decisions.
18。3—1 The basic EOQ model is so popular due to a combination of simplicity and wide applicability.
18。3—2 The model assumes a constant demand rate， that the order quantity to replenish inventory arrives all at once just when desired, and that planned shortages are not allowed. The model is sometimes used when these assumptions are not completely satisfied as in the case where constant demand is only a reasonable approximation.
18.3-3 Lead-time is the amount of time between the placement of an order and the delivery of the order quantity. The reorder point is the inventory level at which an order is placed。
18。3-4 In a continuous—review inventory system the current inventory level is monitored on a continuous basis while in a periodic-review system the inventory level is checked only periodically.
18。3—5 A continuous—review inventory system will not fit the basic EOQ model if shortages often occur and it is necessary to carry safety stock.
18。3-6 The single decision variable for the model is the order quantity, Q。
18。3—7 The shape of the pattern of inventory levels over time is a saw—toothed pattern。
18.4—1 The two types of costs included in the total variable cost are the annual setup cost and the annual holding cost。 These costs are equal at the point where the order quantity is optimal。
18。4—2 The optimal order quantity increases if the demand rate increases in order to avoid overly large increases in the number of setup costs incurred per year。 The optimal order quantity increases if the setup cost increases in order to decrease the number of times this setup cost will be incurred per year. The optimal order quantity decreases if the unit holding cost increases in order to drive down the average inventory level。
18。4-3 The optimal order quantity can change fairly significantly if fairly small changes are made to either the setup cost of the unit holding cost. This change is even larger if the changes are made to both costs in opposite directions.
18.4-4 The optimal order quantity does not change if both the setup cost and the unit holding cost are changed by the same percentage amount in the same direction.
18.4—5 A fairly small error in estimating either the setup cost or the unit holding cost will not increase the total variable cost very much. The same is true if the error occurs in both costs。
18。5-1 If the cost of holding inventory is high relative to the shortage costs then it may make sense to permit planned inventory shortages。
18。5—2 The assumptions for the EOQ model with planned shortages are the same as for the basic EOQ model except that planned shortages are allowed.
18。5-3 The decision variables for the EOQ model with planned shortages are the order quantity Q and the maximum shortage S.
18。5—4 The total variable cost for this model includes annual setup costs, annual holding costs and annual shortage costs。
18.5—5 The optimal order quantity for this model is larger than for the basic EOQ model。 The maximum inventory level for this model always will be less than for the basic EOQ model.
18。5-6 Management objects to planned shortages because they will hurt the reputation the company has for providing good customer service.
18。6-1 Quantity discounts are reductions in the unit acquisition costs of a product that are offered for ordering a relatively large quantity.
18。6—2 When quantity discounts are offered, annual acquisition costs need to be included in the total variable inventory cost.
18。6-3 The unit holding cost is equal to the price paid for the items in inventory multiplied by the inventory holding cost rate （h = Ic）。
18.6-4 The best order quantity for a discount category whose minimum order quantity exceeds Q from the basic EOQ model is the minimum allowed。 When the maximum order quantity is less than the Q from the basic EOQ model then the best order quantity is the maximum allowed。
18。7—1 The replenishment of inventory over time is common with manufacturers who replenish their inventory by conducting intermittent production runs。
18。7—2 The assumptions for this model are the same as those for the basic EOQ model except that a production run is scheduled to begin each time the inventory level drops to 0， and this production run replenishes inventory at a constant rate throughout the duration of the run。
18.7—3 The maximum inventory level is less than the production lot size because products are being withdrawn from inventory as the production run is under way。
18.7—4 The square root formula for this model contains the annual production rate if producing continuously R.
18。7—5 Independent—demand products have demand that does not depend on the demands for all products。 Dependent—demand products have demand that is dependent upon the demand for another product， generally because the former product is a component of the latter product。
18。7—6 Material requirements planning (MRP） is a popular technique for managing inventories of components of a final product.
18。7-7 A just-in—time inventory system places great emphasis on reducing inventory levels to a bare minimum。
18。7—8 In more general terms, the focus of the just—in-time philosophy is on avoiding waste wherever it might occur in the production process.
Problems

18。1 a）

b）

c)

d）

The results are the same as those obtained in part c。
e) = 13 conputers purchased with each order.
f) Number of orders per year = D / Q = 676 / 13 = 52.
Inventory level at which order is placed = ROP = D（LT) = (13)（1/2） = 6。5
g） The optimal policy reduces the total variable inventory cost by $3，840 per year, which is a 33% reduction。
18。2 a)

b）

c）

d）

The results are the same as those obtained in part c.
e) = 41,231 gallons purchased with each order.
18.3 a) The order quantity should be Q = 1，442; the reorder point should be ROP = 100. The corresponding variable inventory cost is TVC = ＄288。

b) The new order quantity should be Q = 1，665 with TVC = $250. The order quantity is increased by 223 LCDs。

If the order quantity obtained in part a is still used the TVC increases to ＄252。

c） The new order quantity should be Q = 1，290 with TVC = ＄322。 The order quantity is decreased by 152 LCDs.

If the order quantity obtained in part a is still used the TVC increases to $324。

d）

e）

f）

18.4 Optimal Order Quantity：

Total Variable Cost （with Q = Q*)：

Total Variable Cost （with Q = 573）:

The conclusions remain basically the same when the estimates could be off by as much as 25%. In the cases where K and h change in the same direction, there is little or no extra cost incurred. When K and h change in the opposite direction extra cost is incurred， and this extra cost is more than it was for the 10％ increase。
18。5 a） Q＊ will decrease by half.
b） Q＊ will double。
c） Q* remains the same.
d) Q* will double。
e） Q＊ remains the same.
18.6 a)
This is 15% of the acquisition cost per month or 180% per year。
b） The optimal order quantity when h = (20%)（$20) = $4 is 150。 The annual TVC = ＄600.

With the current inventory policy （Q = 50）, the annual TVC = ＄1,000.

c) Reorder point is 10 as shown in the above spreadsheet.
d) The new reorder point is 5 + 10 = 15, which adds ＄20 to his TVC (5 hammers x $4 holding cost）。
18.7
18.8 TVC = K(D/Q） + h（Q/2）

18.9 a) The optimal order quantity is Q = 245 with annual TVC = $4，899。

b) The new optimal order quantity is Q = 316, with a maximum shortage of S = 126.

c）
Quantity
Basic EOQ Model
EOQ Model with Planned Shortages
Order quantity
245
316
Maximum shortage
0
126
Maximum inventory level
245
190
Reorder point
0
-126
Annual setup cost
＄2，449
＄1，897
Annual holding cost
＄2,449
＄1，138
Annual shortage cost
＄0
＄759
Total variable cost
＄4,899
$3,795

18。10 a) p = $15

p = $30

p = $60

p = $120

b）
p
TVC
% reduction from basic model
＄15
＄2，128
12%
$30
$2，255
6％
$60
$2，327
3％
＄120
$2，366
1。5%

c)
p
Max。 wait time (days)
Acceptable case
$15
5。9

$30
3。1

＄60
1.6
*
＄120
0。8
＊

18。11 a) The TVC is reduced from $7，800 to ＄3，904， a reduction of ＄3,896 (about 50%）。

b)

c）

18。12
Ratio of
p to h

Maximum
Inventory Level
Maximum
Shortage
1/3
2，000
500
1,500
1
1，414
707
707
2
1，225
816
408
3
1，155
866
289
5
1，095
913
183
10
1,049
953
95

18.13 a）

b） Orders placed per year = D/Q = 5，200 / 500 = 10。4。
Time interval between orders = Q / D = 500 / 5,200 = （0.096 years）(52) = 5 weeks.
18.14 a)

b) Orders placed per year = D/Q = 365/100 = 3.65.
Time interval between orders = Q/D = 100/365 = (0。274 years）（52) = 14。25 weeks.
18.15 a)
Discount Category
TVC = cD + K(D/Q) = h（Q/2）
1
TVC = (＄8。50)（400) + ($80）（400/Q） + (0。2)(＄8。50）（Q/2）
2
TVC = ($8）（400) + （$80)(400/Q） + (0。2)（＄8）(Q/2)
3
TVC = （$7.50)（400） + ($80)(400/Q) + （0。2）(＄7。50）（Q/2）

b）
Discount Category

1

2

3

c)
Discount Category
Feasible
Q
TVC = cD + K（D/Q） = h（Q/2)
1
99
$3，807
2
200
＄3，520
3
1,000
＄3,782

d)

e） Q* = 200 with a TVC of ＄3,520。
f)

g) Since the value of Q that minimizes TVC for discount category 2 is feasible that means that this order quantity minimizes the annual setup and holding costs. Category 1 could therefore not possibly have lower annual setup and holding costs。 Furthermore， since the purchase price per case is higher for category 1， it could not possibly have lower purchasing costs。 Thus, we can eliminate category 1 as a candidate for providing the optimal order quantity。
h) Orders placed per year = D/Q = 400/200 = 2。
Time interval between orders = Q/D = 200/400 = 0.5 years or 6 months。
18.16 a）
Discount Category
TVC = cD + K（D/Q) = h(Q/2）
1
TVC = (＄1)(2,400) + (＄4)（2，400/Q) + (0。17)（$1）(Q/2)
2
TVC = （＄0。95)(2,400） + (＄4）(2，400/Q） + （0。17）（＄0.95）(Q/2)
3
TVC = （＄0。90)（2,400) + （＄4)(2，400/Q） + （0。17）($0.90）(Q/2）

b）
Discount Category

1

2

3

c)
Discount Category
Feasible
Q
TVC = cD + K(D/Q） = h(Q/2)
1
199
＄2，465
2
345
$2,336
3
500
$2，217

d）

e） Q＊ = 500 with a TVC of ＄2，217.
f）

g） Since the value of Q that minimizes TVC for discount category 2 is feasible that means that this order quantity minimizes the annual setup and holding costs. Category 1 could therefore not possibly have lower annual setup and holding costs. Furthermore, since the purchase price per bag is higher for category 1, it could not possibly have lower purchasing costs。 Thus, we can eliminate category 1 as a candidate for providing the optimal order quantity。
h） Orders placed per year = D/Q = 2,400 / 500 = 4.8.
Time interval between orders = Q/D = 500 / 2，400 = (0。21 years）（12) = 2。5 months.
18。17 a) The production lot size should be Q = 1，000. The annual setup cost is $45，000; the annual holding cost is $45，000; the total variable cost is therefore ＄90,000.

b） Production run duration = Q/PR = 1，000 / 24，000 = 0.042 years = 0。5 months
Time interval between production runs = Q/D = 1，000 / 6,000 = 0.167 years = 2 months.
c） Maximum inventory level = Q – （D/PR)Q = 1,000 – （6，000 / 24，000）（1，000) = 750。 This is less than the production lot size since monitors are being withdrawn from inventory while a production run is going on。
18。18 a） The order quantity should be Q = 4，000。
The annual total variable cost is TVC = ＄26,000。

b） Time interval between orders = Q/D = 4,000 / 52,000 = 4/52 years = 4 weeks。
Delivery duration = Q / PR = 4,000 / 104，000 = 2/52 years = 2 weeks.
c） ROP = D(LT） = 52，000 （1/52） = 1，000。
18。19 a) Setup Cost K = $9,000。 The optimal Q decreases from 50，000 to 43，301.

Setup Cost K = ＄15，000. The optimal Q increases from 50，000 to 55，902。

Holding cose h = ＄2。70。 The optimal Q increases from 50,000 to 57，735.

Holding cost h = $4。50。 The optimal Q decreases from 50，000 to 44,721。

b) K = $9，000； h = $2.70。 The optimal Q remains 50，000。

K = ＄9，000； h = $4。50。 The optimal Q decreases from 50,000 to 38,730.

K = $15,000； h = $2。70。 The optimal Q increases from 50，000 to 64，550。

K = $15，000； h = $4。50。 The optimal Q remains at 50，000。

c） The value of Q* is fairly sensitive to the estimates of K and h in all cases except when both K and h are increased or decreased by the same proportional amount.
d) Optimal Order Quantity:

Total Variables Cost （with Q = Q*）:

Total Variable Cost (with Q = 50,000)：

e） Amount that TVC with Q = 50,000 exceeds TVC with Q = Q*：

Unit Holding Cost （h）