Order 1270563: inventory management

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Managing

Inventories

Chapter 9

9-‹#›

What is a Inventory Management?

Inventory Management

The planning and controlling of inventories to meet the competitive priorities of the organization.

9-‹#›

What is Inventory?

Inventory

A stock of materials used to satisfy customer demand or to support the production of services or goods.

9-‹#›

Inventory Trade-Offs

Scrap flow

Inventory level

Output flow of materials

Input flow of materials

Figure 9.1

9-‹#›

Pressures for Small Inventories

Inventory holding cost

Cost of capital

Storage and handling costs

Taxes

Insurance

Shrinkage

Pilferage

Obsolescence

Deterioration

9-‹#›

Pressures for Large Inventories

Customer service

Ordering cost

Setup cost

Labor and equipment utilization

Transportation cost

Payments to suppliers

9-‹#›

Types of Inventory

Accounting Inventories

Raw materials

Work-in-process

Finished goods

9-‹#›

Types of Inventory

Figure 9.2

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Types of Inventory

Operational Inventories

Cycle Inventory

Safety Stock Inventory

Anticipation Inventory

Pipeline Inventory

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Cycle Inventory

Lot sizing principles

The lot size, Q, varies directly with the elapsed time (or cycle) between orders.

The longer the time between orders for a given item, the greater the cycle inventory must be.

Average cycle inventory = =

Q + 0

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Cycle Inventory

09- 11

d = Average demand per time period = Q/P

= slope of the consumption linear function

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Safety Stock Inventory Anticipation Inventory

09- 12

Safety stock inventory is surplus inventory that protects against uncertainties in demand, lead time, and supply changes.

Inventory used to absorb uneven rates of demand or supply, which businesses often face, is referred to as anticipation inventory

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Pipeline Inventory

Average demand during lead time = DL

Average demand per period = d

Number of periods in the item’s lead time = L

Pipeline inventory = DL = dL

Assume that every period P we order a quantity Q.

The pipeline inventory is Q during L and 0 during P-L.

The average inventory is therefore:

[Q⨯L+0⨯(P-L)]÷[L+(P-L)] = (Q/P) L = dL

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Example 9.1

A plant makes monthly shipments of electric drills to a wholesaler in average lot sizes of 280 drills. The wholesaler’s average demand is 70 drills a week, and the lead time from the plant is 3 weeks. The wholesaler must pay for the inventory from the moment the plant makes a shipment. If the wholesaler is willing to increase its purchase quantity to 350 units, the plant will give priority to the wholesaler and guarantee a lead time of only 2 weeks. What is the effect on the wholesaler’s cycle and pipeline inventories?

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Example 9.1

The wholesaler’s current cycle and pipeline inventories are

Cycle inventory = = 140 drills

(70 drills/week)(3 weeks)

= 210 drills

Pipeline inventory = DL = dL =

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Example 9.1

The wholesaler’s cycle and pipeline inventories if they accept the new proposal

(70 drills/week)(2 weeks)

= 140 drills

Pipeline inventory = DL = dL =

Cycle inventory = = 175 drills

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Inventory Reduction Tactics

Cycle inventory

Reduce the lot size

Reduce ordering and setup costs and allow Q to be reduced

Increase repeatability to eliminate the need for changeovers

Safety stock inventory

Place orders closer to the time when they must be received

Improve demand forecasts

Cut lead times

Reduce supply uncertainties

Rely more on equipment and labor buffers

PL = primary lever

SL = secondary lever

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Inventory Reduction Tactics

Anticipation inventory

Match demand rate with production rates

Add new products with different demand cycles

Provide off-season promotional campaigns

Offer seasonal pricing plans

Pipeline inventory

Reduce lead times

Find more responsive suppliers and select new carriers

Change Q in those cases where the lead time depends on the lot size

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What is an ABC Analysis?

A stock-keeping unit (SKU) is an individual item or product that has an identifying code and is held in inventory somewhere along the supply chain.

ABC analysis is the process of dividing SKUs into three classes, according to their dollar usage, so that managers can focus on items that have the highest dollar value.

See Solved Problem 2

09- 19

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What is an ABC Analysis?

ABC Analysis

The process of dividing SKUs into three classes, according to their dollar usage, so that managers can focus on items that have the highest dollar value.

100

Percentage of SKUs

Percentage of dollar value

100 —

90 —

80 —

70 —

60 —

50 —

40 —

30 —

20 —

10 —

0 —

Class C

Class A

Class B

Figure 9.4

9-‹#›

Economic Order Quantity

The lot size, Q, that minimizes total annual inventory holding and ordering costs

Five assumptions

Demand rate is constant and known with certainty.

No constraints are placed on the size of each lot.

The only two relevant costs are the inventory holding cost and the fixed cost per lot for ordering or setup.

Decisions for one item can be made independently of decisions for other items.

The lead time is constant and known with certainty.

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Economic Order Quantity

Don’t use the EOQ

Make-to-order strategy

Order size is constrained

Modify the EOQ

Quantity discounts

Replenishment not instantaneous

Use the EOQ

Make-to-stock strategy with relatively stable demand.

Carrying and setup costs are known and relatively stable

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Calculating EOQ

Inventory depletion (demand rate)

Receive order

1 cycle

On-hand inventory (units)

Time

Average

cycle

inventory

Figure 9.5

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Calculating EOQ

Annual holding cost

Annual holding cost = (Average cycle inventory)  (Unit holding cost)

Total costs = Annual holding cost + Annual ordering or setup cost

Annual ordering cost = (Number of orders/Year)  (Ordering or setup costs)

Annual ordering cost

Total annual cycle inventory cost

9-‹#›

Annual cost (dollars)

Lot Size (Q)

Holding cost

Ordering cost

Total cost

Calculating EOQ

Figure 9.6

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Calculating EOQ

Total annual cycle-inventory cost

where

C = total annual cycle-inventory cost

Q = lot size (in units)

H = holding cost per unit per year

D = annual demand (in units)

S = ordering or setup costs per lot

C = (H) + (S)

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Example 9.2

A museum of natural history opened a gift shop which operates 52 weeks per year.

Top-selling SKU is a bird feeder.

Sales are 18 units per week, the supplier charges $60 per unit.

Ordering cost is $45.

Annual holding cost is 25 percent of a feeder’s value.

Management chose a 390-unit lot size.

What is the annual cycle-inventory cost of the current policy of using a 390-unit lot size?

Would a lot size of 468 be better?

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Example 9.2

We begin by computing the annual demand and holding cost as

D =

H =

C = (H) + (S)

The total annual cycle-inventory cost for the alternative lot size is

= ($15) + ($45)

= $2,925 + $108 = $3,033

390

936

390

The total annual cycle-inventory cost for the current policy is

(18 units/week)(52 weeks/year) = 936 units

0.25($60/unit) = $15

C =

($15) + ($45) = $3,510 + $90 = $3,600

468

936

468

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Example 9.2

3000 –

2000 –

1000 –

0 –

| | | | | | | |

50 100 150 200 250 300 350 400

Lot Size (Q)

Annual cost (dollars)

Current

cost

Lowest

cost

Best Q (EOQ)

Total cost

= (H) + (S)

Ordering cost = (S)

Holding cost = (H)

Figure 9.7

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Calculating EOQ

The EOQ formula:

EOQ =

2DS

TBOEOQ = (12 months/year)

EOQ

Time Between Orders (TBO):

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Example 9.3

For the bird feeders in Example 9.2, calculate the EOQ and its total annual cycle-inventory cost. How frequently will orders be placed if the EOQ is used?

Using the formulas for EOQ and annual cost, we get

EOQ = =

2DS

= 74.94 or 75 units

2(936)(45)

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Example 9.3

Below shows that the total annual cost is much less than the $3,033 cost of the current policy of placing 390-unit orders.

Figure 9.8

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Example 9.3

When the EOQ is used, the TBO can be expressed in various ways for the same time period.

TBOEOQ =

EOQ

TBOEOQ = (12 months/year)

EOQ

TBOEOQ = (52 weeks/year)

EOQ

TBOEOQ = (365 days/year)

EOQ

= = 0.080 year

936

= (12) = 0.96 month

936

= (52) = 4.17 weeks

936

= (365) = 29.25 days

936

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Application 9.1

Suppose that you are reviewing the inventory policies on an $80 item stocked at a hardware store. The current policy is to replenish inventory by ordering in lots of 360 units. Additional information is:

D = 60 units per week, or 3,120 units per year

S = $30 per order

H = 25% of selling price, or $20 per unit per year

What is the EOQ?

EOQ = =

2DS

= 97 units

2(3,120)(30)

9-‹#›

Current Policy		EOQ Policy

Application 9.1

What is the total annual cost of the current policy (Q = 360), and how does it compare with the cost with using the EOQ?

Q = 360 units

Q = 97 units

C = 3,600 + 260

C = $3,860

C = (360/2)(20) + (3,120/360)(30)

C = 970 + 965

C = $1,935

C = (97/2)(20) + (3,120/97)(30)

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Application 9.1

What is the time between orders (TBO) for the current policy and the EOQ policy, expressed in weeks?

TBO360 =

TBOEOQ =

(52 weeks per year) = 6 weeks

360

3,120

(52 weeks per year) = 1.6 weeks

3,120

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Managerial Insights from the EOQ

SENSITIVITY ANALYSIS OF THE EOQ
Parameter	EOQ	Parameter Change	EOQ Change	Comments
Demand		↑	↑	Increase in lot size is in proportion to the square root of D.
Order/ Setup Costs		↓	↓	Weeks of supply decreases and inventory turnover increases because the lot size decreases.
Holding Costs		↓	↑	Larger lots are justified when holding costs decrease.

Table 9.1

2DS

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Continuous Review System

Continuous review (Q) system

Reorder point system (ROP) and fixed order quantity system

Tracks inventory position (IP)

Includes scheduled receipts (SR), on-hand inventory (OH), and back orders (BO)

Inventory position = On-hand inventory + Scheduled receipts – Backorders

IP = OH + SR – BO

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Continuous Review System

Time

On-hand inventory

TBO

Order

placed

Order

placed

Order

placed

Order

received

Order

received

Order

received

Order

received

Figure 9.9

Selecting the Reorder Point

When Demand and Lead Time are Constant

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Example 9.4

Demand for chicken soup at a supermarket is always 25 cases a day and the lead time is always 4 days. The shelves were just restocked with chicken soup, leaving an on-hand inventory of only 10 cases. No backorders currently exist, but there is one open order in the pipeline for 200 cases. What is the inventory position? Should a new order be placed?

R = Total demand during lead time = (25)(4) = 100 cases

= 10 + 200 – 0 = 210 cases

IP = OH + SR – BO

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Application 9.2

The on-hand inventory is only 10 units, and the reorder point R is 100. There are no backorders and one open order for 200 units. Should a new order be placed?

IP = OH + SR – BO =

10 + 200 – 0 = 210

R = 100

Decision: Place NO new order

9-‹#›

Continuous Review Systems

Time

On-hand inventory

TBO1

TBO2

TBO3

Order

received

Order

placed

Order

placed

Order

received

Order

placed

Order

received

Order

received

Figure 9.10

Selecting the Reorder Point When Demand is Variable and Lead Time is Constant

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Example 9.5

A distribution center (DC) in Wisconsin stocks Sony plasma TV sets. The center receives its inventory from a mega warehouse in Kansas with a lead time (L) of 5 days. The DC uses a reorder point (R) of 300 sets and a fixed order quantity (Q) of 250 sets. Current on-hand inventory at the end of Day 1 is 400 sets. There are no scheduled receipts (SR) and no backorders (BO). All demands and receipts occur at the end of the day.

Determine when to order using a Q system

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Example 9.5

Day	Demand	OH	SR	BO	IP	Q
1	50	400
2
3
4
5
6
7

400 + 0 = 400

340 + 0 = 340

250 after ordering

260 < R before ordering

260 + 250 = 510 after ordering

260

220

145

340

250 due

Day 8

250

220 + 250 = 470

145 + 250 = 395

90 + 250 = 340

250+ 250 = 500

after ordering

0 + 250 – 5 = 245 < R before ordering

245 + 250 = 495 after ordering

250 due

Day 12

9-‹#›

Example 9.5

Day	Demand	OH	SR	BO	IP	Q
8
9
10
11
12
13
14

195 + 250 = 445

150 + 250 = 400

250

120 + 250 = 370

120

70 – 60 + 250

= 260

260 – 40 = 220

170

195 – 45 = 150

250

250 after

ordering

250

70 + 250 = 320

220 + 250 = 470

250

260 < R before ordering

260 + 250 = 510 after ordering

250 due

Day 17

170 + 250 = 420

250

0 + 250 – 50 -5 = 195

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Example 9.5

The demands at the DC are fairly volatile and cause the reorder point to be breached quite dramatically at times.

9-‹#›

Continuous Review Systems

Selecting the reorder point with variable demand and constant lead time

Reorder point = Average demand during lead time + Safety stock

= dL + safety stock

where

d= average demand per week (or day or months)

L = constant lead time in weeks (or days or months)

9-‹#›

Continuous Review System

Choosing a Reorder Point

Choose an appropriate service-level policy

Determine the distribution of demand during lead time

Determine the safety stock and reorder point levels

9-‹#›

Continuous Review System

Step 1: Service Level Policy

Service Level (Cycle Service Level) – The desired probability of not running out of stock in any one ordering cycle, which begins at the time an order is placed and ends when it arrives in stock.

Protection Interval – The period over which safety stock must the user from running out of stock.

9-‹#›

Continuous Review System

Specify mean and standard deviation

Standard deviation of demand during lead time

σdLT = σd2L = σd L

Step 2: Distribution of Demand during Lead Time

9-‹#›

σd = 15

Demand for week 1

σdlt = 25.98

225

Demand for 3-week lead time

Demand for week 2

σd = 15

Demand for week 3

σd = 15

Continuous Review System

Figure 9.11

9-‹#›

Continuous Review System

Average demand during lead time

Cycle-service level = 85%

Probability of stockout

(1.0 – 0.85 = 0.15)

zσdLT

Figure 9.12

9-‹#›

Continuous Review System

Step 3: Safety Stock and Reorder Point

Safety stock = zσdLT

where

z = number of standard deviations needed to achieve the cycle-service level

σdLT = stand deviation of demand during lead time

Reorder point = R = dL + safety stock

9-‹#›

Example 9.6

Let us return to the bird feeder in Example 9.3.

The EOQ is 75 units.

Suppose that the average demand is 18 units per week with a standard deviation of 5 units.

The lead time is constant at two weeks.

Determine the safety stock and reorder point if management wants a 90 percent cycle-service level.

Safety stock = zσdLT = 1.28(7.07) =

9.05 or 9 units

Reorder point = d L + Safety stock =

2(18) + 9 = 45 units

9-‹#›

Application 9.3

Suppose that the demand during lead time is normally distributed with an average of 85 and σdLT = 40. Find the safety stock, and reorder point R, for a 95 percent cycle-service level.

Safety stock = zσdLT =

Find the safety stock, and reorder point R, for an 85 percent cycle-service level.

R = Average demand during lead time + Safety stock

R = 85 + 66 = 151 units

1.645(40) = 65.8 or 66 units

Safety stock = zσdLT =

1.04(40) = 41.6 or 42 units

R = Average demand during lead time + Safety stock

R = 85 + 42 = 127 units

9-‹#›

Continuous Review System

Selecting the Reorder Point When Demand and Lead Time are Variable

Safety stock = zσdLT

R = (Average weekly demand  Average lead time) + Safety stock

= d L + Safety stock

9-‹#›

Example 9.7

The Office Supply Shop estimates that the average demand for a popular ball-point pen is 12,000 pens per week with a standard deviation of 3,000 pens.

The current inventory policy calls for replenishment orders of 156,000 pens.

The average lead time from the distributor is 5 weeks, with a standard deviation of 2 weeks.

If management wants a 95 percent cycle-service level, what should the reorder point be?

9-‹#›

Example 9.7

Safety stock = zσdLT =

(1.65)(24,919.87) = 41,117.79 or 41,118 pens

We have d = 12,000 pens, σd = 3,000 pens, L = 5 weeks, and σLT = 2 weeks

σdLT = L σd2 + d 2σLT2 =

(5)(3,000)2 + (12,000)2(2)2

= 24,919.87 pens

Reorder point = d L + Safety stock =

(12,000)(5) + 41,118

= 101,118 pens

9-‹#›

Continuous Review Systems

Two-Bin system

A visual system version of the Q system in which a SKU’s inventory is stored at two different locations.

Calculating Total Q System Costs

Total cost = Annual cycle inventory holding cost + Annual ordering cost + Annual safety stock holding cost

C = (H) + (S) + (H) (Safety stock)

9-‹#›

Continuous Review System

Advantages of the Q System

The review frequency of each SKU may be individualized.

Fixed lot sizes can results in quantity discounts.

The system requires low levels of safety stock for the amount of uncertainty in demands during the lead time.

9-‹#›

Application 9.5

The Discount Appliance Store uses a continuous review system (Q system). One of the company’s items has the following characteristics:

Demand = 10 units/week (assume 52 weeks per year)

Ordering or setup cost (S) = $45/order

Holding cost (H) = $12/unit/year

Lead time (L) = 3 weeks (constant)

Standard deviation in weekly demand = 8 units

Cycle-service level = 70%

9-‹#›

Application 9.5

What is the EOQ for this item?

D =

10/wk  52 wks/yr = 520 units

EOQ = =

2DS

= 62 units

2(520)(45)

What is the desired safety stock?

σdLT = σd L =

8 3 = 14 units

Safety stock = zσdLT =

0.525(14) = 8 units

9-‹#›

Application 9.5

What is the desired reorder point R?

R = Average demand during lead time + Safety stock

R =

What is the total annual cost?

3(10) + 8 = 38 units

($12) + ($45) + 8($12) = $845.42

520

C =

9-‹#›

Application 9.5

Suppose that the current policy is Q = 80 and R = 150. What will be the changes in average cycle inventory and safety stock if your EOQ and R values are implemented?

Reducing Q from 80 to 62

Cycle inventory reduction = 40 – 31 = 9 units

Safety stock reduction = 120 – 8 = 112 units

Reducing R from 150 to 38

9-‹#›

Periodic Review System (P)

Fixed interval reorder system or periodic reorder system

Four of the original EOQ assumptions maintained

No constraints are placed on lot size

Holding and ordering costs

Independent demand

Lead times are certain and supply is known

9-‹#›

Periodic Review System (P)

Protection interval

Time

On-hand inventory

IP3

IP1

IP2

Order

placed

Order

placed

Order

placed

Order

received

Order

received

Order

received

Figure 9.13

9-‹#›

Example 9.8

Refer to Example 9.5

Suppose that the management want to use a Periodic Review System for the Sony TV sets. The first review is scheduled for the end of Day 2. All demands and receipts occur at the end of the day. Lead time is 5 Days and management has set T = 620 and P = 6 days.

Determine how much to order (Q) using a P System.

9-‹#›

Example 9.8

Day	Demand	OH	SR	BO	IP	Q
1	50
2
3
4
5
6
7

400

340 before ordering

280

260 + 280 = 540

260

220

145

90 + 280 – 95 = 275

340

620 – 340 = 280 due Day 7

280

220 + 280 = 500

145 + 280 = 425

90 + 280 = 370

400

280 after

ordering

340 + 280 = 620 after ordering

270 + 0 = 275

9-‹#›

Example 9.8

Day	Demand	OH	SR	BO	IP	Q
8	50
9
10
11
12
13
14

340 + 0 = 340 before ordering

395

150 + 395 = 545

150

100

40 + 395 – 40 = 395

345

180

620 – 225 = 395 due Day 13

395

100 + 395 = 495

40 + 395 = 435

395 + 0 = 395

225

345 + 275 = 620 after ordering

180 + 395 = 575

395

275 after

ordering

395 after

ordering

225 + 0 = 225 before ordering

225 + 395 = 620 after ordering

620 – 345 = 275 due Day 19

9-‹#›

Example 9.8

The P system requires more inventory for the same level of protection against stockouts or backorders.

9-‹#›

Periodic Review System

Selecting the time between reviews, choosing P and T

Selecting T when demand is variable and lead time is constant

IP covers demand over a protection interval of P + L

The average demand during the protection interval is d(P + L), or

T = d (P + L) + safety stock for protection interval

Safety stock = zσP + L , where σP + L =

9-‹#›

Example 9.9

Again, let us return to the bird feeder example.

Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units.

The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent.

What is the equivalent P system?

Answers are to be rounded to the nearest integer.

9-‹#›

Example 9.9

We first define D and then P. Here, P is the time between reviews, expressed in weeks because the data are expressed as demand per week:

D =

(18 units/week)(52 weeks/year) = 936 units

P = (52) =

EOQ

(52) = 4.2 or 4 weeks

936

9-‹#›

Example 9.9

We now find the standard deviation of demand over the protection interval (P + L) = 6:

For a 90 percent cycle-service level z = 1.28:

Safety stock = zσP + L =

1.28(12.25) = 15.68 or 16 units

We now solve for T:

= (18 units/week)(6 weeks) + 16 units = 124 units

T = Average demand during the protection interval + Safety stock

= d (P + L) + safety stock

9-‹#›

Application 9.6

The on-hand inventory is 10 units, and T is 400. There are no back orders, but one scheduled receipt of 200 units. Now is the time to review. How much should be reordered?

IP = OH + SR – BO

The decision is to order 190 units

= 10 + 200 – 0 = 210

T – IP =

400 – 210 = 190

9-‹#›

Periodic Review System

Selecting the Target Inventory Level When Demand and Lead Time are Variable

Simulation

Systems Based on the P System

Single-Bin System

Optional Replenishment System

Calculating Total P System Costs

C = (H) + (S) + HzσP + L

9-‹#›

Periodic Review System

Advantages of the P System

It is convenient because replenishments are made at fixed intervals.

Orders for multiple items from the same supplier can be combined into a single purchase order.

The inventory position needs to be known only when a review is made (not continuously).

9-‹#›

Application 9.7

Return to Discount Appliance Store (Application 9.5), but now use the P system for the item.

Previous information:

Demand = 10 units/wk (assume 52 weeks per year) = 520

EOQ = 62 units (with reorder point system)

Lead time (L) = 3 weeks

Standard deviation in weekly demand = 8 units

z = 0.525 (for cycle-service level of 70%)

9-‹#›

Application 9.7

Reorder interval P, if you make the average lot size using the Periodic Review System approximate the EOQ.

P = (EOQ/D)(52) =

(62/520)(52) = 6.2 or 6 weeks

Safety stock

Target inventory

T = 10(6 + 3) + 13 = 103 units

T = d(P + L) + safety stock for protection interval

9-‹#›

Application 9.7

Total cost

C = (H) + (S) + HzσP + L

= ($12) + ($45) + (13)($12) = $906.00

10(6)

520

10(6)

9-‹#›

Solved Problem 1

A distribution center experiences an average weekly demand of 50 units for one of its items.

The product is valued at $650 per unit. Average inbound shipments from the factory warehouse average 350 units.

Average lead time (including ordering delays and transit time) is 2 weeks.

The distribution center operates 52 weeks per year; it carries a 1-week supply of inventory as safety stock and no anticipation inventory.

What is the value of the average aggregate inventory being held by the distribution center?

9-‹#›

Solved Problem 1

Type of Inventory	Calculation of Average Inventory
Cycle
Safety stock
Anticipation
Pipeline

1-week supply

None

350

= 175 units

= 50 units

dL = (50 units/week)(2 weeks)

Average aggregate inventory

Value of aggregate inventory

= 100 units

= 325 units

= $650(325)

= $211,250

9-‹#›

Solved Problem 2

Booker’s Book Bindery divides SKUs into three classes, according to their dollar usage. Calculate the usage values of the following SKUs and determine which is most likely to be classified as class A.

SKU Number	Description	Quantity Used per Year	Unit Value ($)
1	Boxes	500	3.00
2	Cardboard (square feet)	18,000	0.02
3	Cover stock	10,000	0.75
4	Glue (gallons)	75	40.00
5	Inside covers	20,000	0.05
6	Reinforcing tape (meters)	3,000	0.15
7	Signatures	150,000	0.45

9-‹#›

Solved Problem 2

SKU Number	Description	Quantity Used per Year		Unit Value ($)		Annual Dollar Usage ($)
1	Boxes	500		3.00	=	1,500
2	Cardboard (square feet)	18,000		0.02	=	360
3	Cover stock	10,000		0.75	=	7,500
4	Glue (gallons)	75		40.00	=	3,000
5	Inside covers	20,000		0.05	=	1,000
6	Reinforcing tape (meters)	3,000		0.15	=	450
7	Signatures	150,000		0.45	=		67,500
				Total		81,310

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Solved Problem 2

Figure 9.14

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Solved Problem 2

Figure 9.14

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Solved Problem 3

Nelson’s Hardware Store stocks a 19.2 volt cordless drill that is a popular seller. Annual demand is 5,000 units, the ordering cost is $15, and the inventory holding cost is $4/unit/year.

a. What is the economic order quantity?

b. What is the total annual cost for this inventory item?

a. The order quantity is

EOQ = =

2DS

2(5,000)($15)

= 37,500 = 193.65 or 194 drills

b. The total annual cost is

C = (H) + (S) =

($4) + ($15) = $774.60

194

5,000

194

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Solved Problem 4

A regional distributor purchases discontinued appliances from various suppliers and then sells them on demand to retailers in the region. The distributor operates 5 days per week, 52 weeks per year. Only when it is open for business can orders be received. The following data are estimated for the countertop mixer:

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Solved Problem 4

What order quantity Q, and reorder point, R, should be used?

What is the total annual cost of the system?

If on-hand inventory is 40 units, one open order for 440 mixers is pending, and no backorders exist, should a new order be placed?

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Solved Problem 4

a. Annual demand is

The order quantity is

D = (5 days/week)(52 weeks/year)(100 mixers/day)

= 26,000 mixers/year

EOQ = =

2DS

2(26,000)($35)

$9.40

= 193,167 = 440.02 or 440 mixers

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Solved Problem 4

The standard deviation of the demand during lead time distribution is

A 92 percent cycle-service level corresponds to z = 1.41

σdLT = σd L =

30 3 = 51.96

Safety stock = zσdLT =

1.41(51.96 mixers) = 73.26 or 73 mixers

Reorder point (R) = Average demand during lead time + Safety stock

= 300 mixers + 73 mixers = 373 mixers

With a continuous review system, Q = 440 and R = 373

100(3) = 300 mixers

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Solved Problem 4

b. The total annual cost for the Q systems is

C = (H) + (S) + (H)(Safety stock)

C = ($9.40) + ($35) + ($9.40)(73) = $4,822.38

440

26,000

440

c. Inventory position = On-hand inventory + Scheduled receipts – Backorders

IP = OH + SR – BO =

40 + 440 – 0 = 480 mixers

Because IP (480) exceeds R (373), do not place a new order

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Solved Problem 5

Suppose that a periodic review (P) system is used at the distributor in Solved Problem 4, but otherwise the data are the same.

a. Calculate the P (in workdays, rounded to the nearest day) that gives approximately the same number of orders per year as the EOQ.

b. What is the target inventory level, T? Compare the P system to the Q system in Solved Problem 4.

c. What is the total annual cost of the P system?

d. It is time to review the item. On-hand inventory is 40 mixers; receipt of 440 mixers is scheduled, and no backorders exist. How much should be reordered?

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Solved Problem 5

a. The time between orders is

P = (260 days/year) =

EOQ

(260) = 4.4 or 4 days

440

26,000

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Solved Problem 5

The OM Solver data below shows that T = 812 and safety stock = (1.41)(79.37) = 111.91 or about 112 mixers.

The corresponding Q system for the counter-top mixer requires less safety stock.

Figure 9.15

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Solved Problem 5

c. The total annual cost of the P system is

C = (H) + (S) + (H)(Safety stock)

C = ($9.40) + ($35) + ($9.40)(1.41)(79.37)

100(4)

26,000

100(4)

= $5,207.80

d. Inventory position is the amount on hand plus scheduled receipts minus backorders, or

IP = OH + SR – BO =

40 + 440 – 0 = 480 mixers

The order quantity is the target inventory level minus the inventory position, or

Q = T – IP =

An order for 332 mixers should be placed.

812 mixers – 480 mixers = 332 mixers

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Solved Problem 6

Grey Wolf Lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all room service items, including a special pine-scented bar soap. The daily demand for the soap is 275 bars, with a standard deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year.

a. What is the economic order quantity for the bar of soap?

b. What should the reorder point be for the bar of soap if management wants to have a 99 percent cycle-service level?

c. What is the total annual cost for the bar of soap, assuming a Q system will be used?

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Solved Problem 6

a. We have D = (275)(365) = 100,375 bars of soap; S = $10; and H = $0.30. The EOQ for the bar of soap is

EOQ = =

2DS

2(100,375)($10)

$0.30

= 6,691,666.7 = 2,586.83 or 2,587 bars

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Solved Problem 6

(5)(30)2 + (275)2(1)2 = 283.06 bars

Safety stock = zσdLT =

(2.33)(283.06) = 659.53 or 660 bars

(275)(5) + 660 = 2,035 bars

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Solved Problem 6

c. The total annual cost for the Q system is

C = (H) + (S) + (H)(Safety stock)

C = ($0.30) + ($10) + ($0.30)(660) = $974.05

2,587

100,375

2,587

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(

)

(

)

units

12.6

0.525