13.7 ORDERING CAMERAS WITH UNCERTAI

13.7 ORDERING CAMERAS WITH UNCERTAIN DEMAND AT MACHEY’S

n Example 13.1, we considered Machey’s department store, which sells, on average,

I

1200 cameras per year. The store pays a setup cost of $125 per order, and the holding cost

is $8 per camera per year. It takes 1 week for an order to arrive after it is placed. In that ex-
ample, the optimal order quantity Q was found to be 194 cameras. Now we assume that the

annual demand is normally distributed with mean 1200 and standard deviation 70.

Machey’s wants to know when to order and how many cameras to order at each ordering

opportunity.

Objective To find the (R,Q) policy that minimizes the company’s expected annual cost.

764 Chapter 13 Inventory Models

The key to choosing

the appropriate

reorder point R is the

distribution of demand

that occurs during an

order lead time.

Solution

Suppose the company places an order for Q cameras every time its inventory level drops to

R. Our goal is to find optimal values of Q and R. Two aspects of this model are critical to

its solution: demand during lead time and the “cost” of running out of inventory.

Demand During Lead Time and Safety Stock

The most critical probabilistic quantity is the amount of demand during an order lead time.

To illustrate, suppose that Machey’s uses R 30 as the reorder point. This means that it

places an order as soon as the inventory level drops to 30 cameras. This order arrives

1 week later. If the demand during this lead time is 25 cameras, say, then no shortage will

occur, and 5 cameras will remain when the order arrives. However, if the demand during

this period is 35 cameras, then there will be a shortage of 5 cameras by the time the order

arrives. Therefore, the demand during lead time, in conjunction with the choice of R, de-
termines the extent of shortages. Before we can continue, we need to analyze this quantity

in some detail.

Let DAD be the annual demand, and let DLD be the demand during an order lead time

of length L. (For clarity, we use subscripts AD for annual demand and LD for lead time de-
mand.) From the statement of the problem, DAD is normally distributed with mean AD

1200 and standard deviation AD 70. By making appropriate probability assumptions, it

can be shown that DLD is also normally distributed, and its mean and standard deviation are

LD LAD and LD L

expected demand during lead time is LD (152)(1200) 23 cameras, and the standard

deviation of demand during lead time is LD 152(70) 9.7 cameras.

Given these values, you might think that Machey’s should set its reorder point R equal

to 23, the mean demand during lead time. But then there would be a 50–50 chance of stock-
ing out before the order arrives (because the probability that a normal random variable is

greater than its mean is 0.5). What if the company instead sets R equal to 1 standard devia-
tion above the mean—that is, R 23 9.7 33? Then the probability of a stockout is

P(DLD 33). This can be found with the NORMDIST function in Excel. (It can also be

found with RISKview, but we take advantage of Excel functions here.) The syntax for this

function is NORMDIST(x, , , 1). It returns the probability that a normal random variable

with mean and standard deviation is less than or equal to a specified value x. Therefore,

we find P(DLD 33), the probability of a stockout, with the formula 1NORMDIST

(33,23,9.7,1), which is approximately 0.15 (see Figure 13.13).

AD. Because the lead time is 1 week (L 152), Machey’s

Figure 13.13

Probability Under a

Normal Distribution

probability

1–NORMDIST(33, 23, 9.7, 1)

23 33

In general, suppose that Machey’s decides to set R equal to k standard deviations

above the mean, where k is a multiplier that must be determined. That is, it uses the reorder

level

R LD kLD LD safety stock (13.9)

13.5 Probabilistic Inventory Models 765

In effect, the multiplier k becomes the decision variable. Usually k is positive (as we

require in this section). The term kLD then becomes the safety stock. To summarize the

reasoning, Machey’s expects an amount LD to be demanded during the 1-week lead time.

However, because shortages are undesirable, it orders when the inventory level is kLD

above LD. Therefore, it expects the inventory level to be kLD, a positive value, when the

order arrives. This value, the safety stock, is its cushion against larger-than-expected de-
mand. But although the company plans for this safety stock to exist, there is no guarantee

that it will exist. The previous probability calculation with k 1 shows that there is about

a 15% chance that the safety stock of 10 units will be depleted before the order arrives. In

this case, a stockout occurs. We want to choose k and the order quantity Q in an optimal

manner.

Finding the Expected Costs We now develop an expression for Machey’s expected total

annual cost as a function of the order quantity Q and k. In the following discussion, we refer

to an order cycle, which begins each time an order arrives and ends just before the next

order arrives (see Figure 13.14).

Place

order

R

Place

order

Order cycle

We first consider the annual setup and holding costs. If an order quantity Q is used, it

takes an expected amount of time QAD to deplete this inventory. (Remember that AD is

the expected annual demand. It plays the same role as D in the deterministic EOQ models.)

Therefore, there are an expected ADQ order cycles per year, so the expected annual setup

cost is KADQ. For the holding cost, consider any order cycle. The lowest inventory level

during a cycle is expected to be kLD, the safety stock. The highest inventory level occurs

when the order arrives and the expected inventory jumps up to Q kLD. Therefore, the

expected average inventory level during a typical cycle is [kLD (Q kLD)]2, and we

multiply this by the unit holding cost h to obtain the expected annual holding cost. (Note

that we are now using the letter h to refer to the unit holding cost. Comparing to the EOQ

section, h s ic.) Simplifying the algebra slightly leads to the following expressions for

expected annual setup and holding costs:

Expected annual setup cost KADQ (13.10)

Expected annual holding cost h(Q2 kLD) (13.11)

inventory. However, the

where (for Machey’s) K $125, h $8, AD 1200, LD 9.7, and Q and k need to be

determined.

Two Ways to “Cost” Shortages We now consider two alternative models of “costing”

shortages. Neither of these models is clearly superior to the other, so Machey’s must decide

which model is more in line with the company’s goals. Model 1 assumes that there is a

shortage cost of p per unit short. In this model, a cycle with a shortage of 5 units is 5 times

as costly as a cycle with a shortage of only 1 unit. For example, suppose Machey’s uses

model 1 with p $10. If the average number of shortages during each of its order cycles

is 2, and there are 13 order cycles during the year, then its annual shortage cost is $260.

766 Chapter 13 Inventory Models

Model 2 gets around the difficult problem of assessing dollar shortage costs by instead

specifying a service level. Specifically, it requires that the fraction of demand that can be

met from on-hand inventory must be at least s, where s is a number between 0 and 1. This

fraction is often called the fill rate. For example, if Machey’s uses model 2 with s 0.98,

then it choose its ordering policy so that at least 98% of all customer demand can be met

from on-hand inventory. That is, it tries to achieve a fill rate of 98%.

Before we can solve Machey’s problem on a spreadsheet, we must develop formulas

for the shortage cost (or service level) for these two shortage-costing models.

Expected Shortage Cost for Model 1 In model 1, Machey’s assesses a shortage cost of

p per unit short during any order cycle. Therefore, to evaluate the expected annual shortage

cost, we must find the expected number of shortages per order cycle. Let E(B) be the ex-
pected number of units short during a typical order cycle. Then the expected shortage cost

during this cycle is pE(B), and the expected annual shortage cost is the expected shortage

cost per cycle multiplied by the expected number of cycles per year, ADQ. This leads to

the following expected total annual shortage cost:

The problem is to find an expression for E(B). This expected value is related to a well-
known quantity called the normal loss function. Fortunately, this can be calculated with

built-in Excel functions. The formula for E(B) is8

Here, n(k) is the standard normal density function evaluated at k, and Z is a standard normal

random variable. (Recall that standard normal implies mean 0 and standard deviation 1.)

We now show how to implement model 1 for the camera example.

DEVELOPING THE SPREADSHEET FOR MODEL 1

We assume that Machey’s decides to use model 1 with p $10 as the unit shortage cost.

The spreadsheet solution appears in Figure 13.15. (See the file Ordering Cameras 1.xlsx.)

It can be developed as follows:

1 Inputs. Enter the inputs in the blue range.

2 Lead time demand. Calculate the mean and standard deviation of lead time demand

in cells B12 and B13 with the formulas

=Lead_time*Expected_annual_demand

and

=SQRT(Lead_time)*Stdev_of_annual_demand

(Admittedly, we have created a lot of range names to make the formulas more readable, but

they can all be created in one step with the Create from Selection shortcut.)

3 Decision variables. Enter any values in cells B16 and B17 for the order quantity Q

and the multiplier k. These are the changing cells.

4 Safety stock and reorder point. The decision variables determine the safety stock

and the reorder point. Calculate them in cells B18 and B19 with the formulas

=Multiple_k*Stdev_lead_time_demand

Model 1 expected annual shortage cost pE(B)ADQ (13.12)

E(B) [n(k) kP(Z k)]LD (13.13)

8 This is one of the few times in this book wher

Kết quả (Việt) 1:

13.7 ĐẶT HÀNG MÁY ẢNH VỚI KHÔNG CHẮC CHẮN YÊU CẦU AT MACHEY CỦAn ví dụ 13.1, chúng tôi xem xét các cửa hàng bách của Machey mà bán, Trung bình,Tôi1200 các máy ảnh mỗi năm. Các cửa hàng trả tiền cho chi phí thiết lập $125 cho một đơn đặt hàng, và chi phí giữlà $8 mỗi máy ảnh mỗi năm. Phải mất 1 tuần cho một đơn đặt hàng đến sau khi nó được đặt. Trong đó ex-phong phú, số lượng tối ưu đặt Q đã được tìm thấy là 194 máy ảnh. Bây giờ chúng tôi giả định rằng cácnhu cầu hàng năm được phân phối bình thường có nghĩa là 1200 và độ lệch chuẩn 70.Của Machey muốn biết khi nào để đặt hàng và các máy ảnh bao nhiêu để đặt hàng tại mỗi đặt hàngcơ hội.Mục tiêu để tìm kiếm (R, Q) chính sách giảm thiểu của công ty dự kiến sẽ chi phí hàng năm.764 chương 13 hàng tồn kho mô hìnhChìa khóa để chọnnhaán sắp xếp lại điểm R là cácphân phối của nhu cầuđó xảy ra trong mộtThứ tự thời gian.Giải phápGiả sử công ty nơi một đơn đặt hàng cho Q máy ảnh mỗi khi mức độ hàng tồn kho của nó xuống đếnR. mục tiêu của chúng tôi là để tìm các giá trị tối ưu của Q và R. Hai khía cạnh của mô hình này là quan trọng đối vớigiải pháp của nó: nhu cầu trong thời gian và các "chi phí" chạy ra khỏi hàng tồn kho.Nhu cầu trong thời gian chì và chứng khoán an toànSố lượng xác suất quan trọng nhất là số tiền của các nhu cầu trong một thứ tự thời gian.Để minh họa, giả sử rằng của Machey sử dụng R 30 như khi sắp xếp lại. Điều này có nghĩa rằng nónơi một đơn đặt hàng ngay sau khi mức độ hàng tồn kho xuống đến 30 máy ảnh. Thứ tự này đến1 tuần sau đó. Nếu nhu cầu trong thời gian này là 25 máy ảnh, nói, sau đó không thiếu sẽxảy ra, và 5 máy ảnh sẽ vẫn khi bộ đến. Tuy nhiên, nếu nhu cầu tronggiai đoạn này là 35 máy ảnh, sau đó sẽ có sự thiếu hụt của 5 máy ảnh khi đơn đặt hàngđến. Do đó, nhu cầu trong thời gian, kết hợp với sự lựa chọn của R, de-termines mức độ của tình trạng thiếu. Trước khi chúng tôi có thể tiếp tục, chúng tôi cần phải phân tích số lượng nàytrong một số chi tiết.Giả sử DAD là nhu cầu hàng năm, và giả sử DLD là nhu cầu trong một thứ tự thời gianchiều dài L. (rõ ràng, chúng tôi sử dụng chỉ quảng cáo cho nhu cầu hàng năm và LD cho dẫn thời gian de -Mandvi.) Từ các báo cáo của vấn đề, bố bình thường được phân phối với có nghĩa là quảng cáo 1200 và tiêu chuẩn độ lệch quảng cáo 70. Bằng cách giả định xác suất thích hợp, nócó thể được hiển thị DLD cũng thường được phân phối, và có ý nghĩa và độ lệch chuẩn của nó làLD LAD và LD Lcác nhu cầu dự kiến trong thời gian dẫn đầu là LD (152)(1200) 23 máy ảnh, và các tiêu chuẩnđộ lệch của các nhu cầu trong thời gian dẫn đầu là LD 152(70) 9.7 máy ảnh.Đưa ra những giá trị này, bạn có thể nghĩ rằng Machey của nên đặt của nó sắp xếp lại điểm R bằng23, nhu cầu có nghĩa là trong thời gian. Nhưng sau đó sẽ có một cơ hội 50-50 của chứng khoán-ing ra trước khi bộ đến (bởi vì xác suất mà một biến ngẫu nhiên bình thường làlớn hơn có nghĩa là nó là 0,5). Nếu công ty thay vào đó đặt R tương đương với 1 tiêu chuẩn devia-tion above the mean—that is, R 23 9.7 33? Then the probability of a stockout isP(DLD 33). This can be found with the NORMDIST function in Excel. (It can also befound with RISKview, but we take advantage of Excel functions here.) The syntax for thisfunction is NORMDIST(x, , , 1). It returns the probability that a normal random variablewith mean and standard deviation is less than or equal to a specified value x. Therefore,we find P(DLD 33), the probability of a stockout, with the formula 1NORMDIST(33,23,9.7,1), which is approximately 0.15 (see Figure 13.13).AD. Because the lead time is 1 week (L 152), Machey’sFigure 13.13Probability Under aNormal Distributionprobability1–NORMDIST(33, 23, 9.7, 1)23 33In general, suppose that Machey’s decides to set R equal to k standard deviationsabove the mean, where k is a multiplier that must be determined. That is, it uses the reorderlevelR LD kLD LD safety stock (13.9)13.5 Probabilistic Inventory Models 765In effect, the multiplier k becomes the decision variable. Usually k is positive (as werequire in this section). The term kLD then becomes the safety stock. To summarize thereasoning, Machey’s expects an amount LD to be demanded during the 1-week lead time.However, because shortages are undesirable, it orders when the inventory level is kLDabove LD. Therefore, it expects the inventory level to be kLD, a positive value, when theorder arrives. This value, the safety stock, is its cushion against larger-than-expected de-mand. But although the company plans for this safety stock to exist, there is no guaranteethat it will exist. The previous probability calculation with k 1 shows that there is abouta 15% chance that the safety stock of 10 units will be depleted before the order arrives. Inthis case, a stockout occurs. We want to choose k and the order quantity Q in an optimalmanner.Finding the Expected Costs We now develop an expression for Machey’s expected totalannual cost as a function of the order quantity Q and k. In the following discussion, we referto an order cycle, which begins each time an order arrives and ends just before the nextorder arrives (see Figure 13.14).Place orderRPlace orderOrder cycleWe first consider the annual setup and holding costs. If an order quantity Q is used, ittakes an expected amount of time QAD to deplete this inventory. (Remember that AD isthe expected annual demand. It plays the same role as D in the deterministic EOQ models.)Therefore, there are an expected ADQ order cycles per year, so the expected annual setupcost is KADQ. For the holding cost, consider any order cycle. The lowest inventory levelduring a cycle is expected to be kLD, the safety stock. The highest inventory level occurswhen the order arrives and the expected inventory jumps up to Q kLD. Therefore, theexpected average inventory level during a typical cycle is [kLD (Q kLD)]2, and wemultiply this by the unit holding cost h to obtain the expected annual holding cost. (Note
that we are now using the letter h to refer to the unit holding cost. Comparing to the EOQ

section, h s ic.) Simplifying the algebra slightly leads to the following expressions for

expected annual setup and holding costs:

Expected annual setup cost KADQ (13.10)

Expected annual holding cost h(Q2 kLD) (13.11)

inventory. However, the

where (for Machey’s) K $125, h $8, AD 1200, LD 9.7, and Q and k need to be

determined.

Two Ways to “Cost” Shortages We now consider two alternative models of “costing”

shortages. Neither of these models is clearly superior to the other, so Machey’s must decide

which model is more in line with the company’s goals. Model 1 assumes that there is a

shortage cost of p per unit short. In this model, a cycle with a shortage of 5 units is 5 times

as costly as a cycle with a shortage of only 1 unit. For example, suppose Machey’s uses

model 1 with p $10. If the average number of shortages during each of its order cycles

is 2, and there are 13 order cycles during the year, then its annual shortage cost is $260.

766 Chapter 13 Inventory Models

Model 2 gets around the difficult problem of assessing dollar shortage costs by instead

specifying a service level. Specifically, it requires that the fraction of demand that can be

met from on-hand inventory must be at least s, where s is a number between 0 and 1. This

fraction is often called the fill rate. For example, if Machey’s uses model 2 with s 0.98,

then it choose its ordering policy so that at least 98% of all customer demand can be met

from on-hand inventory. That is, it tries to achieve a fill rate of 98%.

Before we can solve Machey’s problem on a spreadsheet, we must develop formulas

for the shortage cost (or service level) for these two shortage-costing models.

Expected Shortage Cost for Model 1 In model 1, Machey’s assesses a shortage cost of

p per unit short during any order cycle. Therefore, to evaluate the expected annual shortage

cost, we must find the expected number of shortages per order cycle. Let E(B) be the ex-
pected number of units short during a typical order cycle. Then the expected shortage cost

during this cycle is pE(B), and the expected annual shortage cost is the expected shortage

cost per cycle multiplied by the expected number of cycles per year, ADQ. This leads to

the following expected total annual shortage cost:

The problem is to find an expression for E(B). This expected value is related to a well-
known quantity called the normal loss function. Fortunately, this can be calculated with

built-in Excel functions. The formula for E(B) is8

Here, n(k) is the standard normal density function evaluated at k, and Z is a standard normal

random variable. (Recall that standard normal implies mean 0 and standard deviation 1.)

We now show how to implement model 1 for the camera example.

DEVELOPING THE SPREADSHEET FOR MODEL 1

We assume that Machey’s decides to use model 1 with p $10 as the unit shortage cost.

The spreadsheet solution appears in Figure 13.15. (See the file Ordering Cameras 1.xlsx.)

It can be developed as follows:

1 Inputs. Enter the inputs in the blue range.

2 Lead time demand. Calculate the mean and standard deviation of lead time demand

in cells B12 and B13 with the formulas

=Lead_time*Expected_annual_demand

and

=SQRT(Lead_time)*Stdev_of_annual_demand

(Admittedly, we have created a lot of range names to make the formulas more readable, but

they can all be created in one step with the Create from Selection shortcut.)

3 Decision variables. Enter any values in cells B16 and B17 for the order quantity Q

and the multiplier k. These are the changing cells.

4 Safety stock and reorder point. The decision variables determine the safety stock

and the reorder point. Calculate them in cells B18 and B19 with the formulas

=Multiple_k*Stdev_lead_time_demand

Model 1 expected annual shortage cost pE(B)ADQ (13.12)

E(B) [n(k) kP(Z k)]LD (13.13)

8 This is one of the few times in this book wher