13.5 PROBABILISTIC INVENTORY MODELS

13.5 PROBABILISTIC INVENTORY MODELS
In most situations, companies that make ordering and production decisions face uncertainty about the future. Probably the most common and important element of uncertainty is
customer demand, but there can be others. For example, there can be uncertainty in the
amount of lead time between placement and receipt of an order. A company that faces uncertainty has three basic options. First, it can use best guesses for uncertain quantities and
proceed according to one of the deterministic models we developed in the previous section
(or according to one of the many other deterministic models that exist in the literature).
Second, it can develop an analytical (nonsimulation) model to deal with the uncertainty.
The advantage to such a model is that we can calculate bottom line results, such as expected cost, and then use Solver to optimize. The disadvantage is that these analytical
models tend to be mathematically complex. The third possibility is to develop a simulation model. The advantage of a simulation model is that it is relatively easy to develop,
13.5 Probabilistic Inventory Models 759
Why or why not? Then redo the SolverTable again,
this time trying even larger unit shortage costs. How
do the results in this case compare to the results from
the basic EOQ model with no shortages allowed?
7. In Example 13.4, we showed why a company might
invest to reduce its setup cost. It all depends on how
much this investment costs, as specified (in the model)
by the cost of a 10% reduction in the setup cost. Use
SolverTable to see how the results change as this cost
of a 10% reduction varies. You can choose the range
for this cost that makes the results “interesting.”
Within your range, does the lower limit on setup cost
($50) ever become a binding constraint?
8. Modify the synchronized ordering model in Example
13.5 slightly so that you can use a two-way SolverTable
on the fixed costs. Specifically, enter a formula in cell
B9 so that the fixed cost of ordering kings alone is equal
to the fixed cost of ordering queens alone. Then let the
two inputs for SolverTable be the fixed cost of ordering
queens alone and the joint fixed cost of ordering both
kings and queens together. Let these vary over a reasonable range, but make sure that the first input is less than
the second, and the second input is less than twice the
first. (Otherwise, the model wouldn’t be realistic.)
Capture the changing cells and the sum of annual setup
and holding costs as SolverTable outputs. Describe your
findings in a brief report.
Skill-Extending Problems
9. In the basic EOQ model in Example 13.1, suppose that
the fixed cost of ordering and the unit purchasing cost
are both multiplied by the same factor f. Use SolverTable
to see what happens to the optimal order quantity and the
corresponding annual fixed order cost and annual holding cost as fvaries from 0.5 to 5 in increments of 0.25.
Could you have discovered the same results algebraically, using equations (13.2) through (13.4)?
10. In the basic EOQ model, revenue is often omitted
from the model. The reasoning is that all demand will
be sold at the given selling price, so revenue is a fixed
quantity that is independent of the order quantity.
Change that assumption as follows. Make selling price
a decision variable, which must be between $110 and
$150. Then assume that annual demand is a nonlinear
function of the selling price p: Annual Demand

497000p1.24. (This implies a constant elasticity of approximately 1.24 for the demand curve.) Modify the
model in Example 13.1 as necessary and then use
Solver to find the optimal selling price and order quantity. What are the corresponding demand and profit?
Which appears to affect profit more in this model,
order quantity or selling price?
11. In the quantity discount model in Example 13.2, the
minimum total annual cost is region 3 is clearly the
best. Evidently, the larger unit purchase costs in the
other two regions make these two regions unattractive.
When would a switch take place? To answer this question, change the model slightly. First, change the fixed
cost of ordering to $40. Second, keep the unit cost in
region 3 at $26, but change the unit costs in regions 1
and 2 to $26  2k and $26  k, where you can let k
vary. (Currently, k is $2.) Use a two-way SolverTable,
with k varied over some appropriate range to see how
small k must be before it is optimal to order from region 1 or 2. What region is the optimal ordering quantity in if there is no price break at all (k
0). How do
you reconcile this with your SolverTable findings?
regardless of the complexity of the problem. The disadvantage is that it can be difficult, or
at least time-consuming, to find optimal ordering policies from a simulation.5
We already examined one probabilistic inventory model in Chapter 11, the newsvendor model. The essence of a newsvendor model is that a company must place an order for
some product exactly once and then wait to see how large the demand is. If the demand is
larger than expected, the company loses sales it could have made. If the demand is smaller
than expected, the company must dispose of the excess items or sell them at a markeddown price. This presents a classical trade-off between ordering too few and ordering too
many. We used simulation in Chapter 11 to analyze this problem. We now see how it can
be solved analytically.
Besides the newsvendor model, we also examine a continuous review model where a
company orders a product repeatedly through time. The model we examine is basically the
same EOQ model as in the previous section but with one important difference. Now the demand during any period of time is random, and only its probability distribution is known.
This is more realistic, but it complicates the analysis. We assume that the company uses an
(R,Q) ordering policy, which is used by many companies. This continuous review policy is
determined by two numbers, R and Q. The value R is the reorder point. When the company’s inventory level drops to R, an order is placed. The order quantity Q specifies the
amount to order each time an order is placed.
Newsvendor Model
The newsvendor model is one of the simplest probabilistic inventory models, but it is also a
very important one.6 It occurs whenever a company must place a one-time order for a product and then wait to see the demand for the product. The assumption is that after this demand
occurs, the product is no longer valuable. This could be the case for a daily newspaper (who
wants yesterday’s newspaper?), a calendar (who wants a 2005 calendar after 2005?), a fashion product that tends to go out of style after the current “season” (what woman wants last
year’s dress styles?), and so on. Given the single chance to order, the company needs to balance the cost of ordering too much versus the cost of not ordering enough.
To put this problem in a fairly general setting, we let cover and cover, respectively, be the
cost of having 1 more unit or 1 fewer unit on hand than demand. For example, if demand turns
out to be 100 units, cover is the cost if we order 101 units, whereas cunder is the cost if we order
99 units. Each of these is a per unit cost, so if we order, say, 110 units, the cost is 10cover,
whereas if we order 90 units, the cost is 10cunder. The example discussed shortly indicates how
we find c
over and cunder from given monetary inputs. For now, we assume they are known.
Now let D be the random demand. We assume that D has a cumulative probability distribution F(x), so that for any potential demand x, F(x) is the probability P(D  x) that D is
less than or equal to x. In general, this distribution needs to be estimated, probably from historical data on demands for this product or similar products. Then the best order quantity balances the cost of understocking times the probability of understocking with the cost of overstocking times the probability of overstocking. As an example, suppose the unit cost of
understocking, cunder, is four times as large as the unit cost of overstocking, cover. Then it
seems reasonable (and it can be proved) that we need the probability of understocking to be
760 Chapter 13 Inventory Models
5 Fortunately, this is less true now than it used to be. Palisade, for example, has developed a software package
called RISKOptimizer that uses a genetic algorithm to optimize a specified output in a simulation model. This
software is included with the Palisade suite that is bundled with the book. We refer to Winston (1999) for a discussion of simulation models that use RISKOptimizer.
6 The article by Pfeifer et al. (2001) contains an interesting discussion of three alternative methods to analyze the
newsvendor problem: decision trees, simulation, and the critical fractile analysis discussed here. Although the authors provide pros and cons of each method, they appear to prefer simulation.
1
4 as large as the probability of overstocking. If Q is the order quantity, the probability of
overstocking is P(D  Q)
F(Q), and the probability of understocking is 1 minus this,
1  F(Q).7 Because we want the probability of understocking to be 1/4 as large as the probability of overstocking, we set 1  F(Q)
(1
4)F(Q) and solve for F(Q) to obtain F(Q)
4
5.
A similar argument for any values of cover and cunder leads to the following equation that
the optimal order quantity Q must satisfy:
F(Q)

c
over
c
u
nde
rc
under
 (13.8)
The fraction on the right side of this equation is called the critical fractile. This fraction
determines the optimal order quantity through an examination of the demand distribution.
For example, if the cost of understocking is four times as large as the cost of overstocking,
then the critical fractile is 4/5, so there is an 80% chance that demand is less than or equal
to the optimal order quantity value. For any particular demand distribution, we can then appeal to the RISKview program, built-in Excel functions, ta

Results (Vietnamese) 1:

13.5 PROBABILISTIC INVENTORY MODELSIn most situations, companies that make ordering and production decisions face uncertainty about the future. Probably the most common and important element of uncertainty iscustomer demand, but there can be others. For example, there can be uncertainty in theamount of lead time between placement and receipt of an order. A company that faces uncertainty has three basic options. First, it can use best guesses for uncertain quantities andproceed according to one of the deterministic models we developed in the previous section(or according to one of the many other deterministic models that exist in the literature).Second, it can develop an analytical (nonsimulation) model to deal with the uncertainty.The advantage to such a model is that we can calculate bottom line results, such as expected cost, and then use Solver to optimize. The disadvantage is that these analyticalmodels tend to be mathematically complex. The third possibility is to develop a simulation model. The advantage of a simulation model is that it is relatively easy to develop,13.5 Probabilistic Inventory Models 759Why or why not? Then redo the SolverTable again,this time trying even larger unit shortage costs. Howdo the results in this case compare to the results fromthe basic EOQ model with no shortages allowed?7. In Example 13.4, we showed why a company mightinvest to reduce its setup cost. It all depends on hownhiều này chi phí đầu tư, theo quy định (trong mô hình)bởi chi phí giảm 10% thiết lập chi phí. Sử dụngSolverTable để xem làm thế nào kết quả thay đổi như chi phí này10% giảm khác nhau. Bạn có thể chọn phạm vicho các chi phí này mà làm cho các kết quả "thú vị."Trong phạm vi của bạn, có giới hạn thấp hơn về chi phí thiết lập($50) bao giờ trở thành một ràng buộc khó khăn?8. sửa đổi các mô hình đặt hàng đồng bộ trong ví dụ13,5 hơi do đó bạn có thể sử dụng một SolverTable hai chiềutrên các chi phí cố định. Cụ thể, nhập một công thức trong tế bàoB9 do đó chi phí cố định của đặt hàng kings một mình là bình đẳngvới chi phí cố định của đặt hàng queens một mình. Sau đó cho phép cáchai đầu vào cho SolverTable là chi phí cố định của đặt hàngQuyn một mình và chi phí cố định chung đặt cả haivua và hoàng hậu với nhau. Hãy để những thay đổi trong một phạm vi hợp lý, nhưng đảm bảo rằng đầu tiên là ít hơnThứ hai, và các đầu vào thứ hai là ít hơn hai lần cácđầu tiên. (Nếu không, các mô hình sẽ không được thực tế.)Nắm bắt các tế bào thay đổi và tổng hàng năm thiết lậpvà giữ chi phí như SolverTable kết quả đầu ra. Mô tả của bạnnhững phát hiện trong một báo cáo ngắn.Vấn đề mở rộng kỹ năng9. trong mô hình EOQ cơ bản trong 13.1 ví dụ, giả sử rằngchi phí cố định của đặt hàng và các đơn vị mua chi phícả hai được nhân với cùng một yếu tố f. sử dụng SolverTableđể xem những gì sẽ xảy ra với số lượng đặt tối ưu và cáccố định tương ứng hàng năm để chi phí và hàng năm đang nắm giữ chi phí như fvaries từ 0,5 đến 5 trong từng bước của 0,25.Có thể bạn đã phát hiện ra kết quả tương tự đại số, bằng cách sử dụng phương trình (13.2) thông qua (13.4)?10. trong mô hình cơ bản của EOQ, doanh thu thường bỏ quatừ các mô hình. Lý do là rằng tất cả các yêu cầu sẽđược bán tại bán giá nhất định, do đó, doanh thu là một cố địnhsố lượng là độc lập với số lượng đặt.Thay đổi đó giả định như sau. Làm cho giá bánmột biến quyết định, mà phải trong khoảng từ $110 và$150. Sau đó giả định rằng nhu cầu hàng năm là một phi tuyếnchức năng của bán giá p: nhu cầu hàng năm497000p 1.24. (Điều này ngụ ý một tính đàn hồi liên tục của khoảng 1.24 cho đường cong theo yêu cầu.) Sửa đổi cácMô hình trong ví dụ 13.1 nếu cần thiết và sau đó sử dụngNgười giải quyết để tìm tối ưu bán giá và đặt hàng số lượng. Tương ứng nhu cầu và lợi nhuận là gì?Mà dường như ảnh hưởng đến lợi nhuận nhiều hơn trong mô hình này,số lượng đặt hoặc giá bán?11. trong số lượng giảm giá mô hình trong ví dụ 13.2, cáctối thiểu tổng chi phí hàng năm là vùng 3 rõ ràng là cáctốt nhất. Rõ ràng, đơn vị lớn hơn mua chi phí trong cáchai khu vực khác làm cho hai khu vực kém hấp dẫn.Khi nào một chuyển đổi diễn ra? Để trả lời câu hỏi này, thay đổi các mô hình một chút. Trước tiên, thay đổi các cố địnhchi phí của đặt hàng đến $40. Thứ hai, ghi chi phí đơn vịkhu vực 3 tại $26, nhưng thay đổi các chi phí đơn vị trong khu vực 1và 2 $26 2 k và $26 k, nơi bạn có thể cho kkhác nhau. (Hiện nay, k là $2.) Sử dụng một SolverTable hai chiều,với k khác nhau trên một số khu vực thích hợp để xem như thế nàosmall k must be before it is optimal to order from region 1 or 2. What region is the optimal ordering quantity in if there is no price break at all (k
0). How doyou reconcile this with your SolverTable findings?regardless of the complexity of the problem. The disadvantage is that it can be difficult, orat least time-consuming, to find optimal ordering policies from a simulation.5We already examined one probabilistic inventory model in Chapter 11, the newsvendor model. The essence of a newsvendor model is that a company must place an order forsome product exactly once and then wait to see how large the demand is. If the demand islarger than expected, the company loses sales it could have made. If the demand is smallerthan expected, the company must dispose of the excess items or sell them at a markeddown price. This presents a classical trade-off between ordering too few and ordering toomany. We used simulation in Chapter 11 to analyze this problem. We now see how it canbe solved analytically.Besides the newsvendor model, we also examine a continuous review model where acompany orders a product repeatedly through time. The model we examine is basically thesame EOQ model as in the previous section but with one important difference. Now the demand during any period of time is random, and only its probability distribution is known.This is more realistic, but it complicates the analysis. We assume that the company uses an(R,Q) ordering policy, which is used by many companies. This continuous review policy isdetermined by two numbers, R and Q. The value R is the reorder point. When the company’s inventory level drops to R, an order is placed. The order quantity Q specifies theamount to order each time an order is placed.Newsvendor ModelThe newsvendor model is one of the simplest probabilistic inventory models, but it is also avery important one.6 It occurs whenever a company must place a one-time order for a product and then wait to see the demand for the product. The assumption is that after this demandoccurs, the product is no longer valuable. This could be the case for a daily newspaper (whowants yesterday’s newspaper?), a calendar (who wants a 2005 calendar after 2005?), a fashion product that tends to go out of style after the current “season” (what woman wants lastyear’s dress styles?), and so on. Given the single chance to order, the company needs to balance the cost of ordering too much versus the cost of not ordering enough.To put this problem in a fairly general setting, we let cover and cover, respectively, be thecost of having 1 more unit or 1 fewer unit on hand than demand. For example, if demand turnsout to be 100 units, cover is the cost if we order 101 units, whereas cunder is the cost if we order99 units. Each of these is a per unit cost, so if we order, say, 110 units, the cost is 10cover,whereas if we order 90 units, the cost is 10cunder. The example discussed shortly indicates howwe find cover and cunder from given monetary inputs. For now, we assume they are known.Now let D be the random demand. We assume that D has a cumulative probability distribution F(x), so that for any potential demand x, F(x) is the probability P(D  x) that D isless than or equal to x. In general, this distribution needs to be estimated, probably from historical data on demands for this product or similar products. Then the best order quantity balances the cost of understocking times the probability of understocking with the cost of overstocking times the probability of overstocking. As an example, suppose the unit cost ofunderstocking, cunder, is four times as large as the unit cost of overstocking, cover. Then itseems reasonable (and it can be proved) that we need the probability of understocking to be760 Chapter 13 Inventory Models5 Fortunately, this is less true now than it used to be. Palisade, for example, has developed a software packagecalled RISKOptimizer that uses a genetic algorithm to optimize a specified output in a simulation model. Thissoftware is included with the Palisade suite that is bundled with the book. We refer to Winston (1999) for a discussion of simulation models that use RISKOptimizer.6 The article by Pfeifer et al. (2001) contains an interesting discussion of three alternative methods to analyze thenewsvendor problem: decision trees, simulation, and the critical fractile analysis discussed here. Although the authors provide pros and cons of each method, they appear to prefer simulation.
1
4 as large as the probability of overstocking. If Q is the order quantity, the probability of
overstocking is P(D  Q)
F(Q), and the probability of understocking is 1 minus this,
1  F(Q).7 Because we want the probability of understocking to be 1/4 as large as the probability of overstocking, we set 1  F(Q)
(1
4)F(Q) and solve for F(Q) to obtain F(Q)
4
5.
A similar argument for any values of cover and cunder leads to the following equation that
the optimal order quantity Q must satisfy:
F(Q)

c
over
c
u
nde
rc
under
 (13.8)
The fraction on the right side of this equation is called the critical fractile. This fraction
determines the optimal order quantity through an examination of the demand distribution.
For example, if the cost of understocking is four times as large as the cost of overstocking,
then the critical fractile is 4/5, so there is an 80% chance that demand is less than or equal
to the optimal order quantity value. For any particular demand distribution, we can then appeal to the RISKview program, built-in Excel functions, ta