LESSON 1: INVENTORY MODELS (STOCHASTIC)

Outline

• Single-Period Models– Discrete Demand– Continuous Demand, Uniform Distribution– Continuous Demand, Normal Distribution

• Multi-Period Models – Given a Q, R Policy, Find Cost– Optimal Q, R Policy without Service Constraint– Optimal Q, R Policy with Type 1 Service Constraint– Optimal Q, R Policy with Type 2 Service Constraint

LESSON 1: INVENTORY MODELS (STOCHASTIC)

Stochastic Inventory Control Models Inventory Control with Uncertain Demand

• In Lessons 16-20 we shall discuss the stochastic inventory control models assuming that the exact demand is not known. However, some demand characteristics such as mean, standard deviation and the distribution of demand are assumed to be known.

• Penalty cost, : Shortages occur when the demand exceeds the amount of inventory on hand. For each unit of unfulfilled demand, a penalty cost of is charged. One source of penalty cost is the loss of profit. For example, if an item is purchased at $1.50 and sold at $3.00, the loss of profit is $3.00-1.50 = $1.50 for each unit of demand not fulfilled.

p

p

• More on penalty cost, – Penalty cost is estimated differently in different situation.

There are two cases:1. Backorder - if the excess demand is backlogged and

fulfilled in a future period, a backorder cost is charged. Backorder cost is estimated from bookkeeping, delay costs, goodwill etc.

2. Lost sales - if the excess demand is lost because the customer goes elsewhere, the lost sales is charged. The lost sales include goodwill and loss of profit margin. So, penalty cost = selling price - unit variable cost + goodwill, if there exists any goodwill.

p

Stochastic Inventory Control Models Inventory Control with Uncertain Demand

Single- and Multi- Period Models

• Stochastic models are classified into single- and multi-period models.

• In a single-period model, items are received in the beginning of a period and sold during the same period. The unsold items are not carried over to the next period.

• The unsold items may be a total waste, or sold at a reduced price, or returned to the producer at some price less than the original purchase price.

• The revenue generated by the unsold items is called the salvage value.


• Following are some products for which a single-period model may be appropriate: – Computer that will be obsolete before the next order– Perishable products– Seasonal products such as bathing suits, winter

coats, etc.– Newspaper and magazine

• In the single-period model, there remains only one question to answer: how much to order.


• In a multi-period model, all the items unsold at the end of one period are available in the next period.

• If in a multi-period model orders are placed at regular intervals e.g., once a week, once a month, etc, then there is only one question to answer: how much to order.

• However, we discuss Q, R models in which it is assumed that an order may be placed anytime. So, as usual, there are two questions: how much to order and when to order.

Single-Period Models

• In the single-period model, there remains only one question to answer: how much to order.

• An intuitive idea behind the solution procedure will now be given: Consider two items A and B. – Item A

• Selling price $900• Purchase price $500• Salvage value $400

– Item B• Selling price $600• Purchase price $500• Salvage value $100


• Item A– Loss resulting from unsold items = 500-400=$100/unit – Profit resulting from items sold = $900-500=$400/unit

• Item B– Loss resulting from unsold items = 500-100=$400/unit – Profit resulting from items sold = $600-500=$100/unit

• If the demand forecast is the same for both the items, one would like to order more A and less B.

• In the next few slides, a solution procedure is discussed that is consistent with this intuitive reasoning.


• First define two terms:

• Loss resulting from the items unsold (overage cost)co= Purchase price - Salvage value

• Profit resulting from the items sold (underage cost)cu = Selling price - Purchase price

• The Question Given costs of overestimating/underestimating demand and the probabilities of various demand sizes how many units will be ordered?

• Demand may be discrete or continuous. The demand of computer, newspaper, etc. is usually an integer. Such a demand is discrete. On the other hand, the demand of gasoline is not restricted to integers. Such a demand is continuous. Often, the demand of perishable food items such as fish or meat may also be continuous.

• Consider an order quantity Q• Let p = probability (demand<Q)

= probability of not selling the Qth item.• So, (1-p) = probability of selling the Qth item.

Single-Period Models (Discrete Demand)Decision Rule

• Expected loss from the Qth item = • Expected profit from the Qth item = • So, the Qth item should be ordered if

• Decision Rule (Discrete Demand):– Order maximum quantity Q such that

where p = probability (demand<Q)

uo

u

uo

cccpor

cppc

,

)1(

opc ucp1

uo

u

cccp

Single-Period Models (Discrete Demand)Decision Rule

Example 1: Demand for cookies:Demand Probability of Demand1,800 dozen 0.052,000 0.102,200 0.202,400 0.302,600 0.202,800 0.103,000 0,05

Selling price=$0.69, cost=$0.49, salvage value=$0.29a. Construct a table showing the profits or losses for each

possible quantity (Self study)b. What is the optimal number of cookies to make?c. Solve the problem by marginal analysis.

Single-Period Models (Discrete Demand)

Note: The demand is discrete. So, the demand cannot be other numbers e.g., 1900 dozens, etc.

Demand Prob Prob Expected Revenue Revenue Total Cost Profit(dozen) (Demand) (Selling Number From From Revenue

all the units) Sold Sold Unsold Items Items

1800 0.052000 0.12200 0.22400 0.32600 0.22800 0.13000 0.05

a. Sample computation for order quantity = 2200:Expected number sold = 1800Prob(demand=1800) +2000Prob(demand=2000) + 2200Prob(demand2200) = 1800(0.05)+2000(0.1)+2200(0.2+0.3+0.2+0.1+0.05) = 1800(0.05)+2000(0.1)+2200(0.85) = 2160 Revenue from sold items=2160(0.69)=$1490.4Revenue from unsold items=(2200-2160)(0.29)=$11.6

Single-Period Models (Discrete Demand)Self Study

Demand Prob Prob Expected Revenue Revenue Total Cost Profit(dozen) (Demand) (Selling Number From From Revenue

all the units) Sold Sold Unsold Items Items

1800 0.05 1 1800 1242.0 0.0 1242 882 3602000 0.1 0.95 1990 1373.1 2.9 1376 980 3962200 0.2 0.85 2160 1490.4 11.6 1502 1078 4242400 0.3 0.65 2290 1580.1 31.9 1612 1176 4362600 0.2 0.35 2360 1628.4 69.6 1698 1274 4242800 0.1 0.15 2390 1649.1 118.9 1768 1372 3963000 0.05 0.05 2400 1656.0 174.0 1830 1470 360


a. Sample computation for order quantity = 2200: Total revenue=1490.4+11.6=$1502

Cost=2200(0.49)=$1078Profit=1502-1078=$424

b. From the above table the expected profit is maximized for an order size of 2,400 units. So, order 2,400 units.

Self Study

c. Solution by marginal analysis:

Order maximum quantity, Q such that

Demand, Q Probability(demand) Probability(demand<Q), p1,800 dozen 0.052,000 0.102,200 0.202,400 0.302,600 0.202,800 0.103,000 0,05

0cc

cQpu

udemandyProbabilit

o

u

cc


• In the previous slide, one may draw a horizontal line that separates the demand values in two groups. Above the line, the values in the last column are not more than

The optimal solution is the last demand value above the line. So, optimal solution is 2,400 units.


0ccc

u

u

Continuous Distribution

• Often the demand is continuous. Even when the demand is not continuous, continuous distribution may be used because the discrete distribution may be inconvenient.

• For example, suppose that the demand of calendar can vary between 150 to 850 units. If demand varies so widely, a continuous approximation is more convenient because discrete distribution will involve a large number of computation without any significant increase in accuracy.

• We shall discuss two distributions:– Uniform distribution– Continuous distribution


• First, an example on continuous approximation.• Suppose that the historical sales data shows:Quantity No. Days sold Quantity No. Days sold 14 1 21 11 15 2 22 9 16 3 23 6 17 6 24 3 18 9 25 2 19 11 26 1 20 12


• A histogram is constructed with the above data and shown in the next slide. The data shows a good fit with the normal distribution with mean = 20 and standard deviation = 2.49.

• There are some statistical tests, e.g., Chi-Square test, that can determine whether a given frequency distribution has a good fit with a theoretical distribution such as normal distribution, uniform distribution, etc. There are some software, e.g., Bestfit, that can search through a large number of theoretical distributions and choose a good one, if there exists any. This topic is not included in this course.


Mean = 20Standard deviation = 2.49


The figure below shows an example of uniform distribution.

• The decision rule for continuous demand is similar to the decision rule for the discrete demand

• Decision Rule (Continuous Demand):– Order quantity Q such that

where p = probability (demandQ)– Note the following difference

• The word “maximum”• The “= “ in the formula for p• The “ “ in the definition of p

Single-Period Models (Continuous Demand)Decision Rule

uo

u

cccp

• There is a nice pictorial interpretation of the decision rule. Although we shall discuss the interpretation in terms of normal and uniform distributions, the interpretation is similar for all the other continuous distributions.

• The area under the curve is 1.00. Identify the vertical line that splits the area into two parts with areas

The order quantity corresponding to the vertical line is optimal.


right the on

and left the on

uo

o

uo

u

cccp

cccp

1

• Pictorial interpretation of the decision rule for uniform distribution (see Examples 2, 3 and 4 for application of the rule):


8 5 01 5 0D em a n d

Prob

abili

ty

Area

uo

u

ccc

p

uo

o

ccc

p

1 Area

*Q

• Pictorial interpretation of the decision rule for normal distribution (see Examples 3 and 4 for application of the rule):


Prob

abili

ty

D em and

Area

uo

u

ccc

p

*Q

uo

o

ccc

p

1 Area

Example 2: The J&B Card Shop sells calendars. The once-a-year order for each year’s calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated by

a. uniform distribution between 150 and 850

Single-Period Models (Continuous Demand)

Solution to Example 2:

Overage costco = Purchase price - Salvage value =

Underage costcu = Selling price - Purchase price =


p =

Now, find the Q so that p = probability(demand<Q) =

Q* = a+p(b-a) =

uo

u

ccc

Area =

850150 Demand

Prob

abili

ty

Area=

Q*


Example 3: The J&B Card Shop sells calendars. The once-a-year order for each year’s calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated by

b. normal distribution with = 500 and =120.


Solution to Example 3: co =$0.50, cu =$1.50 (see Example 2)

p = = 0.750.501.501.50uo

u

ccc



Now, find the Q so that p = 0.75

Example 4: A retail outlet sells a seasonal product for $10 per unit. The cost of the product is $8 per unit. All units not sold during the regular season are sold for half the retail price in an end-of-season clearance sale. Assume that the demand for the product is normally distributed with = 500 and = 100.

a. What is the recommended order quantity? b. What is the probability of a stockout? c. To keep customers happy and returning to the store later,

the owner feels that stockouts should be avoided if at all possible. What is your recommended quantity if the owner is willing to tolerate a 0.15 probability of stockout?

d. Using your answer to part (c), what is the goodwill cost you are assigning to a stockout?

Single-Period Models (Continuous Demand)Self Study

Solution to Example 4:a. Selling price=$10, Purchase price=$8 Salvage value=10/2=$5 cu =10 - 8 = $2, co = 8-10/2 = $3

p = = 0.4

Now, find the Q so that p = 0.4or, area (1) = 0.4Look up Table A-1 for Area (2) = 0.5-0.4=0.10

322uo

u

ccc

Prob

abili

ty

D em and= 5 0 0

A rea= 0 .4 0

z = 0 .2 5 5

1 0 0A rea= 0 .1 0

(1 )

(2 )

(3 )


z = 0.25 for area = 0.0987z = 0.26 for area = 0.1025So, z = 0.255 (take -ve, as p = 0.4 <0.5) for area = 0.10So, Q*=+ z =500+(-0.255)(100)=474.5 units.

b. P(stockout)=P(demandQ)=1-P(demand<Q)=1-p=1-0.4=0.6


c. P(stockout)=Area(3)=0.15 Look up Table A-1 for Area (2) = 0.5-0.15=0.35

z = 1.03 for area = 0.3485z = 1.04 for area = 0.3508

So, z = 1.035 for area = 0.35So, Q*=+ z =500+(1.035)(100)=603.5 units.

Prob

abili

tyD em and= 5 0 0

A rea= 0 .3 5

z = 1 .0 3 5

1 0 0A rea= 0 .1 5

(1 )

(2 )

(3 )


d. p=P(demand<Q)=1-P(demandQ) =1-P(stockout)=1-0.15=0.85 For a goodwill cost of g cu =10 - 8+g = 2+g, co = 8-10/2 = $3

Now, solve g in p = = =0.85

Hence, g=$15.

uo

u

ccc 32

2

)( gg


READING AND EXERCISES

Lesson 1

Reading: Section 5.1 - 5.3 , pp. 245-254 (4th Ed.), pp. 232-245 (5th Ed.)

Exercise: 8a, 12a, and 12b, pp. 256-258 (4th Ed.), pp. 248-250 (5th Ed.)

Outline

• Multi-Period Models – Lot size-Reorder Point (Q, R) Systems– Notation, Definition and Some Formula– Example: Given a Q, R Policy, Find Cost

LESSON 2: INVENTORY MODELS (STOCHASTIC)INTRODUCTION TO THE Q,R SYSTEMS

Lot Size - Reorder Point (Q,R) Systems

• In the simple EOQ model, demand is known and fixed. However, often demand is random. The lot size-reorder point (Q, R) systems allow random demand.

• There are two decision variables in a (Q, R) system:– Order quantity, Q and– Reorder point, R

• The Q, R policy is as follows:– When the level of on-hand inventory hits reorder

point, R place an order with lot size Q.


• In the simple EOQ model, R is the demand during the lead time.

• However, in presence of random demand, R usually includes a safety stock, in addition to the expected demand during the lead time. So,

Reorder point, R = lead-time demand + safety stock


• In the simple EOQ model, only holding cost and ordering costs are considered.

• In presence of random demand, the demand may sometimes be too high and exceed the inventory on hand. The result is stock-out.

• For each unit of shortage, a penalty cost p is charged. See Lesson 16 for more information on penalty cost.

Penalty cost = p per unit.


• The goal of a lot size-reorder point system is to find Q and R so that the total annual holding cost, ordering cost and stock-out cost is minimized.

• The current lesson only covers how to compute cost from a given policy.

• The next three lessons address the question how to find optimal Q and R so that the total annual cost is minimized.


Whenever the inventory onhand hits R, a quantity Q is ordered.


Too high lead-time demand may cause stock-outs. Safety stock reduces the chance ofstock-outs.


Safety Stock

Lead-Time Demand

The reorder point is computedfrom the lead-time demand and the safety stock.


Safety Stock

Lead-Time Demand

Goal: Find Q and R such that total annual holding cost, orde-ring cost and stock-out cost is minimized.

τ

ττp

y

y

demand time-lead of deviation standard

demand annual of deviation standard demand time-lead mean

yearin time lead unit per cost out-stock

(Q,R) Policy Notation, Definition and Some Formula


zLn

zLzF

zzF

Rz

cycle per units out-stock 786781- pp. 4,- ATable from available

function, loss edstandardiz the time-lead during out stocking not ofy Probabilit

786781- pp. 4,- ATable from available of left the on area the

curve, normal the under area cumulative the ,


Prob

abili

ty

L ead - t im e D em and

zF Area

R

Rz

zF1- Area

P ro b(sto ck o u t )

P ro b(no sto ck o u t )

Qn

Q

RQQR

QR

shortages of number annual Expected

cycles or orders of number annual Expected

stocksafety inventory Average

stocksafety inventory, Average

regular inventory, Average

stockSafety

22

2


Qnp

QKRhhQ

Qnp

QK

Rh

hQ

)(2

)(2

cost annual Total

cost out-stock Annual

cost ordering Annual

stocksafety cost, holding Annual

regular cost, holding Annual


• Type 1 service– Type 1 service level, is the probability of not

stocking out during the lead time.

– F(z) is available from Table A-4, pp. 781-786• Type 2 service

– Type 2 service level is measured by fill rate, which is the proportion of demands that are met from stock

Qn

1

RzzF ,


Example - Given A Q,R Policy, Find Cost

Annual demand for number 2 pencils at the campus store is normally distributed with mean 2,000 and standard deviation 300. The store purchases the pencils for 10 cents and sells them for 35 cents each. There is a two-month lead time from the initiation to the receipt of an order. The store accountant estimates that the cost in employee time for performing the necessary paper work to initiate and receive an order is $20, and recommends a 25 percent annual interest rate for determining holding cost. The cost of a stock-out is the cost of lost profit plus an additional 20 cents per pencil, which represents the cost of loss of goodwill. Currently, a (Q,R) system with Q = 1500, R = 500 is used.

Find

a. The safety stock


stockSafety

demand, time-Lead

time, Lead

R

b. The average inventory level

c. The expected annual number of orders


stocksafety inventory Average22

RQQ

Q cycles or orders of number annual Expected

d. The probability of not stocking out during the lead-time

e. The expected number of units stock-out per cycle


786)781- pp. 4,- ATable (See time-lead during out stocking not ofy Probabilit

zF

Rz

786)781- pp. 4,- ATable (See

zLn

f. The fill rate

g. The expected annual number of shortages


1 Qn rate, fill The

shortages of number annual Expected

Qn

h.The holding cost per unit per year and penalty cost per unit.


cost,Penalty cost, Holding

pIch

i. The average annual holding cost associated with this policy.


)(

2

cost holding annual Total

stocksafety cost, holding Annual

regular cost, holding Annual

Rh

hQ

j. The total annual cost associated with this policy.


)(2

Qnp

QKRhhQ

Qnp

QK

cost annual Total

cost out-stock Annual

cost ordering Annual


Lesson 2

Reading: Section 5.4, pp. 259-262 (4th Ed.), pp. 250-254 (5th Ed.)

Exercise: 13b (use the result of 13a), p. 271 (4th Ed.), p. 261 (5th Ed.)

Outline

• Multi-Period Models – Lot size-Reorder Point (Q, R) Systems

• Optimization without service– Procedure– Example

LESSON 3: INVENTORY MODELS (STOCHASTIC)Q,R SYSTEMS

OPTIMIZATION WITHOUT SERVICE

Goal: Given find (Q,R) to minimize total cost

Step 1: Take a trial value of Q = EOQStep 2: Find a trial value of R = where and

are respectively mean and standard deviation of the lead-time demand and is the normal distribution variate corresponding to the area on the right, 1-F(z) = see Table A-4, pp. 835-841

Step 3: Find the expected number of stock-outs per cycle, where is the standardized loss function available from Table A-4, pp. 835-841

z

z

pQh /

)(zLn )(zL

pKh ,,,,

Procedure to find the Optimal (Q,R) Policy Without Any Service Constraint

Procedure to find the Optimal (Q,R) Policy Without Any Service Constraint

Step 4: Find the modified

Step 5: Find the modified value of R = where is the recomputed value of the normal distribution variate corresponding to the area on the right, 1-F(z) = see Table A-4, pp. 835-841

Step 6: If any of modified Q and R is different from the previous value, go to Step 3. Else if none of Q and R is modified significantly, stop.

Knph

Q 2

z z

pQh /

Example - Optimal (Q,R) Policy

Annual demand for number 2 pencils at the campus store is normally distributed with mean 2,000 and standard deviation 300. The store purchases the pencils for 10 cents and sells them for 35 cents each. There is a two-month lead time from the initiation to the receipt of an order. The store accountant estimates that the cost in employee time for performing the necessary paper work to initiate and receive an order is $20, and recommends a 25 percent annual interest rate for determining holding cost. The cost of a stock-out is the cost of lost profit plus an additional 20 cents per pencil, which represents the cost of loss of goodwill. Find an optimal (Q,R) policy


y

y

pIch

K

demand, time-lead of deviation Standard

demand, time-lead Mean time, Lead

demand, annual of deviation Standard demand, annual Mean

cost,Penalty cost, Holding

cost, ordering Fixed


I t e r a t i o n 1 S t e p 1 :

h

KQ 2EOQ

S t e p 2 :

p

QhzF )(1

z ( T a b l e A - 4 )

zR

Example - Optimal (Q,R) Policy S t e p 3 : )( zL ( T a b l e A - 4 )

)( zLn S t e p 4 :

Knph

Q 2

Q u e s t i o n : W h a t a r e t h e s t o p p i n g c r i t e r i a ?

S t e p 5 : p

QhzF )(1

z ( T a b l e A - 4 )

zR

Example - Optimal (Q,R) Policy Iteration 2S t e p 3 : )( zL ( T a b l e A - 4 )

)( zLn S t e p 4 :

Knph

Q 2

Q u e s t i o n : D o t h e a n s w e r s c o n v e r g e ?

S t e p 5 : p

QhzF )(1

z ( T a b l e A - 4 )

zR

Fixed cost (K) Note: K, h, and p Holding cost (h) are input dataPenalty cost (p)Mean annual demand () inputLead time (in years input dataLead time demand parameters:

Mean, <--- computedStandard deviation, input data

Iteration 1Iteration 2Step 1Q=Step 2Area on the right=1-F(z)

z=R=

Step 3L(z)=n=

Step 4Modified Q=Step 5Area on the right=1-F(z)

z=Modified R =

z

z

EOQpQh /

pQh /

Table A1/A4

Table A1/A4

Table A4)(zL

hKnp /2


Lesson 3

Reading: Section 5.4, pp. 262-264 (4th Ed.), pp. 253-255 (5th Ed.)

Exercise: 13a, p. 271 (4th Ed.), p. 261

LESSON 4: INVENTORY MODELS (STOCHASTIC)Q,R SYSTEMS

OPTIMIZATION WITH SERVICE

Outline

• Multi-Period Models – Lot size-Reorder Point (Q, R) Systems

• Optimization with service– Procedure for Type 1 Service– Procedure for Type 2 Service– Example

Optimization With Service

• In Lesson 18, we discuss the procedure of finding an optimal Q, R policy without any service constraint and using a stock-out penalty cost of p per unit.

• Managers often have difficulties to estimate p.• A substitute for stock-out penalty cost, p. is service

level.• In this lesson we shall not use stock-out penalty cost,

p. Instead , we shall assume that a service level must be met. Next slide defines two major types of service levels.

Optimization With Service

• Type 1 service– The probability of not stocking out during the lead

time is denoted by . In problems with Type 1 service, is specified e.g., = 0.95

• Type 2 service– Fill rate, : The proportion of demands that are

met from stock is called filled rate and is denoted by . In problems with Type 2 service, is specified e.g., = 0.999

Procedure to Find the Optimal (Q,R) Policy with Type 1 Service

Goal: Given find (Q,R) to minimize total costFirst, find mean of the lead-time demand, andstandard deviation of the lead-time demand,

Step 1: Set Q = EOQStep 2: Find z for which area on the left, F(z) =Step 3: Find R =

,,,, Kh

z

Goal: Given find (Q,R) to minimize total costFirst, find mean of the lead-time demand, andstandard deviation of the lead-time demand,

Step 1: Take a trial value of Q = EOQStep 2: Find expected number of shortages per cycle, standardized loss function, and the standard normal variate z from Table A-4, pp.

835-841. Find a trial value of R= z

,/)( nzL

,,,, Kh


),( 1Qn

Step 3: Find area on the right, 1-F(z) from Table A-1 or A-4, pp. 835-841

Step 4: Find the modified

Step 5: Find expected number of shortages per cycle, standardized loss function, and the standard normal variate z from Table A-4, pp.

835-841. Find the modified R=Step 6: If any of modified Q and R is different from the

previous value, go to Step 3. Else if none of Q and R is modified significantly, stop.


2121 )))(/((/))(/( zFnhKzFnQ

),( 1Qn ,/)( nzL

z

Example - Optimal (Q,R) Policy with Service

Annual demand for number 2 pencils at the campus store is normally distributed with mean 2,000 and standard deviation 300. The store purchases the pencils for 10 cents and sells them for 35 cents each. There is a two-month lead time from the initiation to the receipt of an order. The store accountant estimates that the cost in employee time for performing the necessary paper work to initiate and receive an order is $20, and recommends a 25 percent annual interest rate for determining holding cost.

a. Find an optimal (Q,R) policy with Type 1 service, =0.95 and Q=EOQ

b. Find an optimal (Q,R) policy with Type 2 service, =0.999 and using the iterative procedure



y

y

IchK

demand, time-lead of deviation Standard

demand, time-lead Mean time, Lead

demand, annual of deviation Standard demand, annual Mean

cost, Holding cost, ordering Fixed

a. Type 1 service, = 0.95 Step 1. Q = EOQ = Step 2. Find z for which area on the left, F(z) = = 0.95

Step 3. R =

b. Type 2 service, =0.999 This part is solved with the iterative procedure as

shown next.

z



I t e r a t i o n 1 S t e p 1 :

hkQ 2EOQ

S t e p 2 :

)( 1Qn

nzL )(

z ( T a b l e A - 4 )

zR


Step 3: )(zF1 Step 4:

2

12

1

)()( zF

nhK

zFnQ

Question: What are the stopping criteria?


S t e p 5 :

)( 1Qn

nzL )(

z ( T a b l e A - 4 )

zR


Step 3: )(zF1 Step 4:

2

12

1

)()( zF

nhK

zFnQ

Question: Do the answers converge?

Iteration 2


S t e p 5 :

)( 1Qn

nzL )(

z ( T a b l e A - 4 )

zR

Fixed cost (K ) Note: K , and hHolding cost (h ) are input dataMean annual demand () input dataLead time (in years input dataLead time demand parameters:

Mean, <--- computedStandard deviation, input data

Type 2 service, fill rate, input dataIteration 1 Iteration 2

Step 1 Q=Step 2 n=

L(z)=z=R=

Step 3 Area on the right=1-F(z )Step 4 Modified Q=Step 5 n=

L(z)=z=R=

2121 )))(/((/))(/( zFnhKzFn

EOQ)( 1Q

/n

z41-835 pp. A1/A4,Table

41-835 pp. A1/A4,Table

/n

z41-835 pp. A1/A4,Table

)( 1Q

(Q,R) Systems

Remark

• We solve three versions of the problem of finding an optimal (Q,R) policy– No service constraint – Type 1 service– Type 2 service

• All these versions may alternatively and more efficiently solved by Excel Solver. This is discussed during the tutorial.

Multiproduct Systems

• Pareto effect– A concept of economics applies to inventory systems– Rank the items in decreasing order of revenue

generated– Item group A: top 20% items generate 80% revenue– Item group B: next 30% items generate 15% revenue– Item group A: last 50% items generate 5% revenue

Multiproduct Systems

• Exchange curves– Parameters like K and I are not easy to measure– Instead of assigning values to such parameters show

the trade off between holding cost and ordering cost for a large number of values of K/I

– The effect of changing the ratio K/I is shown by plotting holding cost vs ordering cost

– Similarly, instead of assigning a value to type 2 service level , one may show the trade off between cost of safety stock and expected number of stock-outs.


Lesson 4

Reading: Section 5.5, pp. 264-271 (4th Ed.), pp. 255-262 (5th Ed.)Section 5.7 (skim) pp. 275-280 (4th Ed.), pp. 265-270 (5th Ed.)

Exercise: 16 and 17, p. 271 (4th Ed.), p. 262 (5th Ed.)

LESSON 1: INVENTORY MODELS (STOCHASTIC)

Documents

Transcript of LESSON 1: INVENTORY MODELS (STOCHASTIC)