probabilistic inventory models raw materials

8/8/2019 Probabilistic Inventory Models Raw Materials

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Probabilistic inventory models of

raw materials

Topic objectives

Understand the nature of uncertainty in inventory

management Select and apply the appropriate probabilistic inventory

model

Dr Muhammad AlSalamah, Industrial Engineering, KFUPM


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Uncertainties

In reality, demands many not be predictable with relative ease.

Average demands

can

be

adequate

in

some

situations,

but

considering the demands as a random variable can provide far more accura e an reasona e resu s.

Probabilistic inventory models assign a probability function to the demand variable which is characterized b a mean and variance.

When the demand is considered as a random variable, pro uc on can ace wo s ua ons, e er over s oc o ems or under stock.

Both of these situations are not desirable.



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Single period stochastic model A plant uses a certain raw material that is used to

manufacture its product.

The raw material is usable for a single period only, any leftovers are discarded.

The plant incurs a cost for any unit of demand that is not satisfied.

. This model is commonly known as the newsvendor model and

newsboy model.



4/37

Define the following costs

cu : shortage cost (underage cost) The demand D is assumed to be a continuous nonnegative

random variable with a probability density function f(x) and

cumulative function

F(x).

be purchased at the beginning of the period.



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Define G(Q, D) to be the total underage and overage costs at the end of the period.

If it happens Q > D, positive inventory is realized; if Q < D, a shortage happens:

inventory = max(Q D, 0)

shortage = max(D

Q,

0) ,

G(Q, D) = c0 max(Q D, 0) + cu max(D Q, 0)

== )(),()),(()( dx x f xQG DQG E QG


0


6/37

Substitute in:

+=

0 0)()0),max(0),max(()( dx x f x QcQ xcQG

u

+=00

0 )(0),max()(0),max( dx x f x Qcdx x f Q xc u

Notice max(Q D, 0) > 0 only when Q > D and max(D Q, 0) > 0 only when D > Q:

Q

+=

QQ

Qu x x x c x x xc

00


+= Qu

Qu dx x f Qcdx x xf cdx x xf cdx xQc )()()()(

00

00


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To prove G(Q) is convex:

)()()(0

0 =

dx x f cdx x f cdQ

QdG uQ

)(

))(1()(2

=QGd

Q F cQ F c uo

0)()(

02

+=

=

Q f cc

ccdQ

uo

u



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The minimizer Q* of G(Q) is obtained by setting the first derivative of G(Q) to zero:

uo

c F ccQ F cQ F c=+

=0*

0*))(1(*)(

uo

u

ccc

Q F +=*)(

Since F(Q*) is the probability that the demand does not

* + u o u satisfying the demand during the period if Q* is ordered.



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The probability of shortage happening is

)(1)(1)( 0 Q F dx x f dx x f Q == The probability of overstock happening is

Q

0

x x =



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Illustration

A laban producer buys a certain additive which has a shelf life of 40 days.

The demand for laban is uncertain, and hence manufacturing consumes this additive in uncertain amounts.

The total consumption during 40 days is normally distributed with mean = 11.73 tons and standard deviation = 4.74.

, amount of the additive can be sold for 1,000 riyals per ton as animal feed.

Each ton of the additive can generate a profit of 7,500 riyals. How many tons of the additive should be purchased at the



11/37

The cost of unused ton of the additive is 2,500 1,000 = 1,500 riyals, hence c = 1,500 riyals per ton.

The shortage cost is the opportunity cost; hence cu

= 7,500 2,500 = 5,000 riyals per ton.

The critical ratio is equal to 5,000/(1,500+5,000) = 0.77.

The laban producer

should

purchase

enough

tons

of

the

. .

From the normal distribution curve, the area under the curve and to the left of Q* is 0.77.

From the standard normal tables, z = 0.74. Using the standard normal distribution, z = (Q* )/ .



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Therefore,

* =. . .

Q* = 15 tons The probability of shortage is

0.24510.75491)6899.0(174.4

73.11151 ===

F F

The probability of overstock isF(0.6899) = 0.7549



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Multiple period stochastic model When multiple periods are considered, excess items incur

inventory holding cost. The multiple period model is used when items have extended

shelf lives (non perishable). In this case, uo is set to the inventory holding cost per unit per

period. * , ,

ordered at the beginning of the following period.



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Stochastic model with lead time If the lead time is > 0, then the demand during + 1 periods

should be considered. The mean and the standard deviation of the demand should

be replaced by ( + 1) and ( + 1)1/2 .



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Continuous review stochastic demand model with lead

This model is formally referred to as lot size reorder point model.

It is a continuous review; when the level of stock reaches R, an order is places, so it is called (Q, R) model.

Manufacturing is vulnerable to shortage only during the lead time (waiting for the order to arrive), hence the total demand durin the lead time is considered to be a random variable.

The lot size Q is selected so that the setup cost and holding cost are minimum and R is selected that will reduce the expec e s or age cos .

Assume the total demand during the lead time has mean and variance 2.


When the demand rate is , then = .


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The expression of cost that is derived for this case is only an approximation; hence the application of the model in real production planning will require making reasonable assumptions.

n e erm n ng , s mo e s approx ma e y e model.

Hence the demand er eriod demand rate is iven b . The level of inventory at the end of the cycle is

s = R The quantity s is called the safety stock.

The total inventory in a cycle is


s 2+


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The setup cost is K. ,

short during the lead time has to be estimated. Shortage happens when the total demand during the lead

time exceeds R; hence the expected shortage units is

= dx x f R xn )()( The total cost in a cycle is

p : shortage cost per unit

2



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The average total cost is

The quantities Q and R are selected in order to minimize G:

n p, 2=

0/ /2/22

Qn pQ K hG

==

2//) (

2

2 hQn p K =+

( ))1(

2 pn K Q

+=



19/37

0/))(( =+=

Qdx x f p hG

))(1( = hQ R F p

)(1 = p

F

)2(11 =

p F R

e so u on w requ re era ve y so v ng an un convergence is reached, with any reasonable starting value for Q.



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The standardized loss function is given by

(t) : standard normal density function

)(1)()()()( z z z dt t z t z L z

==

If D is a normal random variable with and 2, then the

expected number of shortage units is

)()( dx x f R xn R

=

)(

z L

dt t t R z

=

= =



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Illustration

A manufacturing plant requires a specific raw material, with a unit cost of 100 riyals.

The vendor delivers the requested raw material quantity in 6 months.

The fixed ordering cost is 500 riyals, and inventory interest rate is 20% annual.

250 riyals per output unit when the raw material is in short.

The demand during the lead time is normally distributed with mean of 100 units and standard deviation of 25 units.

How many units of this raw material should be ordered every



22/37

The inventory holding cost h = 0.2 100 = 20 riyals. .

The mean

demand

in

6

months

(lead

time)

is

100

units;

then

the demand rate (the demand per year) is = 2100 = 200 units.

Initially, Q

is

set

to

any

reasonable

quantity,

such

as

200

units

.

The value of z is computed:20020

11

From the standard normal table, z = 1.41.

.200250


Hence, R = 251.41+100 = 135.


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From the L tables, L(1.41) = 0.0359. = = . . .

The modified

value

of

Q

is

computed:

0.89752505002002 +

The rest of the calculations are

20==

Q 1Qh/p z R L n200 0.9200 1.41 135 0.0359 0.8975121 0.9516 1.66 142 0.0201 0.5025

112 0.9552 1.70 142 0.0183 0.4575

111 0.9556 1.70 143 0.0183 0.4575


.


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The safety stock is= = = .

So, when

the

lot

is

received,

the

level

of

inventory

is

Q + s = 111 + 43 = 154 units



25/37

Determining Q and R from a service level

If the shortage cost p is difficult to come by, other indicators of acceptable service are used.

There are two particular indicators in use; shortage probability (type 1) and proportion of demand that is met

ype . In type 1, the probability of shortage not happening during

the lead time is s ecified. The specification of determines the quantity R; hence Q and

R are separately calculated. To determine R, set F(R) = . The lot size is specified by Q = EOQ.



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Illustration

Manufacturing has determined that the best service level is when the probability of no shortage is 0.98.

The total demand during the lead time is normally distributed with mean 100 units and standard deviation of 25 units.

The inventory holding cost is 20 riyals per unit per year, and order setup cost is 500 riyals.

. Then, R is selected so that

100 R From the z tables,

.25 =


25.15105.225 ==

R


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The demand rate is = 100 2 = 200 units per year.

1005002002 ==Q



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In type 2, the proportion of the demand that is met is specified.

The average fraction of the demand that is in short is n/Q. Hence, R is selected so that n/Q = 1. Similarly, Q is set to EOQ.



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Illustration

Manufacturing has determined that the best service level is when the proportion of the demand that is met is 0.98.



. The demand rate is = 100 2 = 200 units per year.

The lot

size

is

10020

5002002 ==Q



30/37

Then, R is selected so that

2

.

=

=

n

n

From the L tables for L(z) = 2/25 = 0.08, z = 1.02. Hence, R is

5.12510002.125 =+= R



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Exact Q under type 2 policy A more accurate value of Q can be found if desired. ,

( ))(1 R F Qh

p = Substituting p in the Q equation leads to

22 n K n)(1)(1

++

= R F h R F

s equat on an t e equat on n = are so ve iteratively until convergence is reached.

The uantit can be initiall set to EO .



32/37

Illustration

Manufacturing has determined that the best service level is when the proportion of the demand that is met is 0.98.



. The demand rate is = 100 2 = 200 units per year.

The initial

value

of

Q

is

10020

5002002 ==Q



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The expected shortage units is

The value

of

R

is

.

12610025/225 1 =+= L The next estimate of Q is

2

114

25

1001261

)220

2005002

25

1001261

2 =

++

=Q

The next value of R is 124; and Q = 114.


and R = 124.


34/37

Imputed shortage cost The use of a service level criterion does not necessitate the

specification of a shortage cost. For a given service level or , there is a corresponding value

for p in the continuous review stochastic demand model, w c w g ve e same an .

Since there is no shortage cost specified, the value of p is called the im uted shorta e cost.



35/37

Illustration

For a certain manufacturer, Q = 100, R = 151, = 0.98, = 200, = 25, and h = 2.

The imputed shortage cost is2100 ==

( )

)151(1002 F



36/37

Scaling of lead time demand It may be possible that the demand is expressed per period

such as a week or month. It is necessary to scale the lead time demand if the lead time

is different than the period of the demand. If the demand follows a normal distribution, then the demand

during the lead time is the sum of the demand normal random variables.

Assume the per period demand has a mean of and variance 2.

If the lead time is , then the demand during the lead time has a normal distribution function with mean = and variance 2 = 2.



37/37

Illustration

The demand for a certain raw material in a week is normally distributed with mean 34 and variance 8.

The lead time is 5 weeks. The total demand during the lead time is normally distributed

with

mean = 34

5 = 170 units = =

probabilistic inventory models raw materials

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