Determining Optimal Level of Product Availability: Uday Venkatadri January 11, 2018

Determining Optimal Level of Product
Availability
Uday Venkatadri
January 11, 2018

IENG 4579: Supply Chain Management 2
Single Period Lost Sales Newsboy Problem

Let:
r = Unit price of the product
c = Unit cost of the product
s = Salvage value per unit of product
Then:
Cost of Understocking, Cu = r−c
Cost of Overstocking, Co = c−s
The question is to decide how much stock of the product to carry

in order to minimize the total expected costs of under and over
stocking.
The Newsboy Critical Probability

1. Start with a stock level of zero and keep adding units until the
marginal cost of overstocking is less than the marginal cost of
understocking.
2. Let p be the probability of not selling the marginal unit.
3. The critical value of p is reached when the two marginal costs
are equal. Stop ordering additional units.
Thus:
pCo = (1 − p)Cu
Cu
⇒p =
Co + Cu
Expected Profit in the Newsboy Model

If demand during the period is normally distributed with mean µ
and standard deviation σ, the optimal order quantity
O∗ = F −1 (p, µ, σ), where F −1 is the inverse of the normal
cumulative probability density function.
Let:
Fs (.) = Standard normal cdf

fs (.) = Standard normal pdf
F (.) = Normal cdf
f (.) = Normal pdf
Then, the expected profit, EP is given by:

EP = (r − s)µFs ( O−µ
σ ) − (r − s)σfs ( O−µ
σ ) − O(c − s)F (O, µ, σ) +
O(r − c)[1 − F (O, µ, σ)]
Newsboy Example
SparesRUs, an auto parts retailer, must decided on the order size of
a model of brakes. The demand is expected to have a mean of 350
and a standard deviation of 100. The retail price of the brakes is
$250. Each brake costs $100. Unsold brakes are disposed of for $80.
How many brakes should be ordered?
Cu = $150
Co
= $20
150
⇒p = , i.e., 0.88
150 + 20
⇒ O∗ = F −1 (0.88, 350, 100), i.e., 468
The expected profit can be shown to be $49,146.

Desired Cycle Service Level for Continuously

Stocked Items
The Backordering Case
Assume: Number of back orders relatively small compared to
average on-hand inventory
• Let X be the demand during lead time and r the reorder point.
• Let H = hC be the holding cost per item per year.
• Increasing the safety stock by one unit is equivalent to
increasing the re-order point by one unit. This is because the
reorder point r = D.L + SS, where D is the demand, L the
lead time, and SS the safety stock.

We use marginal analysis to figure out by how much to increase
safety stock.
• By increasing the safety stock by one unit, an additional
holding cost of H is incurred.
• However, this decreases the probability of stockout and as a
result, backordering cost.
If Cu be the backordering cost, then at the critical point, we should
have:
H = P (X ≥ r) × Cu × n, which gives us:
H
P (X ≥ r) = nCu

D
Substituting n = Q, we obtain:
HQ
P (X ≥ r) = DCu
The customer service level (CSL) is the probability of not stocking

out during a cycle.
In other words, it is the probability of X being less than or

equal to r.
Therefore:
HQ
CSL = 1 − P (X ≥ r) = 1 − DCu
For this formula to be valid, HQ should be less than equal to DCu .
Generally speaking, Q < D, and H < Cu and therefore,
HQ < DCu .
Desired Cycle Service Level for Continuously

Stocked Items
The Lost Sales Case
Assume: Number of lost sales relatively small compared to average
on-hand inventory. Again, let X be the demand during lead time
and r the reorder point. Let H = hC be the holding cost per item
per year.
• Again, we use marginal analysis to figure out by how much to
increase lead-time.
• By increasing the lead time by one unit, an additional holding
cost of H(1 − P (X ≥ r)) is incurred. This is because the
increase in holding cost is partially offset by the loss in sales
which will eventually lead to a reduction in inventory.
• Additional inventory decreases the expected lost sales cost.
The Lost Sales Case

If Cu be the cost of lost sales, then at the critical point, we should
have:
H(1 − P (X ≥ r)) = P (X ≥ r) × Cu × n, which gives us:
H
P (X ≥ r) = H+nCu
D
Substituting n = Q, we obtain:
HQ
P (X ≥ r) = HQ+DCu
The customer service level, CSL is the probability of not stocking

out during a cycle. In other words, it is the probability of X being
less than or equal to r. Therefore:
HQ DCu
CSL = 1 − P (X ≥ r) = 1 − HQ+DC u
= HQ+DCu
CSL will always be less than 1.
Continuously Replenished Items Example

The demand for a continuously replenished product is 100 units per
week. The standard deviation of demand per week is 20 units. The
value of the item is $3 the holding rate used is 20%. The EOQ for
the product is 400 units. Since the lead time is 2 weeks, the
product reorder point is set at 200 units.
1. For the back ordered case, if Cu is $20 per unit what will be
the resulting CSL and reorder point?
2. For the lost sales case, if Cu is $80 per unit what will be the
resulting CSL and reorder point?
3. Determine the relationship between the optimal reorder point
and the cost of understocking Cu for both the backordering
and lost sales cases.
Continuously Replenished Items Solution

Please take a look at the tab “Continuous-1” in the Excel file
Availability.xls.
The relationship between Cu and the reorder point is shown in the

form of a table.
Strategies to Improve Supply Chain Efficiency

• Improved Forecasting: By using better forecasting methods and
collaborating with customers, it may be possible to reduce the
measure of uncertainty (standard deviation) in demand, i.e., σ.
• Quick Response: The lead time L in continuous review models
and L + T in periodic review models could be reduce to reduce
the level of safety stock inventory.
• Postponement: Postpone product differentiation closer to the
point of sale
• Tailored Sourcing: Use a low lead-time supplier at higher cost
as a backup for the low-cost perhaps long lead-time supplier.
Impact of Improved Forecasting Example

Question: For the example just discussed, show the impact of
reducing the standard deviation from the current level of 20 units
per week to 0 units per week. Assume that the cost of backordering
or lost sales, Cu , is $100 per unit.
Answer: Refer to the tab “Continuous-2” in the Excel file

Availability.xls. It can be seen that decreasing uncertainty reduces
the level of required safety stock.
Impact of Reducing Standard Deviation through

better Forecasting Methods
Figure 1: Effect of Reducing Standard Deviation

Impact of Reducing Lead-time

Question: For the example just discussed, show the impact of
reducing lead time from the current level of 2 weeks to 1 week.
Assume that the cost of backordering or lost sales, Cu , is $100 per
unit.
Answer: Refer to the tab “Continuous-2” in the Excel file

Availability.xls. It can be seen that decreasing the lead time
reduces the level of required safety stock.
Impact of Reducing Lead-time
Figure 2: Effect of Lead-Time

Postponement Example
United Colors of Benetton can either dye and knit (Option 1) or
postpone dyeing until after a garment is knitted (Option 2). The
retail price of each garment is $50. The manufacturing costs for the
two options are $20 and $22 respectively. The salvage value of each
garment is $10. Knitting takes 20 weeks. Garments are sold in four
colours, each with a mean (independent) demand of 1,000 and a
standard deviation of 500. With Option 1, Benetton makes the
buying decision for each colour 20 weeks in advance and holds
separate inventories for each colour. With Option 2, Benetton
forecasts only the aggregate uncoloured thread to purchase 20
weeks in advance. The held inventory is based on the aggregate
demand across all four colours. The quantity for individual colours
is made after the demand is known.
Quantify the impact of postponement for Benetton.

Benetton Example
Refer to the tab “Benetton” in the Excel file Availability.xls.
Option 1
Clearly, Cu = 30 and Co = 10.
30
Therefore, CSL = 30+10 = 0.75
The optimal ordersize is given by:
O∗ = F −1 (0.75, 1000, 500) = 1337
Expected profits are: $94,578
Option 2
28
Since Cu = 28 and Co = 12, CSL = 28+12 = 0.70
Over all colours, the mean demand is 4000 and standard deviation
p
is 1000 (= 500 ∗ (4)).
The optimal ordersize is given by:
O∗ = F −1 (0.70, 4000, 1000) = 4524
Expected profits are: $98,092

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Determining Optimal Level of Product

January 11, 2018

Single Period Lost Sales Newsboy Problem

The question is to decide how much stock of the product to carry

The Newsboy Critical Probability

Expected Profit in the Newsboy Model

Fs (.) = Standard normal cdf

Then, the expected profit, EP is given by:

The expected profit can be shown to be $49,146.

Desired Cycle Service Level for Continuously

The Backordering Case

The Backordering Case

The customer service level (CSL) is the probability of not stocking

In other words, it is the probability of X being less than or

Desired Cycle Service Level for Continuously

The Lost Sales Case

The customer service level, CSL is the probability of not stocking

Continuously Replenished Items Example

Continuously Replenished Items Solution

The relationship between Cu and the reorder point is shown in the

Strategies to Improve Supply Chain Efficiency

Impact of Improved Forecasting Example

Answer: Refer to the tab “Continuous-2” in the Excel file

Impact of Reducing Standard Deviation through

Figure 1: Effect of Reducing Standard Deviation

Impact of Reducing Lead-time

Answer: Refer to the tab “Continuous-2” in the Excel file

Impact of Reducing Lead-time

Figure 2: Effect of Lead-Time

Quantify the impact of postponement for Benetton.

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