Inventory Optimization Model

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EXECUTIVE SUMMARY
Three Jays Corporation, a subsidiary of Fremont Jams and Jellies (FJ&J) was started by Jana
Fremont in 2005. A significant increase in production due to growing demand for organic foods
was observed over the years since its inception. However, to maintain this growing trend,
company needs to run some attractive marketing campaigns. The company feels that reducing
the inventory level would save them a major chunk of the revenue which can be invested in
these campaigns. Therefore, the need to make immediate changes to the existing model of
inventory handling was imminent.
This report looks at the various parameters and approaches being considered in the current
inventory policy and identifies the potential drawbacks. Starting by analyzing the current model
which had a total annual inventory cost of $11,300 for the given sample of which majority was
penalty cost for backlogging, the report goes on to recommend a Q-R model (weekly inventory
review) which reduces the total annual inventory costs to $5,175 and increases the service level
to 99.8% from current level of 79.2%. Analyzing further, we find that a continuous review of
inventory level in the proposed Q-R model, the total inventory cost further drops down to
$4,522 with marginal increase in service level to 99.8%.
The incorporation of demand forecast in the Q-R model will help business anticipate the future
trend of demand and thus help them plan inventory especially safety stock optimally. This is can
be potential next step in inventory system redesign journey of the organization.

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1 Contents
EXECUTIVE SUMMARY .................................................................................................................................2
2 Introduction..........................................................................................................................................4
3 Cost Structure.......................................................................................................................................5
3.1 Present Cost Structure .................................................................................................................6
3.1.1 Setup Cost.............................................................................................................................6
3.1.2 Unit Cost ...............................................................................................................................7
3.1.3 Carrying cost.........................................................................................................................7
3.2 Revised Cost Structure .................................................................................................................7
3.2.1 Setup Cost.............................................................................................................................7
3.2.2 Unit cost & Carrying cost......................................................................................................8
3.2.3 Penalty cost ..........................................................................................................................9
4 EOQ-ROP model ...................................................................................................................................9
5 Current Process Followed ..................................................................................................................11
6 Q-R Model with Periodic Inventory Review......................................................................................13
6.1 Q-R with Continuous Review .....................................................................................................15
7 Conclusion and Recommendation.....................................................................................................15
8 Future Scope.......................................................................................................................................16
9 APPENDIX ...........................................................................................................................................17

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2 Introduction
FJ&J was started by Alex Fremont in 1954 which produced high-quality jams and jellies under
their own label, Fremont Preserves. Owing to its surplus capacity, it went on to produce for
other supermarket chains under private labels.
Convinced of a growing market for organic products, Jana Fremont started an independent
production facility under the name ‘Three Jays Corporation’ which manufactured organic jams
and jellies. Complying with federal regulations required for labelling products organic, Three
Jays’ entire production line, starting from purchase of the raw materials to warehousing of the
finished products, was made separate from that of FJ&J’s. Their product line has a total of 141
SKUs which comprised of four different jar sizes - 2, 4, 8, and 12 ounces - with the 12-ounce size
being most popular. Each stock-keeping unit (or SKU) is defined by a set of three parameters –
the kind of jam/jelly, the size of the jar, and the label.
The company has setup an EOQ-ROP system that reviews the weekly inventory levels of each
SKU. It sends a report with the list of SKUs whose inventory level has fallen below the reorder
point (ROP) and amount that needs to be produced for each. This system closely resembles the
Q-R model, the only difference being that the former treats demand to be constant over time
and thus EOQ and ROP are estimated without considering possible chances of stock outs. The
order for various raw ingredients is placed through FJ&J’s purchasing department. The

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production cycle starts with weighing and prepping the ingredients which is taken care by
Emma & Julia. Once the ingredients are cooked properly, the cooked mixture is sent down to
the filling stations where the jars are filled and later taken off to the nearby warehouse for
storage. This downstream production line is overseen by Jake and Josh who are assisted by
three other part-time workers. While prepping takes almost an hour, the remaining part of the
cycle is completed in an additional 90 min. Additional time is spent in cleaning the equipment
between production of different flavors, and during the size changeovers.
The company feels that they hold high inventory levels and thus incur high inventory holding
costs. Considering the growing demand for organic foods, the company wishes to launch a
major marketing campaign for which funds are required. It appears to the company that one of
the quickest ways to obtain these funds is by reducing their inventory levels of finished goods.
But at the same time, the company wants to increase their efficiency to hold onto customer
good will. However, reducing the inventory levels may not be the only way to obtain savings as
reduced inventory level may lead to significantly low service levels. Thus, it is necessary to
conduct thorough analysis of the current system and suggest suitable alternatives to maximize
savings and attain a desirable service level.
3 Cost Structure
In this section we studied the different components of cost that are used by company in
EOQ_ROP system. The study is divided into two segments – the first one looks at the current
components while the second section highlights anomalies and required changes in the present
cost structure. The cost structure plays an important role in determining various parameters of

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inventory models like optimal quantity to produce, reorder point etc. Thus, it becomes
necessary to vet the current estimates and perform necessary corrections.
3.1 Present Cost Structure
The cost structure used by company in current inventory system is being discussed in detail in
this section. The method used to estimate each component along with the reason for inclusion
of each is presented below.
3.1.1 Setup Cost
Order processing cost: This is the cost of the administrative labor in FJ&J’s purchasing
department for placing orders for 3Js which was estimated at $5.25/batch.
Product preparation, cooking & cleaning cost: Emma & Julia, the two full time workers were
paid on an hourly basis at a constant $30.25/batch for this operation.
Size changeover cost: It took Jake & Josh about 30 min. every time there was a change in the
jar size. Their contribution amounted to $4.7/SKU (i.e. $23.5/hr x 2 workers x 30 min =
$23.5/batch and $23.5/5 = $4.7/SKU taking an average production of 5 SKUs).
Production-line cleaning cost: Every time there is a change in the flavor, it took Jake & Josh
another 30 min. for cleaning the equipment. They were being paid an additional $23.5/batch
(i.e. $23.5/hr x 2 workers x 30 min = $23.5/batch).

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3.1.2 Unit Cost
This included the variable components which varied with the number of units being produced
like the material cost, production labor cost which was set at $1.29/person/batch, and the fixed
and variable overheads set at a constant $2.55 and $1.45 per case.
3.1.3 Carrying cost
Since warehousing bore no costs, the annual carrying cost was revealed at 9% which combined
a 6% capital cost with an additional 3% charges on maintenance and depreciation.
3.2 Revised Cost Structure
Three Jays reaped savings on production equipment & storage space which were provided free
of cost by FJ&J’s on the account of surplus capacity. But a thorough analysis of the present cost
structure reveals several discrepancies that need to be corrected.
3.2.1 Setup Cost
Order processing cost: Managing the orders was taken care of by buyers at FJ&J’s who placed
the orders for both 3Js’ and FJ&J’s. Since this cost is not linearly dependent on number of
orders generated, we can safely remove this component from set up cost. Moreover, this
person is employed by FJ&J, thus a detailed analysis on maximum order capacity was not
conducted as Three Jays don’t bear cost of this person.
Product prep, cooking & cleaning cost: Both the workers work full-time for Three Jays and are
said to perform a variety of other tasks when not working on the production line which may not
be directly related to the production itself like, for example, working on new recipes.
Irrespective of the number of setups performed over a week, which can be down to zero during

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a given backup period, their pay remains unaffected which justifies our decision of not including
their contribution towards the setup cost.
Production-line cleaning cost: Jake & Josh, who are employed full-time by FJ&J and this cost
being independent of number of batches run annually, we neglected it from the setup cost.
However, during this period (30 mins), the three part-time workers remain idle simply by virtue
of the production line itself being idle. So, the cost born due to this idle time for the workers
needs to be considered here. This yields an amount of $18.75 per batch (i.e. $12.5/hr x 3
workers x 0.5 hrs).
Size changeover cost: Going with the previous argument, we can eliminate Jake & Josh’s
contribution to this cost. According to the system being presently followed, size change occurs
once per every week (for 0.5 hrs) and hence it can be considered that this activity doesn’t make
the part-time workers idle. Though in any other model where the production is not limited to
one size per week, the frequency of size changeover would be higher thereby leaving the
workers idle during the changeover period which would entail an average cost of $3.75/SKU
considering an average of 5 SKU’s per production run (i.e. $12.5/hr x 3 workers x 0.5 hrs =
$18.75/batch; 18.75/5 = $3.75/SKU).
3.2.2 Unit cost & Carrying cost
The estimates given for the unit cost appear to be consistent except that some further details
are needed to better validate the Fixed and Variable Overhead. The carrying cost used in
current system doesn’t account for the opportunity cost. The company believes that this capital

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if invested in marketing campaigns will yield a return of 20%. Thus, carrying cost rate should be
0.29 (0.09+0.20).
3.2.3 Penalty cost
A major cost which wasn’t being considered in the original structure is the cost of being stocked
out. The company has plenty of production capacity to meet the significantly growing demand,
thus any unmet demand should be associated with a penalty cost as it causes a loss of good will
amongst customers and causes unplanned setups and labor costs. Since it is a crucial growing
face for the company, loss of any good will can be considered a big setback. So, we assume a
penalty worth 60 times the holding cost.
Setup
Setup Cost
Penalty Cost
Inventory
Carrying
Cost
Order
Processing
cost
Prep,
cooking &
cleaning cost
Size
changeover
cost
Production-
line cleaning
cost
Old $5.25 $30.25 $4.70 $23.50 - 0.09
New - -
$3.75
(except in the
current
process)
$18.75
60*Holding
Cost
0.20
Table 3.1 (Cost Structure New vs Old)
4 EOQ-ROP model
The model initially suggested by the inventory team of Three Jays was an EOQ-ROP model. The
two-important components of this model are: 1) Economic order quantity (EOQ): The amount
to be manufactured every cycle. 2) Reorder point(ROP): If the inventory level drops below the
ROP, the department should produce quantity equal to the Economic order quantity.

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For calculating these values, the demand data of the year 2010 was used. Thus, EOQ is
estimated assuming that demand is not evolving over time which is not clearly the case. The
basic intention of this model is to produce the right amount of quantity at the right time.
However, this model has some serious drawbacks. The demand data used was that of the year
2010. On comparing the demand data for the year 2010 and current year, i.e. 2013, we see that
there is almost a 130% percent increase in demand. The EOQ-ROP levels suggested by the
model were extremely low considering the current demand. Thus, the probability of a stock out
is extremely high, reducing the service level significantly. The total inventory costs are low if
there is no penalty associated for the unmet demand. Since the probability of stocking out is
high, consideration of penalty cost skyrockets the total inventory cost. Thus, performance of
EOQ and ROP calculated based on 2010 demand was assessed using 2013 data. The
management feels that updating the EOQ-ROP model with the demand data of 2012 would give
better results. However, since the demand in 2013 is almost 78% more than the 2012 demand,
it might not lead to much better results. Thus, this updated EOQ and ROP were assessed using
similar approach. A low service level is naturally undesirable since it causes a loss of goodwill
and thus with growing business, the company should not be content with this service level.
(Detailed calculations in Appendix Exhibit 3)

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EOQ (2010 Demand data) EOQ (2012 Demand data)
Total inventory cost: $1,36,786 $92,942
Inventory carrying cost: $5,969 $2,821
Setup cost: $1,275 $3,371
Service level: 41.1% 59.78%
Table 4.1 (Performance of EOQ-ROP (2010) and EOQ-ROP (2012) in 2013)
Although this EOQ model optimizes all components of inventory related costs, on-floor workers
feel that majority of the inventory cost can be reduced by reducing the number of setups. Thus,
they implemented a different method that reduces the frequency of setups which has been
discussed in detail in the next section.
5 Current Process Followed
We now move ahead in analyzing the current system used by the on-floor workers which
resembles the conventional R-T model (details in ‘Reference’ Section). R-T model is an
operating strategy for periodic review systems also referred to as “order up to R” doctrine.
Every time there is an inventory review an order is placed/produced which will bring the
inventory position up to R. The decision variables for the system are the “order up to level” R,
and the time between reviews (production) T.
The current system in use follows a weekly manufacturing cycle. The production cycle is split
into 4 parts: Week one manufactures all jams of size 12oz, week two manufactures all jams of
size 8oz, week three manufactures all jams of size 4 and 2 oz, and week four is a backup week
which can be used to manufacture any size jams which run out during the previous weeks. Each

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size is manufactured to meet the demand of the next four weeks with a safety stock worth two
week. Thus, production cycle is scheduled every four weeks for each size. While determining
the demand for the next six weeks, the demand data of the previous month is used.
Drawing parallel of current process using sample R-T model graph (Figure 5.1) shows that point
A is the point where inventory level of each SKUs belonging to a particular size say 12oz is
reviewed and thus SKUs that need to be replenished are identified. The production takes 1
week represented by  in the graph. Then this SKU is again reviewed after 4 weeks making T = 4
in our case. In a conventional R-T model, the R-value is always fixed irrespective of the change
in demand. However, in the method followed by on-floor workers, the R value fluctuates with
the demand since R-level is forecasted using the demand of the previous month. Thus,
conventional R-T model is a close approximation of the current process used.
Figure 5.1 Sample R-T Model Graph
The biggest advantage in this system is the reduction in the number of setups as a size
changeover happens once in a week. This saves the cost required for changing the production
line and reduces the idle time of the part-time workers, thus reducing the overall setup cost. It

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also increases the overall efficiency of the machine usage since frequent changes pertaining to
size can cause machine breakdown which can lead to unwanted expenditure.
However, there are some down sides to the current process in use. Since the quantity to be
manufactured is purely based on intuition, the probability of stock out is significant as they are
not accounting for the variance in demand. The stock up to level is 6 weeks demand forecasted
by workers which is assumed to be 6 times the average weekly demand of the SKU in 2013. The
estimation of cost and service level based on R and T give following results (Refer Appendix
Exhibit 4 for details). Since the company has plenty of manufacturing capacity and a growing
business, the loss of goodwill due to a stock out should ideally be as low as possible. Thus, we
explore some other models to reduce total inventory costs and increase the service level of the
company.
Total inventory cost: $33,701
Inventory carrying cost: $8,522
Setup cost: $1,219
Service level: 79.2%
Table 5.1 (Performance of Current Process approximated using R-T model)
6 Q-R Model with Periodic Inventory Review
Although the current process followed by Jake Evans and Josh Francis is better than the
conventional EOQ-ROP inventory system, there is a need to build a robust model to account for
variance in demand in a structured way. The demand of products is random and thus requires
organization to devise framework to capture these patterns effectively. Q-R model considers

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the variance in demand and recommends the optimal quantity to order as well as reorder point
for a desired service level.
Q-R model utilizes parameters of demand along with the various costs associated with
inventory to calculate the optimum inventory level that should be produced in each cycle (Refer
Appendix Exhibit 5). The analysis of demand data for sample SKUs reveals that demand within a
year is somewhat normally distributed while the yearly demand witnesses stepwise increase
every year. The demand distribution for the sample 12oz SKUs was captured using the latest 5
months data of 2013 to avoid undue inflation of mean and variance as demand sees step
increase from 2012 to 2013. The penalty associated with the unsatisfied demand was also
considered here.
Total inventory cost: $12,809
Inventory carrying cost: $9,865
Setup cost: $2,368
Service level: 99.8%
Table 6.1 (Results of Periodic Q-R Model)
The recommended model closely resembles the EOQ-ROP model that company planned to
implement, the only difference being that the fluctuations in demand is considered while
deciding the order quantity and reorder point. Thus, the model can be easily implemented with
the current infrastructure of Three Jays Corporation and won’t require any extra investment.
We have already seen that changes made by workers generally tends to decrease one of the
cost components but doesn’t necessarily optimize the entire process. Thus, company must

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focus on convincing them to follow the recommendations of the proposed inventory system to
realize potential savings.
6.1 Q-R with Continuous Review
The proposed Q-R model was designed keeping in mind that there are no drastic changes
required in infrastructure for implementation of the model. The analysis shows that proposed
model could be further improved by having continuous review inventory system. This means
that investment must be made by the company to track inventory level on daily or real time
basis. This will reduce the variance of lead time demand as total lead time reduces by one
period (added due to weekly review) and thus reduce the safety stock. This improvised Q-R
model will lead to an additional savings of around 17.16% and reduce average safety stock level
by around 29.07% for the sample SKUs (Refer Appendix Exhibit 6).
7 Conclusion and Recommendation
The current process followed by workers as well as the EOQ-ROP inventory system has
shortcomings which can hinder Three Jay’s objective of capturing the growing market of organic
products. They need to work swiftly towards adopting the robust inventory management
systems which can reduce their operation cost as well as improve the service level.
The proposed Q-R model designed for periodic review inventory system will help them
implement the new system without any changes to infrastructure. The company can then work
on setting up continuous inventory review system and can smoothly transition to the robust
continuous review Q-R model. Any change in organization is bound to face opposition from
various sections due to complexity and huge capital at stake. Thus, savings generated from

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periodic Q-R model can be used to strengthen the case for moving towards continuous review.
The improvement in service level as well as reduction in operating cost will not only benefit
Three Jays in immediate future but will also give them competitive edge over others in the long
run.
Model Total Cost Safety Stock Average Service Level
Current Process $33,701 338 units 79.20%
EOQ-ROP (2010 Demand) $1,36,786 (305%) 32.84 (-90.28%) 41.1% (-38.1 pp)
EOQ-ROP (2012 Demand) $92,942 (175.78%) 83 units (-75.4%) 59.78% (-19.42 pp)
Q-R Periodic Review $12,809 (-62%) 821 units (142.9%) 99.8% (20.6 pp)
Q-R Continuous Review $10,610 (-68.52%) 582 units (72.2%) 99.9% (20.7 pp)
Table 6.1 (Summary of Results)
* Values in brackets indicate comparison with current process; pp: Percentage Points
8 Future Scope
The proposed Q-R model uses latest realized demand data to estimate the demand distribution.
However, as the annual demand of products for Three Jays is not stationary over time, there is
a need to build a forecasting model which can accurately predict the systematic components of
the demand. The forecast from this model can be used to ascertain optimal base stock level and
the residual for previous years will determine the safety stock for a product.
The implementation of forecast model requires demand data for each SKU at a monthly level
for at least 3 years so that residuals can be examined to precisely estimate random pattern in
demand.

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Exhibit 1: Glossary of Terms
10
12
13
Annual demand for a SKU in 2010,
Annual demand for a SKU in 2012,
Annual average demand for a SKU in 2013,
Mean of weekly demand,
Variance of weekly demand,
Standa
D
D
D
D
D
D
V

 rd Deviation of weekly demand,
Variance of monthly demand,
Lead Time (weeks),
Lead time demand,
Mean of lead time demand,
Standard deviation of lead time demand,
M
LTD
LTD
V
L
LTD
Q


Order quantity (whereas * represents the Optimal order quantity),
Re-order point (alternately it is the order-upto level in (R,T) model);
Also, *represents the optimal re-order poin
Q
R
R t,
( ) Expected number of unmet/missed demand given level R;
( ) *[ ( ) * ( )] where and
( ) & ( ) are the pdf and cdf of the normalized variable z,
Setup co
D
D
D
n R
R
n R P z z F z z
P z F z
K




  
st/batch ($),
Annual setup costs ($),
Unit cost/case ($),
Annual carrying cost (%),
Holding cost/case (= ) ($),
Annual holding costs ($),
Penalty cost/c
s
h
C
c
i
h ic
C
p ase ($); here, it is taken as 60* ,
Annual penalty costs ($),
Total inventory-related cost ($)
p
p h
C
TC


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Exhibit 2: Sample Parameters (SKU 1)
10
12
13
For sample calculation, Product 1 is considered whose parameters are discussed below:
2993 cases,
3869 cases,
6888 cases;
[Cumulative demand for product 1 over 5 months in 20
D
D
D
13 (i.e. Jan - May) = 2868;
2868
Then, avearge annual demand = *12 6888 cases],
5
132 cases ( = 2868/(5*4.33) as there are 4.33 weeks in a month on an average),
12646 sq. cases [
D
MV V


2
1
1
= ( ) 12646 ],
( 1)
2921 sq. cases [ 2921],
4.33
54.0462 cases [= ],
$28.34/case,
29%,
$8.218/case,
$493.116/case (i.e. 60*8.218)
n
M i D
i
M
D D
D D
D
n
V
V V
V
c
i
h
p



 

 


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Exhibit 3: EOQ-ROP Model
Parameter
EOQ Model (2010 parameters) EOQ Model (2012 parameters)
SKU 1 SKU 2 SKU 3 SKU 4 SKU 5 SKU 1 SKU 2 SKU 3 SKU 4 SKU 5
Demand 2993 2335 1492 886 625 3869 3006 1970 1211 832
Q* (cases) 387 329 280 208 183 146 124 107 80 70
R* (cases) 173 135 86 51 36 223 173 114 70 48
Cs ($) 404.99 473.84 115.27 138.12 142.94 1075.88 1261.36 302.98 356.83 374.2
Ch ($) 1588.84 1456.35 1332.21 891.63 699.7 598.1 547.09 870.39 513.3 291.71
Cp ($) 47582.47 72639.6 689.05 2350.64 6250.64 27511.5 55622.6 51.65 239.31 3324.52
TC ($) 49576.32 74569.8 2136.54 3380.4 7123.23 29185.5 57431.1 1225.03 1109.45 3990.44
Σ $136,786.29 $92,941.56
* *
10 10
Setup cost for Product 1: K = $22.5/batch,
Periodic Review & L=1 week: ( 1)* 264, 1* 76.432,
1. 387, 173,
2. Annual setup
LTD D LTD DL L
Q R
        
 
Sample Calculation :
EOQ - ROP based on 2010 estimates :
13*
10
*
*10
10
*
10
22.5
cost C = * *6888 $404.65,
387
3. Annual holding cost C *[ max{ ,0}]
2
387
8.218*( max{173 264,0}) $1588.47,
2
173 132
4. =
s
h LTD
LTD
LTD
K
D
Q
Q
h R
R
z



 
  
   
 
 *
10
*
10
= 0.758. Then, ( ) *( ( ) * ( )) 96,
54.04
Then, annual penalty costs C * ( ) $47,582.47,
5. Total inventory cost TC = C +C +C $49,576.
LTD
p
s h p
n R P z z F z
p n R
  
 


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* *12
12 12 12
*
* 12
12
*
12
13*
12
2* * 2*22.5*3869 3
1. 146, * 223,
8.218 52
2. ( ) *( ( ) * ( )) 96 where = 0.536,
3. Total Cost TC C +C +C * *[ ma
2
LTD
LTD
LTD
s h p
K D
Q R D
h
R
n R P z z F z z
K Q
D h
Q



    

    
   
EOQ - ROP based on 2012 estimates :
* *
12 12x{ ,0}] * ( )
= 1075.88 + 598.1 + 27511.5 = $29,185.48.
LTDR p n R 

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Exhibit 4: Current Process in Use
SKU R (cases) T (yrs) Cs ($) Ch ($) Cp ($) TC ($)
1 794 0.08 234.37 3263.66 4121.5 7619.53
2 791 0.08 234.37 3500.01 13358.62 17093.01
3 164 0.08 234.37 638.49 2956.53 3829.3
4 146 0.08 234.37 612.62 881.51 1728.5
5 133 0.08 234.37 507.28 2642.15 3383.8
Σ $1171.85 $8522.06 $23960.33 $33654.24
Setup cost for product 1: K = $18.75,
1. Average production upto level R = 6* 794 (since six weeks inventory is always maintained),
4
2. Time period between production cycles = 0
52
D
T
 

13
.08 yrs
(since SKUs of a particular size are produced once in every 4 weeks),
1
3. Time taken for production = 0.02 yrs,
52
4. For ( 0.1 yrs) period of uncertainity:
Mean demand ( )*P
T
T D


 

 
  
13
662, Variance = ( )*( )*52 = 14591.88,
4.33
Standard deviation 120.79,
5. 1.0958, then ( ) *( ( ) * ( )) 8.4,
6. Total cost TC = C +C +C *[ * ] * ( )
2
M
P
P P
P
p P P
P
s h p D P
V
V T
V
R
z n z P z z F z
K T
h R D p n z
T







 

    
    
= 234.37 + 3263.66 + 4121.5 = $7619.53.

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Exhibit 5: (Q,R) Model with Periodic Review
Setup cost for product 1: K = $22.5, Lead time: L = 1 week,
1. Mean of LTD: ( 1)* 265,
Variance of LTD: ( 1)* 5842, Standard deviation 76,
2. To find (Q*,R
LTD D
LTD D LTD LTD
L
V L V V
 

  
    
13
1
1
1
1
*) for a given (Q,R) system: we start with an initial assumption of Q
from the basic EOQ model:
2* * 2*22.5*6888
Iteration 1: 194;
0.29*28.34
*
From ( ) 1
*
K D
Q
h
h Q
P D R
p D
  
  
1
3
1
1
1
349*0.29*28.34
1 0.9991,
493.11*6888
we get 518 assuming D ~ N( , );
then 3.328 which gives ( ) 0.0094,
Iteration 2: Taking
LTD LTD
LTD
R
LTD
R
R
z n R
 


  


  
1
1 13
2
1
1 2
13
n(R ) from the previous iteration, recalculating Q:
2*[ * ( )]* 2*[22.5 493.11*0.01763]*6888
213.34;
0.29*28.34
*
From ( ) 1 ,we get
*
K p n R D
Q
h
h Q
P D R R
p D
 
  
    2515.74 and ( ) 0.0104,n R 
Product Q* (cases) R* (cases) n(R*) Cs ($) Ch ($) Cp ($) TC
1 215.45 515.53 0.0105 719.35 2944.88 166.19 3830.43
2 219.76 641.97 0.0163 702.3 4319.4 271.07 5292.78
3 98.25 134.52 0.0084 325.61 1004.54 57.07 1387.23
4 86.39 93.19 0.0046 329.03 738.66 34.37 1102.07
5 89.02 112.12 0.0081 291.37 857.1 48.4 1196.88
Σ $2367.9 $9865.16 $577.7 $1,810.76

of 25
3 3 3
4 4 4
Iteration 3: 215.26, 515.55, ( ) 0.0105,
Iteration 4: 215.45, 515.53, ( ) 0.0105,
Since values in the last two iterations coincide, we can take *=215.45, *=515.53
and
Q R n R
Q R n R
R Q R
  
  
13
13
( *)=0.0105.
*
3. Total cost TC = C +C +C = * *[ * ] * ( *)*
* 2 *
= 719.35 + 2944.88 + 166.19 = $3830.42
s h p LTD
n R
K Q D
D h R p n R
Q Q
   

of 25
Exhibit 6: (Q,R) Model with Continuous Review
SKU Q* (cases) R* (cases) n(R*) Cs ($) Ch ($) Cp ($) TC ($)
1 208.99 310.12 0.007 741.59 2318.93 117.28 3830.42
2 209.55 400.4 0.0109 736.52 3303.65 191.07 4231.24
3 95.95 83.92 0.0058 333.42 814.46 40.28 1188.16
4 85.14 55.88 0.0032 333.83 623.87 24.28 981.98
5 87.04 70.2 0.0056 298.01 698.83 34.17 1031.01
Σ $2443.6 $7759.75 $407.1 $6362.39
Setup cost for product 1: K = $22.5, Lead time: L = 1 week,
1. Mean of LTD: * 132,
Variance of LTD: * 2916, Standard deviation 54,
2. To find ( *, *) for a
LTD D
LTD D LTD LTD
L
V L V V
Q R
 

 
   
13
1
1
1
13
given (Q,R) system: we start with the EOQ model:
2* * 2*22.5*6888
Iteration 1: 194;
0.29*28.34
* 194*0.29*28.34
From ( ) 1 1 0.9995,
* 493.11*6888
K D
Q
h
h Q
P D R
p D
  
     
1
1
1
1
1
we get 311 assuming D ~ N( , );
then 3.314 which gives ( ) 0.0066,
Iteration 2: Taking n(R ) from the previous iteration, recalculating Q
LTD LTD
LTD
R
LTD
R
R
z n R
 




  
1 13
2
1
1 2 2
13
3
:
2*[ * ( )]* 2*[22.5 493.11*0.0066]*6888
207.92;
0.29*28.34
*
From ( ) 1 ,we get 310.2 and ( ) 0.0071,
*
Iteration 3: 208.92,
K p n R D
Q
h
h Q
P D R R n R
p D
Q
 
  
    
 3 3
4 4 4
310.13, ( ) 0.0072,
Iteration 4: 208.99, 310.13, ( ) 0.0072,
Since values in the last two iterations coincide, we can take *=208.99, *=310.13
and ( *)=0.0072.
3. Total cost
R n R
Q R n R
R Q R
n R
 
  
13
13
*
TC = C +C +C = * *[ * ] * ( *)*
* 2 *
= 741.59 +2318.93 + 117.28 = $3177.78.
s h p LTD
K Q D
D h R p n R
Q Q
   

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