Optimal Service Level in Production and Warehousing. How sales forecasts can be used to set levels in production or in multi-level warehousing

1. Optimal Service Level in Production and Warehousing
Introduction
In the following we will show how sales forecasts can be used to set levels in production or in multi-level
warehousing. Since the problem discussed is the same for both production and warehousing, the two
terms will be used interchangeably.
The calculation will be based on knowledge of the sales distribution, both expected sales and its
variation. In addition will sales usually have a seasonal variance creating a balance act between
production, logistic and warehousing costs.
In the example given below the sales forecasts will therefore have to be viewed as a periodic forecast
(month, quarter, etc.).The production lead time will then determine the production planning (timing).
Purposes of Inventory
1. To maintain independence of operations
2. To meet variation in product demand
3. To allow flexibility in production scheduling
4. To provide a safeguard for variation in raw material delivery time
5. To take advantage of economic purchase-order size
Inventory Costs
1. Holding (or carrying) costs
2. Costs for capital, storage, handling, “shrinkage,” insurance, etc.
3. Setup (or production change) costs
4. Costs for arranging specific equipment setups, etc.
5. Ordering costs
6. Costs of someone placing an order, etc.
7. Shortage costs
8. Costs of canceling an order, etc.
Inventory Systems
1. Single-Period Inventory Model
a. One time purchasing decision (Example: vendor selling t-shirts at a football game)
b. Seeks to balance the costs of inventory overstock and under stock
2. Multi-Period Inventory Models
a. Fixed-Order Quantity Models
b. Event triggered (Example: running out of stock)
3. Fixed-Time Period Models
a. Time triggered (Example: Monthly sales call by sales representative)
The “too much/too little problem”
1. Order too much and inventory is left over at the end of the season
2. Order too little and sales are lost.
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2. To maximize expected profit order Q units so that the expected loss on the Qth unit equals the expected
gain on the Qth unit:
I. Co F(Q) Cu 1 FQ ,
Where Co =The cost of ordering one more unit than what would have been ordered if demandhad been
known – or the increase in profit enjoyed by having ordered one fewer unit,
Cu = The cost of ordering one fewer unit than what would have been ordered if demandhad been
known– or the increase in profit enjoyed by having ordered one more unit, and
F(Q) = Probability Demand for q<= Q
Rearrange terms in the above equation
Cu
II. Prob{Deman d Q} F(Q)
Co Cu
The ratio Cu / (Co + Cu) is called the critical ratio (CR).
The usual way of solving this is to assume that the demand isnormal distributed N(m,s)giving Q as:
III. Q = m + z * s, where: z= (Q-m)/s is normal distributed N(0,1)
Demand however has seldom a normal distribution and to make things worse we usually don’t know
the exact distribution at all. We can only ‘find’ it by Monte Carlo simulation and thus have to
numerically find the Q satisfying equation I.
The optimal service level
The warehouse (or production) level should be set to maximize profit given the sales distribution. This
implies that the probability for stock out (lost sales) should be weighed against warehousing, logistic and
production costs.
If we for the moment assume that all thesecosts can be regarded as a variable cost, will the product
markup (%) determine the optimal warehouse level.
Expected sales
The figure below indicates the sales distribution. Expected sales are 1819 units, but the distribution is
heavily skewed to the right so there is a possibility of sales exceeding expected sales:
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3. By setting the product markup – in the example below it is 300% - we can calculate profit and loss based
on the sales forecast.
Profit and Loss of opportunity
The loss is calculated as the value of lost sales (stock-out) and the cost of having produced and stocked
more than can be expected to be sold.
The profit is calculated as value of sales less production costs of both sold and unsold items.
The figure below indicates what will happen as we produce and stock at different levels of probability of
stock-out. We can see as we successively move to higher production (from left to right on the x-axis)
that expected profit will increase to a point of maximum, the same point where loss is minimized:
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4. At that point we can expect to have some excess stock and in some cases also lost sales. But regardless,
it is at this point that profit is maximized, so this is the optimal stock (production) level.
Product markup
The optimal stock or production level will be a function of the product markup. A high markup will give
room for a higher level of unsold items while a low level will necessitate a focus on cost reduction and
the acceptance of stock- out:
If we put it all together we get the chart below. In this the green curve is the cumulated sales
distribution giving the probability of the level of sales and the red curve give the optimal stock or
production level given the markup.
The Optimal stock and production level
The optimal stock level is then found by drawing a line from the right markup axis (right y-axis) to the
curve (red) for optimal stock level, and down to the x-axis giving the stock level. By continuing the line
from the markup axis to the probability axis (left y-axis) we find the probability level for stock-out (1-the
cumulative probability) and the probability for having a stock level in excess of demand:
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5. By using the sales distribution we can find the optimal stock/production level given the markup and this
would not have been possible with single point sales forecasts – that could have ended up almost
anywhere on the curve for forecasted sales.
Even if a single point forecast managed to find expected sales – as mean, mode or median – it would
have given wrong answers about the optimal stock/production level, since the shape of the sales
distribution would have been unknown.
In this case with the sales distribution having a right tail the level would have been to low – or with low
markup, to high. With a left skewed sales distribution the result would have been the other way around:
The level would have been too high and with low markup probably too low.
In the case of multi-level warehousing, the above analyses have to be performed on all levels and solved
as a simultaneous system.
We can do this!
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6. Risk and Reward
Increased profit comes at a price: increased risk. The graph below describes the situation; the blue curve
shows how profit increases with service level. The spread between the green and red curves indicates a
band where the actual profit will fall, and this shows how the uncertainty in profit increases with service
level. There is no such thing as a free lunch.
On the other hand will the uncertainty band around loss as the service level increases decrease. This of
course lies in the fact that losses due to lost sales diminishes as the service level increases and the fact
that markup is positive (300%) and will easily cover the cost of over-production.
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7. Coefficient of variation
Actual sales
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8. Profit
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9. Loss
Under- production
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10. Over-production
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11. Data and analysis
Data
The data needed to perform the analysis sketched above will be found in the internal accounts:
1. Production costs
2. Data on distribution structure (existing and proposed)
3. Logistic costs from production plants to warehouses
4. Warehousing costs
5. Logistic costs from warehouses to shops
6. Product group prices
7. Markup on product groups
8. Sales forecasts for product groups and regions (countries or cities etc.) (will have to be done by
S@R in corporation with Rappala)
Results
1. Optimal warehousing levels in a multi-level structure per product group
2. Optimal production level per product group
3. Probability distribution for profit/loss
4. Optimal warehousing structure (given proposed alternatives)
Further analysis
This study and program can be a basis for an EBITDA/Budgeting model for Rappala, that again can be
used for Balance simulation and further decision making and valuation.
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