Recent Researches in Applied Mathematics, Simulation and Modelling
Expansion of the basic EOQ model with
inclusion of trim-loss costs
Jure Erjavec, Luka Tomat, Miro Gradišar
In the ideal case, the company could have infinite stock at
stock costs equal to zero. If the order costs were equal to zero,
the company could order the minimum amount at minimal
intervals. However, such situations are not possible in practice
and it is therefore necessary to determine what the order size is
at which the total costs are minimized. This problem can be
solved by the model of optimal order size under certain
limitations. The presumptions of the basic model of optimal
order size are as follows:
Abstract—We suggest an expansion of the basic economic order
quantity model by including costs of trim-loss into the model. The
expansion is base on the findings about trim loss costs from our
previous work. The ratio between the stock size and order size
significantly affects the trim loss costs thus it is important to take that
ratio into consideration when proposing the expanded model. In the
first part of the contribution we conducted a literature research of
current model expansions and have not found any expansions that
would include trim loss costs. In the second part the proposal for
expansion is presented.
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Keywords—costs optimization, cutting stock problem, economic
order quantity, inventory management.
I. INTRODUCTION
In relationship to some costs related to stock, a model for
optimal order size (economic order quantity – EOQ model)
was developed at the beginning of the last century [1], [2].
This model is still used today in many varieties and
expansions. The model is suitable for solving the problem of
the optimal order for one product [3], and particularly
adequate for solving a one-dimensional cutting stock
optimization problem in which companies order one type of
item. There is a limitation for companies that order the same
type of items but different lengths. It is possible to address the
problem by averaging order lengths and transforming them
into a unified product so that the EOQ model can be applied.
This paper first presents the basic model of optimal order
size with some of the most common expansions. This is
followed by a proposal to expand the model of optimal order
size with trim-loss costs.
The costs of stock per unit are constant in the basic model
because the company pays an equal price for all units
purchased. The viability of including the cost of stock per unit,
which does not affect the final result in the basic model, is
presented later in the explanation of expanded models. Stock
costs increase with the growth of individual order size. When
orders are larger, the intervals between individual orders are
also larger. This means that companies will have higher
average stocks, which consequently lead to higher stock costs.
In the basic model of optimal order size, constant growth of
stock cost is presupposed.
Stock cost falls with larger order size. The larger the order
is, the fewer times the company will have to order new
deliveries. Because certain costs do not depend on order size
(e.g., administrative costs of the order) and some costs do not
grow linearly with the order size (e.g., equipment-use costs,
transport cost, quality-control cost, etc.), the total cost of
ordering decreases. Due to these components of ordering
costs, these costs do not fall linearly but are dependent on each
individual instance.
All three types of costs are shown in Fig. 1. From the figure
it is evident that the costs per stock unit do not affect the
optimal order size, which is found when the total costs of
stock, total cost of ordering, and total cost of stock units are
minimal.
II. MODEL OF OPTIMAL ORTER SIZE
The model of optimal order size focuses on the question of
when and what amount should be ordered to keep the total
stock costs as low as possible. It was developed at the
beginning of the twentieth century and it is still used in many
versions to solve problems today. The description of the basic
model of optimal order size is recapitulated from [4] – [6].
The following costs are included in the basic model:
• Stock costs;
• Order costs;
• Costs per unit in stock.
ISBN: 978-1-61804-016-9
Demand is known and constant in time;
Costs are known and constant in time;
Supply time is equal to zero;
Stock shortages are not allowed;
Each order is delivered from a single supply;
Purchase price is constant regardless of the amount
ordered.
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Recent Researches in Applied Mathematics, Simulation and Modelling
optimal order size with regard to payment shift was first
introduced by [12]. The opportunity costs of interest were
included later [13]. This extension is also present in current
expansions of the model [14] – [17].
By removing the assumption about pre-known supply times
and by introducing uncertainty, running out of stock became
an important factor of the model. Costs that arise from
inability to meet demand due to insufficient stocks were
introduced to the model by [18]. They were followed by many
other researchers [19] – [21].
There are many other expansions of the model, but these are
not mentioned separately in this paper. What is important is
that in the literature it is not possible to find an extension of a
model of optimal order size that takes trim-loss costs into
consideration. The proposal for expanding the basic model of
optimal order size with costs of trim loss that arise from
cutting is presented in the following section.
III. PROPOSITION OF EXPANSION
The proposed expansion of the basic model of optimal order
size is presented by introducing trim-loss cost with the
assumption that the trim-loss costs are similar to the costs of
holding the unit in stock. The costs of the order should be
about the same size as well. If the costs are smaller, the trimloss costs will not significantly affect the total stock cost. It is
therefore reasonable to include them in the model only for
theoretical purposes because it is not useful in practice.
The ratio between the stock size and order size significantly
affects the trim-loss costs. It is therefore important to take this
ratio into consideration when proposing an expanded model
[22]. With increasing individual order size to suppliers in
relation to individual order size from customers, the average
stock also grows in relation to the average order. Based on
findings in [22], it is assumed that trim-loss costs will be
lower by increasing the average stock (if the order size
remains unchanged). This phenomenon occurs due to the
greater variety of stock items from which the company can cut
orders received from its customers. Hence it is necessary to
appropriately adjust the horizontal axis on the graph of the
basic model of optimal order size.
Therefore it is not possible to find the absolute order size
for suppliers on the horizontal axis of the expanded model, but
the relative size of average stock, which represents half of the
size of individual order to suppliers due to size of individual
customer orders. By converting the values on the horizontal
axis, the initial shapes of the curves of costs do not change in
the basic model.
This explanation of the uniformity of values on the
horizontal axis of the costs presented can also be expressed
with equations. The order size for suppliers from Fig. 1 can be
reformulated as:
Fig. 1: Basic EOQ model [4]
The basic model of optimal order size can offer an answer
to optimal order size, either in the number of units that need to
be ordered or in the order value.
The basic model of optimal order size therefore provides a
result that presents the optimal order. However, the result may
not always be an integer, which can lead to a useless result
because certain types of items are only able to be delivered in
discrete amounts. It is possible to find many cases in practice.
For example, a company cannot order 3.71 computers or 18.9
trucks. Thus, rounding up or down and the sensitivity analysis
are necessary due to the optimal result [7]. The second
limitation of the result arises from packing or transport.
Assume, for instance, that it is possible to load a container
with 13 tons of loose cargo. Based on computation, it is
determined that the optimal order size is equal to 14 tons. This
means that the second container would be almost empty and
the cost of transport would rise. In this case it is also necessary
to take other costs into account.
Due to these limitations and because of the limitations on
the basic EOQ model that arise from basic assumptions that
indicate the narrowness of use of the model in practice, many
expansions of the model have been developed. The most
important and recent models are introduced below.
An important extension of the basic model of optimal order
size is represented by taking quantity discounts from suppliers
into account [8] – [10]. Suppliers offer quantity discounts for
larger orders, which results in a decrease of average cost per
stock unit. Hence the cost per stock unit curve, which is
horizontal in Fig. 1, acquires a downward character. The most
common situation is stepped-curve drops, which also lead to
stepped drops in the total-cost curve. The reasons for stepped
drops are stepped quantity discounts, in which a discount is
applied to all items at a certain order quantity. The topicality
of expansion is also reflected in its recent discussion by
various researchers [11].
In practice, payments are not carried out immediately but
are shifted for various reasons. Extending the basic model of
ISBN: 978-1-61804-016-9
(1)
The ratio defined in [22] can be reformulated as:
(2)
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Recent Researches in Applied Mathematics, Simulation and Modelling
REFERENCES
For the purposes of expanding the basic model of optimal
order size, it is possible to presuppose that the average order
size for customers is constant, and so the following relation
can be established:
[1]
[2]
[3]
(3)
[4]
The graph of the proposed extended basic model of optimal
order size is presented in Fig. 2.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
Fig. 2: Graphic presentation of the proposed expanded EOQ model
[15]
The basic shape of trim-loss costs curve is assumed based
on simulations carried out in [22]. The position of the trimloss costs curve in relation to the other cost curves in Fig. 2 is
presented symbolically. The position depends on the relative
value of material in a ratio with other costs. The higher the
value of cut material is (if other costs remain unchanged), the
higher the curve of trim-loss cost will be positioned. The
influence on the total-cost curve will also rise. The closer the
global maximum of the trim-loss costs curve to optimal order
size in the basic model (which means that optimal order size in
the basic model will be close to the value of the ratio between
stock size and order size), the larger the influence trim-loss
costs have on the minimum of total cost in the proposed
expanded model.
In proposing an expansion of the model, we relied on
findings about the optimal ratio between stock size and order
size [22]. Because the cutting problem is NP hard and
including the cutting problem into the problem of optimal
order size makes the problem mathematically more
demanding, we suggest using simulations for searching for the
optimal order size in the proposed expansion of the model.
Further studies could therefore involve the use of simulations
for the proposed model.
ISBN: 978-1-61804-016-9
[16]
[17]
[18]
[19]
[20]
[21]
[22]
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