Proceedings of the 2013 Industrial and Systems Engineering Research Conference A. Krishnamurthy and W.K.V. Chan, eds.
An EOQ Binary Model with All Units Discounts for Multiproducts
Valdecy Pereira,
[email protected] Escola de Engenharia, Universidade Federal Fluminse Niterói, Brazil
Helder Gomes Costa,
[email protected] Escola de Engenharia, Universidade Federal Fluminse Niterói, Brazil Abstract The following work presents a new solution for EOQ (Economic order Quantity) models that takes into account the discounts that can be done by suppliers, by acquiring a larger volume of a given product or products. The discount presented in this work is an all units quantity discount, which is a discount given on every unit that is purchased after the purchasing quantity exceeds a given level (breakpoint). To solve this problem taking into account multiple products, an integer non-linear model with binary variables is developed and the presented method differs from other classical methods, because it uses a binary strategy to obtain the optimal solution. Numerical examples are presented throughout the work in order to improve the understanding of the model. The results were compared with another method of resolution proposed by Chopra e Mendl (2001) and our solution had a reduced total cost.
Keywords Project Management Office; Implementation; Framework
1. Introduction The economic order quantity (EOQ) is an important tool for companies that need to minimize costs and reinvest the capital that was previously wasted on poorly planned inventory, to other areas that need more budget or could take advantage of new opportunities in the market. The EOQ model finds a balance between the number of orders per year and holding costs. The first EOQ model, the basic model, presented by F.W Harris in 1913, find the optimal point between the number of orders and the holding costs [1]. Despite being a robust model, it cannot be applied to all managerial situations. For example, the model does not take into account the discounts that can be done by suppliers, by acquiring a larger volume of a given product. Some authors developed some kind of contribution such as Maiti, Bhunia, & Maiti [2] that proposed to solve a mixed-integer non-linear programming problem with constraints by a real coded genetic algorithm (RCGA). Shin & Benton [3] developed a quantity discount model for supply chain coordination between a supplier and a buyer, Transchel & Minne [4] considered a quantity model, of an economical order, where the supplier offers an all-units quantity discount and a price sensitive customer demand. Toptal [5] formulated a model for the replenishment decisions under a general replenishment cost structure that includes a generalized transportation costs and all unit quantity discounts, and finally Lin [6] studied the inventory models with imperfect quality in the pull system, in which the buyer is powerful. Despite the advances since the pioneer model of Harris, there are still some gaps in the EOQ universe of problems, and this work tries to fill a part of this gap proposing an original set of models that includes dealing with the single product or multiproduct environment with all units discounts. With this concern, we describe a step by step propose for modeling EOQ with discounts for one or more products. The discount presented in this work is about the all units discount. An all units quantity discount is a discount given on every unit that is purchased after the purchasing quantity exceeds a given level (breakpoint) [7]. In Section 1 is exposed to modeling developed by Harris and the different behaviors of the total cost functions through the use of all units discount. In section 2, the modeling for one or more products with all
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Pereira & Costa units discounts will be explained step by step, along with numerical examples. In Section 3 numerical examples are given for all the proposed models and finally this article concludes with an analysis of the modeling performed.
1.1. Basic model The EOQ model developed by Harris has the following expression as the Total Cost function:
And EOQ itself as: √
( )
where: = Fixed order cost; = Annual demand; = Purchase price per unit; = Cost of storage, which is a percentage of the purchase price; = EOQ; = Total cost of the annual orders; = Total cost of the annual holding; = Total cost of the annual purchase of products. The total cost curve is convex and presents a single point of minimum that can be found by taking the first derivative of equation 1 [8]. In the basic model the total purchase cost is not necessary to obtain the EOQ (equation 2).
1.2. Model with all units discount When the supplier promotes discounts on the total quantity of items, the total annual cost curve (Figure 1) suffers changes and the EOQ can’t be calculated in the same way, as is show in Equation 1. The new curve has the same number of minimum points as there are breakpoints (discounts), but there is only one global minimum point, which can be anywhere in the curve. $
EOQ
Q
Figure 1 - EOQ with All Units Discounts
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2. Model for one product with all units discount The objective function of the model is based on the minimization of the function of the total annual all units discount (equation 3). But this same function features one peculiarity, the use of binary variables, , that multiplies the holding costs and the purchasing costs. The quantities of items purchased, is represented by . Whereas , we obtain:
* (
)
+
∑
∑
[
]
where, = Purchase price within a range of discount . = Amount to be purchased within a range of discount . To illustrate this function, assume the following example: a certain product has the purchase price of $ 1.00 for amounts between one and 999 items (range ), a price of $ 0.95 for quantities between 1000 and 2499 items (range ) and a purchase price of $ 0.90 for quantities above 2500 items (range ): Table 1 – Discount ranges
where, = Lower limit of the range k; = Upper limit of the range k. And considering that the annual demand (D) is 8,000 units, the cost of storage (i) is 20% and the fixed order cost (A) is $ 150.00, we have:
*
(
[
)
+ (
[
)] (
)]
Each binary variable is connected to a buying strategy, which is responsible for accept (y = 1) or reject (y = 0) each strategy, and only one should be choose. But the model will only be able to make that choice if the constraints are well defined. Without appropriate constraints the model will assume that all binary variables are equal to zero, thereby obtaining as cheaply as possible, only the total order cost (
). The following restrictions, will made the
model to choose the strategy with lower costs and therefore find the optimal economic lot. The first constraint represents the quantity to be purchased in each range of : [
]
( )
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Pereira & Costa If the economic lot is within a certain range , the binary variable is equal to one, the first member of the left side ] ; will indicate the quantity to be bought , and of the equation 5: [ and ; the second member of equation 5: , will be zero. But if the economic lot size is found in another range of , the binary variable , takes the value of zero and the first member of the left side of equation 5, will also be zero. The second member of the equation, will represent the maximum amount of items that can be purchased in that specific range . For example, for the same values used in Figure 4, if the economic lot were 1,500 units, would be equal to 999 units (purchase price = $ 1.00) and = 501 (purchase price = $ 0.95). The value of should be ignored because the economic lot was found in the previous range. The second constraint represents the last range : [
]
( )
This restriction has the same formulation of the equation 5, but the second member of the left side is missing, because there is no limit to buy units with that discount. The third constraint concerns the binary variables. It forces the model to choose just one strategy of the objective function: ∑ The fourth constraint ensures that the chosen strategy is accepted when at least one item is purchased in the range :
The non-negativity constraints also apply to the model, which can be also programmed to find deterministic solutions, by adding that is an integer. This constraint is optional and will increase the difficulty to find the optimal solution of the model, but empirical tests show that the solution of the relaxed model, it´s a very good approximation of the deterministic optimal solution. It may be noted that if the values are all equal, the model will calculate the Basic EOQ model (equation 2). If the vendor could only sell items in a specific batch size, another optional constraint could be added:
Where: The quantity of batchs to be bought, this a variable that can only assume integer numbers; The vendor´s batch size. If the buyer possesses a limited space for the inventory, another optional constraint could also be added: (10) Where: Space required for one unit or one batch of Q; Total avaiable space. All these optional constraints are referred in this paper as: Optional Constraints Sets. For the complete model we have:
* ( S.A
[
)
+
∑
] [
]
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∑
[
]
Pereira & Costa ∑
Optional Constraints Sets
2.1 - Model with All units discount for more than one product The multi-product model below is based on the work of [9] which seeks to synchronize the replenishment of multiple products in the same time (Figure 4b), to prevent the disorganized arrival of products that could generate cost regarding the management of the cargo and the storage (Figure 4a). These authors developed a methodology for solving this problem, which finds a solution very close to optimal. Their solution basically, modifies the amount of the EOQ, in a way that a synchronization of the product arrivals occurs in the same period, with the penalty of very high EOQ values for each product. The model proposed in this paper does the same, but the new batches for synchronization have values closer to the economic lot, because the synchronization is done according to the cycle of each product. The penalty for receiving non-synchronized products, are considered in this work an intangible very hard to measure and therefore it will not be included in the total cost.
T
T
Figure 2a - Example of Non-Synchronized Products
Figure 2b - Example of Synchronized Products
In Figure 2a, the products arrive on different dates because each product has its own economic lot and ideal consumption rate, in Figure 2b we have the same dates of arrival. For the synchronization of a product, it´s necessary to first find the most ordered product and for this we must add in the objective function, the following expression: ∑
(
)
where, = Binary variable responsible for choosing the shorter cycle; = Quantity of products with discounts; = A sufficiently large number. Equation 11 indicates, along with a new set of restrictions that will be detailed below, which is a product with the largest number of orders through the lower cycle . , with a large enough number, will not change the minimal value of the total cost function. Whereas that suffer discounts, we obtain: ∑
{
*
(
)
+
, being an index that indicates the set of products
∑
∑
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*
+}
∑
(
)
Pereira & Costa And the following restrictions must be contained in the modeling: ∑ ∑(
)
where: = Value of the largest number of orders; = Variable that must assume only integer values. Equation 13 indicates that only the product with a lower cycle and therefore the larger number of orders is chosen, equation 14 indicates the value of this order and finally the equation 15 ensures that the cycles of all other products are sub-multiples of , so the synchronization can occur. Given the necessary changes in the objective function and the addition of new constraints, we get: ∑ ∑{ S.A
*
[
]
(
)
+
∑
[
∑
[
]}
(
)
]
∑
∑ ∑(
)
Optional Constraints Sets
3 Numerical examples This section will give solved examples for the multiproduct models a demo version of LINGO 11.0 64-bit was used on a computer with the Windows 7 x64platform, a Quad Core Processor 2.40 GHz and 8 GB of DDR 2 memory.
For the following example, which only demonstrates the synchronization power of the proposed model, consider the following products P1, P2 and P3. All the costs and the calculation results are shown in Table 2.
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Table 2 informs the proposed model synchronized quantities: (SQ) for each product: P1 – 492.1834 items (with a cost of $48,990.33); P2 – 147.6921 items (with a cost of $15,509.62) and P3 – 49.21834 items (with a cost of $4,889.03). The total cost (TC) was $69,398,97. The model developed by [4] has found the following synchronized quantities: (SQ(C&M)) for each product: P1 – 1147.0787 items (with a cost of $67,815.29); P2 – 114.70787 items (with a cost of $16,196.75) and P3 – 45.88315 items (with a cost of $4,909.50). The total cost (TC) was $88,921,54. The solution proposed by our model has assured a reduction of 23.74%. For the next example consider the following products in Table 3: Table 3 - Products A and B
a) All Units discount - Single Product: Product A and B Tables 4 and 5 show the results from the modeling of two single products (A and B, respectively) with all units discounts. Also for this case the results for product A and B are very similar, except by the fact that the number of interations for product B is higher than for product A, as also occurred for incremental discounts. Table 4 - Product A- All units discount
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In Figure 3, there is no synchronization, occurring for every order of product B, a fractional number of orders for A. And the total cost for both products is equal to $74,233.34.
Figure 3 - Products A and B Non-Synchronized – All Units Discount
b) All units discount: Product A and Product B While synchronizing both products and under all units discount, the model found basically the same solution for the problem with non-integer quantity (Q) and for the problem with integer quantity, the only difference was in the number of iterations that was increased in the integer case, as one could expect. The binary variables
and
with the values of 1, shows that the best strategy that minimizes the total cost was to order from product A, a quantity that lies in the third range of discount (more than 2,500 units) and from product B, a quantity that lies in the second range of discount (between 15,000 and 35,000 units). Table 6 shows these results. Table 6 - Product A and B - All Units discount
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Pereira & Costa In Figure 4 we observe, that for every n requests for A occur exactly orders for Product B ( integers). In this case, and . And the total cost is equal to $75,925.00.
and
always
Figure 4 - Products A and B Synchronized - Discounts On All Units
4. Conclusion This work proposes a model to find out the optimal minimal cost solutions for one or more products, with all units type of discounts, or even products with no discounts at all. In the multiproduct case the all models seek to synchronize the replenishment of the products to prevent their disorganized arrival and so generate costs regarding the management of the cargo and storage. When the supplier promotes all units discounts on the total quantity of items, the total annual cost curve (a convex curve) suffers changes not having only one minimum point. The new curve has the same number of minimum points as there are breakpoints (discounts), but there is only one global minimum point, which can be located anywhere in the curve. The same considerations can be made about products with incremental discounts; the only difference being an overlap of curves that depend also on the number of breakpoints (discounts). The proposed models are nonlinear integer programs with binary variables, which are connected to a buying strategy (a range of discount), and are also responsible for accepting (y = 1) or rejecting (y = 0) each strategy; only one strategy should be chosen and, this way, an optimal quantity is defined. The constraint alongside with the objective function defines which discount each product has, and if one or more products don’t have any type of discount, the model can be easily modified to cover this aspect too. Many other constraints can be added to the model, like: limited space for the inventory, fixed batch sizes and/or integrality constraint. But the integrality constraint is not recommended to be added, because the non-integer models can find a solution very close to the integer one and such optional constraints in some cases can make the model infeasible or more difficult to solve.
References 1. 2.
3. 4. 5.
Erlenkotter, D., 1990. “Ford Whitman Harris and the Economic Order Quantity Model”. Operations Research. Vol 38, Nº 6, November-December. Maiti, A. K., Bhunia, A. K., & Maiti, M. 2006.! An application of real-coded genetic algorithm (RCGA) for mixed integer non-linear programming in two-storage multi-item inventory model with discount policy. Applied Mathematics and Computation”, 183, 13. doi: 10.1016/j.amc.2006.05.141 Shin, H., & Benton, W. C. 2007. “A quantity discount approach to supply chain coordination”. European Journal of Operational Research, 180, 16. doi: 10.1016/j.ejor.2006.04.033Xx Transchel, S., & Minne, S. 2008. “Coordinated Lot-sizing and Dynamic Prizing under a Supplier All-units Quantity Discount. Business Research, 17(1), 125-141. Lin, T.-Y. 2010. “An economic order quantity with imperfect quality and quantity discounts”. Applied Mathematical Modelling, 34(10), 3158-3165. doi: 10.1016/j.apm.2010.02.004
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Toptal, A. 2009. “Replenishment decisions under an all-units discount schedule and stepwise freight costs”. European Journal of Operational Research, 198(2), 504-510. doi: 10.1016/j.ejor.2008.09.037 Mendoza, A & Ventura, J.A., 2008. “Incorporating quantity discounts to the EOQ model with transportation costs”. Int. J. Production Economics Nº 113, pages 754–765, November. Benton, W.C. & Park, S. 1996. “A classification of literature on determining the lot size under quantity discounts”. European Journal of Operational Research, nº92. Chopra, S. Meindl, P. 2001. “Supply Chain Management”. London: Prentice Hall.
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