Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of a given item to order. A great deal of literature as dealt wit tis problem (unfortunately many of te best books on te subject are out of print). Many formulas and algoritms ave been created. Of tese te simplest formula is te most used: Te EOQ (economic order quantity) or Lot Size formula. Te EOQ formula as been independently discovered many times in te last eigty years. We will see tat te EOQ formula is simplistic and uses several unrealistic assumptions. Tis raises te question, wic we will address: given tat it is so unrealistic, wy does te formula work so well? Indeed, despite te many more sopisticated formulas and algoritms available, even large corporations use te EOQ formula. In general, large corporations tat use te EOQ formula do not want te public or competitors to know tey use someting so unsopisticated. Hence you migt wonder ow I can state tat large corporations do use te EOQ formula. Let s just say tat I ave good sources of information tat I feel can be relied upon. Te Variables of te EOQ Problem Let us assume tat we are interested in optimal inventory policies for widgets. Te EOQ formula uses four variables. Tey are: D: Te demand for widgets in quantity per unit time. Demand can be tougt of as a rate. Q: Te order quantity. Tis is te variable we want to optimize. All te oter variables are fixed quantities. C: Te order cost. Tis is te flat fee carged for making any order and is independent of Q. : Holding costs per widget per unit time. If we store x widgets for one unit of time, it costs us x@. Te EOQ problem can be summarized as determining te order quantity Q, tat balances te order cost C and te olding costs to minimize total costs. Te greater Q is, te less we will spend on orders, since we order less often. On te oter and, te greater Q is te more we spend on inventory. Note tat te price of widgets is a variables tat does not interest us. Tis is because we plan to meet te demand for widgets. Hence te value of Q as noting to do wit tis quantity. If we put te price of widgets into our problem formulation, wen we finally ave finally solved te optimal value for Q, it will not involve tis term.
Te Assumptions of te EOQ Model Te underlying assumptions of te EOQ problem can be represented by Figure 1. Te idea is tat orders for widgets arrive instantly and all at once. Secondly, te demand for widgets is perfectly steady. Note tat it is relatively easy to modify tese assumptions; Hadley and Witin [1963] cover many suc cases. Despite te fact tat many more elaborate models ave been constructed for inventory problem te EOQ model is by far te most used. Figure 1 Te EOQ Process An Incorrect Solution Solving for te EOQ, tat is te quantity tat minimizes total costs, requires tat we formulate wat te costs are. Te order period is te block of time between two consecutive orders. Te lengt of te order period, wic we will denote by P, is Q/D. For example, if te order quantity is 0 widgets and te rate of demand is five widgets per day, ten te order period is 0/5, or four days. Let T p be te total costs per order period. By definition, te order cost per order period will be C. During te order period te inventory will go steadily from Q, te order amount, to zero. Hence te average inventory is Q/ and te inventory costs per period is te average cost, Q/, times te lengt of te period, Q/D. Hence te total cost per period is: T P C Q Q C Q D D If we take te derivative of T p wit respect to Q and set it to zero, we get Q = 0. Te problem is solved by te device of not ordering anyting. Tis indeed minimizes te inventory costs but at te small inconvenience of not meeting demand and terefore going out of business. Tis is wat many people, peraps most people do, wen trying to solve for te EOQ te first time. Te Classic EOQ Derivation Te first step to solving te EOQ problem is to correctly state te inventory costs formula. Tis can be done by taking te cost per period T p and dividing by te lengt of te period, Q/D, to get te total cost per unit time, T u : T u CD Q Q
In tis formula te order cost per unit time is CD/Q and Q/ is te average inventory cost per unit time. If we take te derivative of T u wit respect to Q and set tat equal to 0, we can solve for te economic order quantity (were te exponent * implies tat tis is te optimal order quantity): Q * T u * CD If we plug Q * into te formula for T u, we get te optimal cost, per unit time: CD Te Algebraic Solution It never urts to solve a problem in two different ways. Usually eac solution tecnique will yield its own insigts. In any case, getting te same answer by two different metods, is a great way to verify te result. If we multiply te formula for T u on bot sides by Q, we get, after a little rearranging, a quadratic equation in Q: Q QT u CD Next we divide troug by / to cange our leading coefficient to 1. Completing te square, we get: 0 Q T u CD Tu 0 Note tat tis equation as two variables. T u is a function of Q (and could well be written as T u (Q)). Hence we ave a curve in te Cartesian plane wit axes labeled Q and T u. We want te value of Q tat minimizes T u. Notice tat te term rearranging te equation, we get: T u CD Q T u is a constant. Writing tat constant as k, and k 3
If we set te quadratic term to zero, ten T u k increases te size of T u. Hence te optimal size of T u is 3050 Q *. 36 6. Any cange in te quadratic term from zero k wic just appens to be CD Q T u wic is te value we found earlier. Te quadratic term is zero if and only if. Q * * T u Tis gives us te identity. An Example It is useful at tis point to consider a numerical example. Te demand for klabitz s is 50 per week. Te order cost is $30 (regardless of te size of te order), and te olding cost is $6 per klabitz per week. Plugging tese figures into te EOQ formula we get: Tis brings up a little mentioned drawback of te EOQ formula. Te EOQ formula is not an integer formula. It would be more appropriate if we ordered klabitz s by te gallon. Most of te time, te nearest integer will be te optimal integer amount. In tis case, te total inventory cost T u is $134.18 per week wen we order klabitz s. If instead, we order 3 klabitz s te cost is $134.. Total Cost 1600 1400 100 1000 800 600 400 00 0 Min 1 6 11 16 1 6 31 36 41 46 51 56 61 66 71 76 Order Quantity Figure Total Cost as a Function of Order Quantity 4
Order Cost 1600 1400 100 1000 800 600 400 00 0 Min 1 6 11161631364146515661667176 Order Quantity Holding Cost Figure 3 Order and Holding Costs A grap of tis problem is illuminating: Figure. Because te grap is so flat at te optimal point, tere is very little penalty if we order a sligtly sub-optimal quantity. We can better understand te grap if we view te combined graps of te order costs and te olding costs given in Figure 3. Te basic sapes of all tree graps (total costs, order costs, olding costs) are always te same. Te grap of order costs is a yperbola; te grap of olding costs is linear; and as a dtu result te grap of te total costs (T u ) is convex. Tis can also be seen in tat te function is dq increasing and te function dt dq CD Q 3 is positive every were. If we plug te optimal quantity, Q *, into tis last formula we get: dt dq * 5 CD 4CD
Ordinarily tis last quantity is very small, wic indicates tat te total cost of inventory T u canges very slowly wit Q (in te optimal region). Hence te assumptions of te EOQ model do not ave to be accurate because te problem usually is tolerant of errors. If you study closely te graps in Figure 3, it may seem clear to you tat teir sum, T u, reaces a minimum precisely were te two graps intersect; tat is at te point were order costs and olding costs are equal. Te gives us te equality way to derive te EOQ formula. Q CD Q. Solving tat equality is te easiest Wy Use te EOQ Formula At All? A problem tat occurs in applied matematics more tan pure is tat we ang onto formulas and tecniques tat ave been made (mostly) obsolete by tecnology. Try to tink wat it was like to solve a problem like te one ere forty years ago. Te simple operation of division was eiter done by and, or by use of logaritms out of a table, or less exactly by using a slide rule. It was not practical to simply calculate te total cost of inventory for a large set of order quantities and to compare answers. Te EOQ formula simplified te problem to a minimal number of calculations. However, now it is quite simple to calculate total costs of inventory for undreds of order quantities, and tis can be done from scratc in less time tan it use to take to employ te EOQ formula. We can do it wit a spreadseet. Te spreadseet in Figure 4, takes te problem from te previous example, and computes for a large variety of quantities te order costs, olding costs, and total costs. It took about 15 minutes to set tis spreadseet up, and most of tat time was spent on formatting (for example putting in lines). Te formulas for order costs and olding costs were inserted to calculate te respective entries for te order quantity of one. Te total cost entry is defined to be te sum of te first two entries (altoug its column comes first). Ten te formulas are copied down te lengt of te order quantity column. Te only trick is tat you must remember to use absolute addresses for te fixed parameters. Once te spreadseet is set up, te values for C, D, and can be canged and te entire spreadseet will recalculate in less tan a second. Note also, tat te EOQ formula itself is calculated at te top of te spreadseet: its result can be compared wit te columns. Te spreadseet also provided te graps in Figure and Figure 3. Spreadseets are superb for many problems in discrete matematics. Knowing tis, software companies ave endowed current spreadseets wit undreds of scientific functions. Just like formulas, te spreadseet can be made to incorporate assumptions more realistic tan tose in te EOQ model. In many cases it is easier to do tis wit spreadseets and more illuminating. 6
Demand/unit time order cost olding cost item /unit time 50 30 6 EOQ =.3606797749979 Order Quantity Total Cost Order Cost Holding Cost 1 1503.00 1500.00 3 756.00 750.00 6 3 509.00 500.00 9 4 387.00 375.00 1 5 315.00 300.00 15 6 68.00 50.00 18 7 35.9 14.9 1 8 11.50 187.50 4 9 193.67 166.67 7 10 180.00 150.00 30 11 169.36 136.36 33 1 161.00 15.00 36 13 154.38 115.38 39 14 149.14 107.14 4 15 145.00 100.00 45 16 141.75 93.75 48 17 139.4 88.4 51 18 137.33 83.33 54 19 135.95 78.95 57 0 135.00 75.00 60 1 134.43 71.43 63 134.18 68.18 66 3 134. 65. 69 4 134.50 6.50 7 5 135.00 60.00 75 6 135.69 57.69 78 7 136.56 55.56 81 8 137.57 53.57 84 9 138.7 51.7 87 30 140.00 50.00 90 31 141.39 48.39 93 3 14.88 46.88 96 33 144.45 45.45 99 34 146.1 44.1 10 35 147.86 4.86 105 36 149.67 41.67 108 37 151.54 40.54 111 38 153.47 39.47 114 39 155.46 38.46 117 40 157.50 37.50 10 Figure 4 A Spreadseet Analysis of te Inventory Problem 7
Hadley, G. and T. M. Witin. Analysis of Inventory Systems. 1963. Englewood Cliffs, New Jersey: Prentice-Hall. 8