College and Research Libraries Research N ote_s Economic Joint Ordering of Consumable Items for a University Library S. K. Goyal In this paper the problem of joint ordering of consumable items for a university library is for- mulated. A simple heuristic method is pro- posed for determining the most economical or- dering policy. An example involving two items is solved to illustrate the method. A modern library consumes a large number of items in its day-to-day opera- tions, i.e., loan slips, interlibrary loan forms, photocopying paper of d~ferent kinds and sizes, standard stationery items order forms, etc. Very often anum- ber of items are procured from a single supplier. For example, our l?cal public ~i­ brary purchases photocopymg paper m two sizes from one supplier. The demand for the photocopying paper in the two sizes is fairly constant over time. Th.e problem facing the library management ts how to procure photocopying paper in the most economical manner. · Items that are purchased exclusively from a single supplier can be ordered in two simple ways: (1) each item is ordered on its own, and (2) items are ordered jointly. Whenever a purchase order is initiated, a fixed cost of placing an order is incurred. This fixed cost primarily consists of ad- ministrative costs of preparing a purchase order. Some additional cost may be in- curred depending on the particular item(s). Therefore, in the first instance, where each item is ordered on its own, every time an item is ordered the fixed cost of placing the purchase order and the vari- able cost of ordering that particular item must be incurred. On the other hand, in the second instance, where items are or- dered jointly, the cost of placing the order consists of the fixed cost of placing the or- der plus the total of individual ordering cost of the items included in the purchase order. Therefore, as a result of ordering items jointly, considerable saving in the total ordering cost can be achieved. In the operations research/management scienc.e literature, a considerable number of publi- cations exist that deal with the problem of determining an economical ordering pol- icy (see Shu, 1 Goyal/'3 Silver,4 and Kaspi and Rosenblate.) For the consumable items having a fairly constant rate of demand the Economic Or- der Quantity Model (EOQ model) is con- sidered to be most appropriate for applica- tion in the context of a library inventory of consumable items. Let S = fixed cost associated with a replen- ishment; s. K. Goyal is professor in the Department of Quantitative Methods at Concordia University, Montreal, Que- bec, Canada, H3G 1MB. 275 276 College & Research Libraries n = number of consumable items or- dered from the supplier and for the i th item; Si = variable cost of including the item on the replenishment order; Di= demand per year; h i = stock holding cost per item per year. In the first instance, where each item is or- dered on its own, the EOQ and the mini- mum annual cost (MAC(EOQ)), for the th items are given by formulas 1 and 2. The I 2Di (5+5J (EOQ)i hi (MAC(EOQ))i I 2Dihi (5+5i) n May 1986 minimum annual cost for all the consum- able items is given by formula 3. In the second instance where items are ordered jointly, let Trepresent the time in- terval between replenishments. The order quantity for the ith item will be TDi. Hence the annual cost, Z, of ordering and stock holding in case of joint replenishment is given by formula 4. The total annual cost has a least value at T =To as given in formulas 5 and 6. l (1) (2) (MAC(EOQ)) E I 2Dihi(5+5i) (3) i=l n (5 + l: 5) i=l z + T n 2(5+ l: 5i) i =l To n l: Dihi i=l (MAC(JOINT)) (EOQ(JOINT))i T n l: Dihi 2 i=l n 2(5 + l: 5i) i=l 2( 5 + l: 5i) i=l (4) (5) n l: Dihi (6) i=l (7) The economic order quantity for the ith item is given by formula 7. If the total annual cost (MACGOINT)) as obtained from (6) is lower than the (MAC(EOQ)) evaluated from (3), then it is more economical to order items jointly. Very often, the (EOQGOINT)); for some items may be lower than the EOQ when the item is ordered on its own without in- curring the fixed cost, S, (in such a situa- tion [ (EOQ); = /WiA) ]. For these items it makes sense to order less frequently, every second replenishment or every third, and so on. In short, the ith item may be ordered in every K/h replen- ishment where K;= 1, 2, 3, and so on. In or- der to determine the value of K;, we select the nearest non-zero integer number from the ratio shown as formula 8. See Silver6 for determining the value of K;. The time interval between replenish- ments, the order quantity for the ith item and the (MAC(JOINT)) are given by for- . mulas 9, 10, and 11. Therefore, for items purchased from a single supplier, the following steps j (EOQGOINT)); n Research Notes 277 should be implemented for determining the economic ordering policy. A SYSTEMATIC PROCEDURE FOR DETERMINING THE ECONOMIC POLICY Step 1. For each item i=1, 2, ... , n, evaluate (EOQ); from (1), and (MAC(EOQ)) from (3). Step 2. Obtain (EOQ(JOINT)); from (7}, evaluate (EOQ(JOINT)); and select the nearest non-zero integer as the value of K; for i=1, 2, ... , n. Step 3. Determine (MAC(JOINT)} from (11), if (MAC(EOQ)) ~ (MAC(JOINT)) then select (EOQ); as the order quantity for each item and the policy is to order items individually. If (MAC(EOQ)) > (MAC(JOINT)) then items are ordered jointly. AN EXAMPLE A library orders photocopying paper in two sizes from a supplier. The various es- (8) 2(5 + I: 5/K;) i=1 T(JOINT) (9) (EOQGOINT)); D; • TGOINT) (10) n n (MACGOINT)) 2(5 + I: 5;/K;) I: D;K;h; (11) i=1 i=1 278 College & Research Libraries May 1986 Step 1. From (1): I 2D1(5 + 5t) j (EOQh h1 237boxes I 2D2(S + 52) J (EOQh h2 87boxes From (3): (MAC(EOQ)) / 21D1h1(5 + 51) + / / 2 X 1000 X 1(20 + 8) +I 236.64 + 96.12 $332.76 per year Step 2. From (7): 2(5 + 51 + 52) (EOQGOINT)h Dt 1000 249boxes . . (EOQGOINT)h D2 37boxes. (EOQGOINT))t (EOQGOINT)h Step 3. From (11): (MACGOINT)) / D1h 1 + D2h2 V 2 X 1000 8/1 249 v' 2x150x8/1.1 37 / 2(20+8+8)(1000x1x1+ 150x1.1x1) = $289.62 per year FIGURE 1 2 X 1000(20 + 8) 1 2 X 150(20 + 8) 1.1 2D2h2(S + S2) 2 X 150 X 1.1(20 + 8) 2(20 + 8 + 8) (1000x1 + 150x1.1) 0.51, hence K1 = 1 1.26, hence K2 = 1 Research Notes 279 2(20 + 8 + 8) TO OINT) 1000 + 165 0.249 years · (EOQOOINT))t 249 boxes; (EOQOOINT)h = 37 boxes FIGURE2 timates for the problem are given below: S = $20 per order For size 1 : i = 1 D1 = 1000 boxes/per year sl = $8 per order h1 = $1 per box per year For size 2: i=2 D2 = 150 boxes per year 52 = $8 per order h2 = $1.10 per box per year We apply the three steps to this problem as shown in figure 1. Because (MAC(JOINT)) is lower than (MAC(EOQ)), the economic policy is to order jointly. The orders are placed at in- tervals obtained from (9). (See figure 2.) The reduction in cost as a result of order- ing jointly (MAC(EOQ))- (MACOOINT)) 332.76-289.62 $43.14 per year The percentage reduction in cost 100 X [(MAC(EOQ))-(MACOOINT))] (MAC(EOQ)) 100(332.76- 289.62) 332.76 12.96% CONCLUDING REMARKS As a result of ordering items from a sin- gle supplier in an economical manner, sig- nificant cost savings can be achieved. The procedure given in this paper can help in achieving such savings. REFERENCES 1. F. T. Shu, "Economic Ordering Frequency for Two Items Jointly Replenished, " Management Science 10:406-10 (1971). 2. S. K. Goyal, "Determination of Economic Packaging Frequency for Items Jointly Replenished," Management Science 20:232-35 (1973). 3. S. K. Goyal, "Determination of Optimum Packaging Frequency of Items Jointly Replenished," Management Science, 21 :436-43 (1974). 4. E. A . Silver, "A Simple Method of Determining Order Quantities in Joint Replenishment under Deterministic Demand," Management Science 12:1351-61 (1976). 5. M. Kaspi and M. J. Rosenblatt, "An Improvement of Silver's Algorithm for the Joint Replenish- ment Problem," liE (Transactions), 14:264-67 (1968) . 6. Silver, "A Simple Method .... " Swets ... an attractive, many facetted and transparent subscription seiVice. We would be pleased to send you our informative brochure as well as detailed documentation of our seiVices.