I. INVENTORY CONTROL

I. INVENTORY CONTROL

I.7.1. BACKORDERINGEOQ WITH SHORTAGES (1/5) A backorder s demand that wll be flled later than desred. In the backorderng stuaton, a frm does not lose the sale when ts nventory s depleted. Instead, loyal, patent, or captve customers wat to have ther demand flled when the frm receves ts next order. Fgure 3-5 depcts the backorderng nventory model. An order for Q unts s placed when the stock on hand reaches the reorder pont. The sze of the stockout s J unts, and the maxmum nventory level s Q J unts. The backorderng cost per unt per year s K, and t s drectly proportonal to the length of the tme delay.

I.7.1. BACKORDERINGEOQ WITH SHORTAGES (/5) Durng the tme perod t3 one order s placed, so the order cost s C. There s a postve nventory balance durng the tme perod t1, and the average holdng cost durng t1 s gven as follows: H( Q J) t1 H( Q J) where t1 (Q-J)/R R The stockout tme perod s t, and the average backorderng cost durng t s as follows: where t J/R KJt KJ R Total annual cost purchase cost + order cost + holdng cost + backorder cost H ( Q J ) KJ PR + + + TC (Q,J) CR Q Q Where:Jmaxmum backorderng quantty n unts, Kbackorderng cost per unt per year. Q

I.7.1. BACKORDERINGEOQ WITH SHORTAGES (3/5) To obtan optmal values for Q and J, partal dervatves of the total annual cost functon wth respect to Q and J are equated to zero. The followng optmum formulas result: CR Q* H H + K K J* HQ* H + K The mnmum total cost per year s obtaned by substtutng Q* for Q n the total annual cost equaton. A smplfed formula for the mnmum total cost per year results: TC(Q*,J*) PR + HKQ * PR + KJ * H + K

I.7.1. BACKORDERINGEOQ WITH SHORTAGES (4/5) When backorders are permtted, the reorder pont calculaton s modfed (reduced by the sze of backorder). The reorder pont s the lead tme demand mnus the number of unts backordered, or: Reorder pont lead tme demand backorders RL B J * N Wth a backorderng polcy, the longest tme a customer wll have to wat s a vtal pece of nformaton. The longest delay tme n years for a backorder s calculated as follows: J * Longest delay tme R

I.7.1. BACKORDERINGEOQ WITH SHORTAGES (5/5) Example: From the nformaton gven n example The Santoso Manufacturng Company, what happens to the economc order quantty f backorderng s possble and the stockout cost per unt per year s $1.00? CR H + K (30)8000 3 + 1 Q* 800 H K 3 1 unts, HQ * H + K 3(800) 3 +11 J* 600 unts, B RL N 8000() J* 600 9 5 unts, TC(Q*,J*) PR + KJ* (10)8000 + 1(600)$80,600.00, J * R 600 8000 Longest delay 0.075 years or 3.9 weeks.

I.7.1.3 QUANTITY DISCOUNT It s a common practce for supplers to offer lower unt prces on orders for larger quanttes as an economc ncentve to buyers to purchase n larger lot szes. The seller benefts from sales of larger quanttes by reducng per unt order processng and setup costs and by (at least temporarly) ncreasng volume. The buyer benefts both by havng reduced per unt orderng costs and by payng the lower unt prce, but at the cost of havng to hold more nventory. The problem faced by the buyer s to dentfy the lot sze that mnmzes total costs. There are two general types of quantty dscount schedules offered by supplers: the all-unts dscount and the ncremental dscount. Wth the allunts dscount, purchasng larger quanttes results n a lower unt prce for the entre lot. Incremental dscount, however, apply the lower unt prce only to unts purchased above a specfed quanttes.

I.7.1.3.1 ALL-UNITS QUANTITY DISCOUNT (1/4) The followng procedure, as outlned n fgure 3-8, ndcates how to obtan the mnmum cost order quantty when one or more all-unts quantty dscounts are avalable: 1. Startng wth the lowest unt cost, calculate the EOQ at each unt cost untl a vald EOQ s obtaned.. Calculated the total annual cost for the vald EOQ and all prce break quanttes larger than the vald EOQ. (A prcebreak quantty s the lowest quantty for whch the prce dscount s avalable). 3. Select the quantty wth the lowest total cost n step above.

I.7.1.3.1 ALL-UNITS QUANTITY DISCOUNT (/4) Example: The M Company purchases 8000 unts of a product each year. The suppler offers the unts for sale at $10.00 per unt for orders up to 500 unts and at $9.00 per unt for orders of 500 unts or more. What s the economc order quantty f the order cost s $30.00 per order and the holdng cost s 30% of per unt cost per year? The EOQ for each unt prce s as follows: CR (30)8000 unts, Q * 9 4 PF 9(0.3) CR (30)8000 unts. Q * 10 400 PF 10(0.3) CR PFQ + Q TC(Q) PR + TC(400) 10(8000) + TC(500) 9(8000) + 30(8000) 10(0.3)400 + $81,00.00, 400 30(8000 ) 10(0.3)500 + $73,155.00, 500 Consder the total costs of the sngle prce-break quantty and the vald EOQ, the mnmum cost order quantty s 500 unts.

I.7.1.3.1 ALL-UNITS QUANTITY DISCOUNT (3/4)

I.7.1.3.1 ALL-UNITS QUANTITY DISCOUNT (4/4) Another example: Consder an tem wth the quantty dscounts wth R 1,000 unts per year, f 0. per year, and C $100 per order. The another nformaton s shown n the table: Fnd the EOQ for ths problem! Q* (P$.50) 63 (nvald, vald 5,000 unts), Q* (P$3.00) 577 (nvald, vald,000 unts), Q* (P$3.50) 535 (vald). TC(535) $3,874.16 TC(,000) $3,650.00 TC(5,000) $3,770.00 The mnmum of the three costs calculated s $3,650.00 assocated wth Q,000 unts. Therefore Q*,000 unts. Lot sze Between 100 500,000 5,000 and 499 1,999 4,999 over Prce/unt $4.00 $3.50 $3.00 $.50

I.7.1.3. INCREMENTAL QUANTITY DISCOUNT(1/) The followng procedure wll determne the optmum lot sze wth ncremental quantty dscount: 1. Calculate the EOQ for each unt purchase cost.. Determne whch EOQs are vald. 3. Calculate the total cost for each vald EOQ. 4. Select the vald EOQ wth the lowest total cost.

I.7.1.3. INCREMENTAL QUANTITY DISCOUNT(/) M D + PQ, Where: D e 1 ( U e 1)( P e 1 P e ). Therefore, D s, n effect, an addtonal orderng cost, snce t s ncurred M each tme an order s placed. The purchasng cost per unt s: D Q Q + P Total cost per year of a lot sze of Q unts s: TC(Q) purchase cost + order cost + holdng cost P + D Q R + CR Q + FQ C + D ) R P FQ P R + + + Q P + D Q ( FD Q* R( C + D ) PF R C + e 1 ( Ue 1)( Pe 1 Pe ) PF

I.7.1.3. INCREMENTAL QUANTITY DISCOUNT(3/) Example: The annual demand for an tem s 4800 unts, the orderng cost s $40 per order, and the annual holdng cost fracton s 0.5. What s the optmum lot sze f the frm faces the ncremental dscount schedule below: Lot Sze Unt Prce <400 $10.00 400-1199 9.00 100-4799 8.50 >4799 8.00

I.7.1.3. INCREMENTAL QUANTITY DISCOUNT(4/) Q * R ( C + D ) P F (4800 )( 40 + 0) Q * 0 39 unts (vald), 10 (0.5 ) (4800 )(40 + 399 ) Q * 1 1369 unts (not vald), 9(0.5) (4800 )( 40 + 998.5) Q * 166 unts (vald), 8.50 (0.5 ) (4800)(40 + 3398) Q * 3 406 unts (not vald). 8.00(0.5) P U ( e e 1 e 1 D U 1)( P P ). e 0 $10.00 1 0 1 9.00 400 399($10.00-9.00)399.00 8.50 100 399+1199($9.00-8.50)988.50 3 8.00 4800 998.50+4799($8.50-8.00)3398.00

I.7.1.3. INCREMENTAL QUANTITY DISCOUNT(5/) Calculatng the total cost for each vald EOQ, TC(Q*) C + D ) R P FQ * P R + + + Q * ( FD TC(39) 10(4800) + (40+ 0)4800 10(0.5)39 (0.5)0 + + $48,979.80, 39 TC(166) 8.50(4800) + (40+ 998.50)4800 8.50(0.5)166 (0.5)998.50 + + $45,57.57. 166 The optmum lot sze s the vald EOQ wth the least annual cost. The best polcy s to order 166 unts at a unt prce of $8.50.