Sp ace IOE 202: Lecture 2 ou tline Annou ncements Las t tim e... Inventory manage ment prob lems and models: Econ om ic Order Quantity models: con tinued Adiﬀerent inventory manage ment situation (and a Linear Progr amming model) IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 1 Sp ace Last time Prob lems of maintaining and replenishing inve nto ry One of the issues: balancing holding costs vs. ordering and shortage costs First example: a prob lem with Lon g-term planning: Known, steady demand Known costs (setup and per unit ordering, holding) that do not chang e over time Kno wn lead time tha t do es not cha nge ov er time No (planned) shortage s allowed Con tinuou s review Approa ch to managing inventory: Se lect a bat ch size Q ite ms Order a batch of size Q just as you are abou t to run ou t W hat value of Q maximizes net profi ts? W hat is the “E con om ic Order Quantity”? IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 2 Sp ace Inve ntory leve l ove r time Q/a Q Time I n v e n to r y L e v e l - a IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 3 Sp ace Inputs an d ou tputs of EOQ models Definition Unit s a Demand per unit of time unit/u nit of time L Le ad tim e units of time K Setup cost for ordering on e batch $ c Cost for purchasing on e unit $/u nit h Holding cost per unit per unit of time held $/( unit×unit of time) Q Order Qua ntit y (ba tch size) unit Q /a Tim e between orde rs unit of time T (Q ) Cost per unit of time $/u nit of time Note: sales revenue does not depend on ordering policy, as lon g as we never run ou t of inventory. So, to maximize net profi t, we simply need to minimize the cost of ordering and holding inventory. IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 4 Sp ace One ordering cycle: details Cycle duration = Q /a units of time Production /ordering cost per cycle = K + cQ dollars Holding cost per cycle = Total cost per cycle = IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 5 Sp ace Descriptive model: Expression of cost Thus, the cost per unit of time is: T (Q ) = Total cost per unit time = Total cost per cycle Duration of the cycle = aK Q +ac + hQ 2 This formula is a de scriptive mode l: wh at happen s wh en ordering quantit y is Q ? ✻ ✲ IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 6 Sp ace Prescriptive model: EOQ We wa nt a prescriptive model: what value of Q is op timal, i.e ., the best? Optimization prob lem: “m inimize T (Q ) ov er all Q ≥ 0” To find the minimum, com pute the derivative of T (Q )andsetit to 0: T (Q )= aK Q + ac + hQ 2 , so T (Q )= − aK Q 2 + h 2 The value of Q tha t minimizes the annua l invento ry ordering and holding cost: Q = 2aK h (the “E con om ic Order Quantity”) Tim e between orde rs: t = Q a IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 7 Sp ace Bac k to CubicleMin’s prob lem: Par ameters (with units): a = 100 cameras per mon th = 1200 cameras per year L =1 week K = $35 c = $100 per camera h = $10 per camera per year Hence, the op timal solution is to set up the facility to order cameras in batches of Q = 2aK h = 2 · 1200 · 35 10 = 91 .6 on ce every t = Q /a = 91 .6 1200 =0.076 years ≈ 4 weeks We shou ld place an order when the inventory level is ≈ 23 cameras IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 8 Sp ace Exam ple: Man ufac turing speakers for TV sets A TV manufacturing com pany produces its own speakers, which are then used in the production of its TV sets. The TV sets are produced on a con tinuou s production line at a rate of 8,000 per mon th, and each set needs on e speaker. The speakers are produced in batches, because relatively large quantitie s can be pro duc ed in a short tim e. The refore, speakers are plac ed into inve nto ry until the y are ne ede d for assembly into TV sets. The com pany is interested in determining when to produce a batch of speakers and how many to produce in each batch. Several costs must be con sidered in making the abov e decision : Each time a batch of speakers is produced, a setup cost of $12, 000 is incurred The unit production cost of a single speaker (excluding the setup cost) is $10, independent of the batch size produced The estimated holding cost of keeping a speaker in stock is $0 .30 per mon th IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 9 Sp ace Solution of the TV speaker prob lem EOQ model is appropriate in this prob lem Par ameters: a =8,000 units per mon th (de mand rate ) L =0 (lag tim e between orde r and de live ry) K =$12, 000 (fixed/setup cost for producing a batch) c =$10 per unit (unit ordering cost) h =$0. 30 per unit per mon th (unit holding cost per unit of time) Hence, the op timal solution is to set up the facility to produce speakers in batches of Q = 2aK h = 2 ∗ 8000 ∗ 12000 0.30 = 25, 298 on ce every t = 25298 /8000 = 3.2 mon ths. Total production /storage cost of speakers T (Q )= aK Q + ac + hQ 2 = $87589 .47 per mon th IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 10 Sp ace EOQ extension : Ordering with quan tity discou nts So far, we assumed that the unit cost of an item is the same regardless of quantity in the batch. In fact, as a result, the op timal order quantity and frequency are independent of this unit cost That is not the case, for example, if we take into accou nt the time value of mon ey that is tied up in inventory. If the interest rate is i% per unit of time, than Q = 2aK h+ic . See hom ework for derivation of this formula. Often, however, discou nts and econ om ies of scale are available if large batches are ordered. In the TV example, suppose that the unit cost of eve ry sp eaker is c 1 = $11 if fewer than 10, 000 speakers are produced, c 2 = $10 if production is at least 10, 000, but fewer than 80, 000 speakers, and c 3 = $9 if production is 80, 000 speakers or more. W hat is the op timal inventory policy? IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 11 Sp ace Quan tity discou nts, step I From ou r discussion of the EOQ model, the total cost per mon th if the unit cost was c j can be com puted as follows: T j (Q )= aK Q + ac j + hQ 2 , for j =1, 2, 3. Let’s plot these three curves! The value of Q tha t minimizes T j (Q ) can be fou nd by using the ba sic EOQ mo del: Q = 2aK h = 25, 298. Q is the sam e for all the curve s. This is because the cost of capital is not taken into accou nt, and so the eﬀect of diﬀerent values of inventory is not felt. If the cost of capital is not 0, on e must use the formulas Q j = 2aK h+ic j — in this case, the points at which each cost curve is minimized wou ld be diﬀeren t! IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 12 Sp ace Cost per unit of time with quan tity discou nts 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 $75,000 $80,000 $85,000 $90,000 $95,000 $100,000 $105,000 Batch size Q Total cost per unit of time Cost per unit of time with quantity discounts $11.00 $10.00 $9.00 IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 13 Sp ace Quan tity discou nts, step II The minim izing value Q =25, 298 is feasible for the cost function T 2 (Q ), whose value at this point is T 2 (Q ) =$87, 589. Note that for any value of Q , T 1 (Q ) > T 2 (Q ), so orde ring at the cost c 1 can be eliminated from con sideration . T 3 (Q ) is minimized (ov er its feasible range Q ≥ 80 , 000) at Q 3 =80, 000, with T 3 (Q 3 ) =$85, 200. Com paring these three values, we con clude that the quantity Q =80, 000 is op timal. If the discou nts were less steep, for example, if c 3 = $9 .5, then it wou ld be op timal to stick with Q =25, 298! IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 14 Sp ace Finding op timal order quan tity with discou nts: 1. For each value of unit cost c j ,usetheEOQformulaforthe EOQ model to calculate its op timal order quantity, Q j . 2. For each c j wh ere Q j is within the feasible range of order quantitie s for c j , calcualte the correspon ding total cost per unit tim e, T j (Q j ). 3. For each c j wh ere Q j is not within this feasible range , de term ine the orde r quantit y Q j that is feasible for this cost and closest to Q j . Calculate the cost T j (Q j ). Note the diﬀeren ce between Q j and Q j . 4. Com pare the costs ob tained for all c j ’s and choose the minimum, with the order quantity ob tained in step 2 or 3 that gives this minimum cost. IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 15 Sp ace Another inve ntory man age ment prob lem 1 • You are the Michigan distributor of Nature’s Peak, a high-end brand of frozen dog fo od. You have (pre-paid) contracts with local “boutique” pet stores to deliver, in each of the next 4 months, resp ectively, 50, 65, 100, and 70 lb of fo od, and these orders must be filled on time. • You obtain the fo od from the manufacturer at wholesale prices which vary month to month. In the next four months, unit prices are $5, $8, $4, and $7 per pound, resp ectively, and you can buy at most 80 lb each month. • Food needs to be delivered to the stores at the end of each month. You place your order with the manufacturer in the beginning of each month, receive your order at the end of the month, and immediately deliver fo od to the local stores. • If you have fo od remaining after the demand has been satisfi ed, you can keep some of in your wearhouse at a cost of $2 per pound per month until the next delivery, and donate the rest to the Humane So ciety. • In 4 months, Nature’s Peak is planning to change the recipe and packaging for this fo od. If you have any fo od left at that time, you can sell it to discount pet stores in the area for $6 per pound. (Until then, the company wants to ma inta in the pro duc t’s high -end ima ge .) • How should you manage your inventory for the next 4 months? 1 Se ction 4.12 of De nardo describes a problem in which such inventory management issu es are part of the decisions IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 16 Sp ace Operation al decision s in the pet food distributor’s prob lem Is an EOQ model appropriate for this prob lem? We need a diﬀerent type of mathematical model! W hat decision s do you need to make for the com ing 4 mon ths? W hat performance measure wou ld you use to com pare diﬀerent decision s? W hat con straints (restriction s) must you r decision s satisfy? W hat assumption s are being made? IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 17 Sp ace Formulation of a mathematical model for the pet food distributor’s prob lem Decision variab les: represent decision s by variables. Objective function : express the performance criterion in terms of the decision variables; shou ld it be minimized of maximized? IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 18 Sp ace Formulation of a model for the pet food distributor’s – con t. Con straints: express all (explicit and implicit) con straints and restriction s on the values of the decision variables. IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 19 Sp ace Mathematical model for pet food distributor’s prob lem Decision variab les: x 1 , x 2 , x 3 , x 4 : lbs of food ordered in each of the 4 mon ths s 1 , s 2 , s 3 , s 4 : lbs of food stored/sold at the end of each of the 4 mon ths y 1 , y 2 , y 3 , y 4 : lbs of food don ated in each of the 4 mon ths Mathematical model: mi nimi ze 5x 1 +8x 2 +4x 3 +7x 4 +2(s 1 + s 2 + s 3 ) − 6s 4 (Ne t) cost objective subject to x 1 = 50 + s 1 + y 1 Inven tory balan ce in mon th 1 con strain t x 2 + s 1 = 65 + s 2 + y 2 Inven tory balan ce in mon th 2 con strain t x 3 + s 2 = 100 + s 3 + y 3 Inven tory balan ce in mon th 3 con strain t x 4 + s 3 = 70 + s 4 + y 4 Inven tory balan ce in mon th 4 con strain t x 1 , x 2 , x 3 , x 4 ≤ 80 Orde ring capa city cons tra int( s) x 1 , x 2 , x 3 , x 4 , No nnegativity constraint(s) s 1 , s 2 , s 3 , s 4 , y 1 , y 2 , y 3 , y 4 ≥ 0 IOE 202: Op eration s Mo deli ng, Fall 2009 Pa ge 20 Marina Epelman L2