Aalyss o Mult-product Break-eve wth Ucerta Iormato* Lazzar Lusa L. - Morñgo María Slva Facultad de Cecas Ecoómcas Uversdad de Bueos Ares 222 Córdoba Ave. 2 d loor C20AAQ Bueos Ares - Argeta lazzar@eco.uba.ar Abstract We revse the classc methodology to d the mult-product break-eve pot. I the curret paper we propose a soluto to the problem uder ucertaty codtos, based o Durá Herrera s crsp approach [] ad applyg uzzy lear programmg. We troduce the cocept o approxmate break-eve or cases whch, due to the ature o the problem, t s ot possble to d the break-eve pot. We propose two stuatos to deal wth approxmate break-eve whe the ormato regardg margal cotrbutos ad /or total xed costs s ucerta. A uzzy lear programmg method s appled to the study o a real case a small eterprse. The characterstcs o ths case make t ecessary to use teger lear programmg. Keywords: break-eve pot, multproducg rm, ucertaty, uzzy lear programmg.. Itroducto Ater the producto process begs a rm, as log as sale prces are hgher tha varable costs, sale comes keep absorbg part o the xed costs up to the pot where certa volumes o producto ad sales cover the xed ad varable costs. The pot where sales equal expeses s called break-eve pot. There s o prot made or loss curred at the break- eve pot. The calculato o ths pot provdes a useul tool to estmate prot based o sales volume, the short term, sce t allows to easly estmate expeses or ay busess operato level. The ucertaty about xed ad varable producto costs s a key aspect o admstratve decso makg. By calculatg the break-eve pot we ca aalyze the mportace o each o these varables ad the way they aect each other. The majorty o the exstg papers o the subject deal exclusvely wth compaes producg ad sellg a sgle product or several products wth a sgle margal cotrbuto. I practce these cases are a morty compared to rms wth a wder varety o products ad/or servces. The aalyss o the break-eve pot s ot usually easy these stuatos. We revse the classc methodology to d the multproduct break-eve pot. I the curret paper we propose a soluto to the problem uder ucertaty codtos, based o Durá Herrera s approach [4] applyg uzzy lear programmg. We troduce the cocept o approxmate break-eve or cases whch, due to the ature o the problem, t s ot possble to d the break-eve pot. By usg a geeral model or uzzy lear programmg developed by Delgado, M., Verdegay, J.L ad Vla, M.A. [2] we study approxmate breakeve whe the ormato regardg margal cotrbutos ad/or total xed costs s ucerta. The proposed method s appled to the study o a real case a small eterprse. The characterstcs o ths case make t ecessary to use teger lear programmg. The ormato was obtaed by tervewg the owers o the compay. Data was processed usg a specalzed sotware applcato or teger lear programmg. Let us ow troduce the otato eeded the rest o the paper. We wll place a symbol over a captal letter t represets a uzzy umber so F, M, W are all uzzy umbers, p, q, m wll deote real umbers. * Ths paper belogs to the UBACyT Project E009: El tratameto de la certdumbre e las pequeñas y medaas empresas medate el empleo de herrametas matemátcas ovadoras.
2. Mult-product Break-eve uder Certaty Codtos Cases o mooproducg or mooservcg rms are ot as requet as those o rms wth a wder rage o products ad/or servces. I act, what we are lookg or s that the margal cotrbuto obtaed by sellg all the products - whchever ther combato s - tured out to be equal to the total xed costs. Break-eve aalyss s geerally ot smple whe t must be doe multproducg rms, especally whe there are techcal codtogs, ud shortage or other type o restrctos. It s possble to d deret combatos o the products satsyg the break-eve codto. May authors such as Yard [], Drmer [] ad Durá Herrera [4] have studed crsp mult-product break-eve. The curret proposal s based o Durá Herrera s geeral approach [4] to the problem o calculatg the break-eve pot a multproducg rm. The ollowg crsp lear programmg problem models ths stuato. Mmze: = p. q Subject to: ( p w =. q = ( A. q : : B (2 q 0 ( where: p : ut prce or product q : amout o product I: sale come rom the set o products currecy uts : total xed costs w : varable ut cost or product Aalyzg the meag o the restrctos we d: ( geeral break-eve codto or the set o products the rm produces. (2 restrctve techcal, commercal or acal codtos, there were ay. ( o-egatvty o the decso varables We look or the combato o the products satsyg the break-eve codto that mmzes the sale come ad ullls the costrats.. Mult-product Break-eve uder Ucertaty Codtos We study stuatos where the ormato s ot kow wth accuracy ad t ca be expressed by uzzy umbers. As total xed costs or ut varable costs could be mprecse, we use the geeral model or uzzy lear programmg proposed by Delgado, M., Verdegay, J.L. ad Vla, M.A. [2]. Ths model o uzzy lear programmg allows to deal wth the approaches whch volve uzzy costrats ad those whch volve uzzy coecets ad gves a resoluto method or them ad all partcular problems that may be deduced rom t. The problem to cosder s: Maxmze c. x Subject to A. x b (4 x 0 Where A s a m matrx o uzzy umbers, m, b a colum vector o uzzy umbers ad c R. The mprecso o the coecets ca be modeled by meas o uzzy umbers. O the other had, the decso maker tolerates volato the accomplshmet o the costrats. The auxlary problem to solve (4 s: Maxmze c. x Subject to A. x < b + t. ( α x 0, α ( 0; ] (5 The relato < s ay the decso maker chooses. Accordg to the kd o relato <, whch we assume, deret models o covetoal lear programmg problems wll be obtaed.[2].. Frst approach We cosder that the ut prce ad the varable ut cost or each product a rm produces are represeted by real umbers, other words, they are well-kow values, whle the total xed costs are mprecse. The crsp model we have proposed 2. turs to a uzzy lear programmg model wth crsp objectve
ucto, crsp techologcal coecets ad mprecse resources. Sce break-eve codto ( wll have some mprecse coecets, we caot state the equalty remas vald. For ths reaso equato ( wll be replaced by the equaltes: ( p w = ( p w =. q (6. q m (7 (6 margal cotrbutos must cover at least total xed costs. (7 m s the optmal margal cotrbuto, soluto to a maxmzg crsp lear programmg problem that cludes the costrats o the orgal oe. It s boudary to (6. We wll ame (6 approxmate break-eve codto. Geeralzg our approach 2., we have a uzzy lear programmg model wth crsp objectve ucto ad uzzy costrats. Mmze: Subject to: = = = p. q m. q (8 m. q m (9 A. q : : B (0 q 0 ( (8 approxmate break eve codto, where m = p w s the ut margal cotrbuto o the product. (9 boudary to uzzy equalty (8 (0 crsp or uzzy restrctos ( o egatvty o the decso varables The aalyss o ths stuato s very useul to the rm because t allows to d the mmum sale come so that margal cotrbutos cover at least the total xed costs. There are several methods to solve the uzzy lear programmg model wth uzzy costrats such as the oes proposed by Delgado, M., Verdegay, J.L. ad Vla, M.A. [2], Taaka, H., Okuda, T. ad Asa, K. [7], Verdegay, J.L. [9].... Study o a Case A ursery school Bueos Ares oers ull-tme servce, part-tme servce (ether morg or ateroo ad extra hours (two at the most that ca be added to the part tme servces. The maxmum capacty o the school s 6 pupls per tur. It s also kow that o more tha 20% o the chldre stay at school a extra hour, o more tha 5% o the pupls stay at school two extra hours ad at the most 0% o the chldre are ull tme pupls. Meals are provded by a caterg servce rm. Luch ut cost s $4 per moth ad breakast/ateroo sack ut cost s $8. These are the oly ut varable costs. It s requred to calculate the umber o servces o each type eeded to cover at least the total xed ad varable costs. We cosder the ollowg decso varables, q : ull-tme servce, q 2 : part-tme servce (morg, q : part-tme servce (ateroo, q : oe extra hour, 4 q 5 : two extra hours The ees appled to each servce oered are p = 50 p2 = 90 p = 90 p4 = 20 p 5 = 40 I all the cases the prce cludes luch, breakast ad/or ateroo sack. Usg the above ormato, varable ut costs are w = 0, w2 = 22, w = 22, w4 = 0, w5 = 0 The correspodg ut margal cotrbutos are m = 20, m2 = 68, m = 68, m4 = 20, m5 = 40 The total xed costs are o less tha 2862 ad o more tha 58. Replacg the rst approach we have the ollowg uzzy lear programmg problem, Mmze 50. q + 90. q2 + 90. q + 40. q5 Subject to q Ζ 5 (2 q 0 5 ( q + q2 6 (4 q + q 6 (5.0, 2 ( q.0, 05 q + (6 4 q q + (7 5 2 q
( q + q.0, q 2 + q (8 20. q + 68. q2 58 (9 20. q + 68. q2 5296 (20 (2 addtoal costrat: the decso varables must be teger umbers ( o-egatvty o the decso varables (4, 5 the maxmum capacty o the school s 6 pupls per tur (6 o more tha 20% o the chldre stay at school a extra hour (7 o more tha 5% o the chldre stay at school two extra hours (8 at the most 0% o the chldre are ull-tme pupls (9 the maxmum tolerace or s t = 72 (20 ts rght had sde s the optmal soluto to the crsp lear programmg problem o maxmzg the margal cotrbuto subject to the costrats (2,8. We use the geeral model or uzzy lear programmg proposed by Delgado, M., Verdegay, J.L. ad Vla, M.A. [2] the specal case whe oly the coecets are uzzy. The problem turs to the ollowg classc parametrc lear programmg problem wth crsp soluto. Mmze 50. q + 90. q2 + 90. q + 40. q Subject to q Ζ 5 q 0 5 q + q2 6 q + q 6 q 4 + q q 5 + q q 2 + q.0, 2.0, 05 ( q + q.0, The uzzy equalty (7 s replaced by the crsp equalty 20. q + 68. q2 58 72.β 0 β 20. q + 68. q2 5296 A specalzed sotware applcato or teger lear programmg s used to solve ths problem. The soluto or some values o β s show Table. From the table we see that t s a approxmate break-eve stuato. 5 β Table : Soluto o the case q 2 q q + 4 q 5 q Icome Prot 0 6 4600 5.0.2 2 28 5 0 4420 5.2.4 27 5 4220.4.6 0 27 4 4050 5.6.8 0 25 4 870.8 9 25 5 0 700 8.0.2. Secod approach Let us cosder the case where, accordg to the avalable ormato, the ut prce or each product a rm produces s a crsp umber, the varable cost or each product s estmated by the tragular uzzy umber W = ( w, w2, w ad the tragular uzzy umber F = (, 2, represets the total xed costs. The crsp model we proposed 2. turs to a uzzy lear programmg model wth crsp objectve ucto, uzzy techologcal coecets ad uzzy resources. I ths stuato t s also ecessary to replace breakeve codto ( by a approxmate break- eve codto. Geeralzg our approach 2., the problem o dg the approxmate break-eve whe margal cotrbutos ad xed costs are tragular uzzy umbers s expressed by: Mmze: Subject to: = = = p. q M. q M. q m A. q : : B (2 (22 q 0 (2 approxmate break eve codto, where M = p W s the ut margal cotrbuto o the product (22 m s the hghest margal cotrbuto, optmal soluto to a maxmzg uzzy lear programmg problem that cludes the costrats o the orgal problem. As the
ormato s gve by tragular uzzy umbers, t ca be solved by usg a multple objectve lear programmg method such as the oe proposed by Youg Jou La ad Chg- La Hwag [6]. s a uzzy rakg method. Note that the rght had sde o equalty (2 s because the hghest total xed costs must be covered every case. Sce there are may methods to order uzzy umbers, the choce o oe o them wll geerate a method o uzzy lear programmg to solve ths model. Amog these methods we ca meto the oes developed by Delgado, M., Verdegay, J.L. ad Vla, M.A. [2], Taaka H., Ichhash H. ad Asa K [8], Klr, J.G ad Yua, B. [5]. Ths model would be useul to deal wth the case o study meals were cooked at school. I ths stuato, varable costs would be mprecse ad tragular uzzy umbers would be sutable or expressg ther vagueess. 4. Remarks There are stuatos where the avalable ormato s uzzy ad the break-eve pot caot be oud usg the classc methods. Ater our aalyss we coclude that these cases are satsactorly solved usg the approxmate break-eve codto, whch allows to calculate the umber o servces or products o each type that must be sold to cover at least the total xed costs. By usg uzzy lear programmg t s qute smple to deal wth the problem o calculatg break-eve uder codtos o ucertaty. Data s processed usg a specalzed sotware applcato or real or teger lear programmg. I the secod approach margal cotrbutos ad total xed costs ca be expressed by ay type o uzzy umber. I every case the rght-had sde o equalty (2 s the supremum o the support o the total xed costs. We could cosder a thrd approach where ut prces are mprecse. I ths case t would be useul to study prot. Ackowledgemet The authors would lke to thak a aoymous reeree whose commets led us to a mprovemet o ths paper. Reereces [] Buckley, J. J. Qu, Y. (990: Solvg Lear ad Quadratc Fuzzy Equatos, Fuzzy Sets ad Systems 8. North Hollad, pp. 4 59. [2] Delgado, M. Verdegay, J.L. Vla, M.A. (989: A Geeral Model or Fuzzy Lear Programmg, Fuzzy Sets ad Systems 29. North Hollad, pp. 2-29. [] Drmer, R.L. (200: Fazas de empresa, Osmar D. Buyatt, Lbrería Edtoral, Bueos Ares, chapter VII. [4] Durá Herrera, J. J. (992: Ecoomía y dreccó acera de la empresa, Prámde. Madrd, chapter 4. [5] Klr, G. J. Yua, B. (995: Fuzzy Sets ad Fuzzy Logc. Theory ad Applcatos. USA Pretce Hall Iteratoal, New Jersey, chapter 5, pp. 408 45. [6] La Y. J, Hwag C.L. (992: A ew approach to some possblstc lear programmg problems, Fuzzy Sets ad Systems 49. North-Hollad, pp 2-. [7] La, Y. J. Hwag, C. L. (992: Fuzzy Mathematcal Programmg. Methods ad Applcatos. Sprger Verlay. Berl. [8] Taaka, H., Ichhash, H. ad Asa, K. (984: A Formulato o Fuzzy Lear Programmg Problems based o the Comparso o Fuzzy Numbers. Cotrol ad Cyberetcs, pp. 86-94. [9] Taaka, H., Okuda, T. ad Asa, K (974: O Fuzzy Mathematcal Programmg, Joural o Cyberetcs 4, pp. 7 46. [0] Verdegay, J.L. (982: Fuzzy Mathematcal Programmg Fuzzy Iormato ad Decso Processes (M.M. Gupta ad E. Sachez Eds.. North-Hollad, pp. 2-27. [] Yard, A. (995: Puto de equlbro multproducto Costos para empresaros Gmeez, C. M. (comp. Edcoes Macch. Bueos Ares, chapter XII.