Arch. Min. Sci., Vol. 58 (2013), No 3, p

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1 Arch. Mi. Sci., Vol. 58 (2013), No 3, p Electroic versio (i color) of this paper is available: DOI /amsc BEATA TRZASKUŚ-ŻAK*, ANDRZEJ ŻAK** BINARY LINEAR PROGRAMMING IN THE MANAGEMENT OF MINE RECEIVABLES BINARNE PROGRAMOWANIE LINIOWE W ZARZĄDZANIU NALEŻNOŚCIAMI KOPALNI This paper presets a method of biary liear programmig for the selectio of customers to whom a rebate will be offered. I retur for the rebate, the customer udertakes paymet of its debt to the mie by the deadlie specified. I this way, the compay is expected to achieve the required rate of collectio of receivables. This, of course, will be at the expese of reduced reveue, which ca be made up for by icreased sales. Customer selectio was doe i order to keep the overall cost to the mie of the offered rebates as low as possible: K x k mi cr j j j1 where: K cr total cost of rebates grated by the mie; k j cost of gratig the rebate to a j th customer; x j decisio variables; j = 1,, particular customers. The calculatios were performed with the Solver tool (Excel programme). The cost of rebates was calculated from the formula: (j) k j = P j K k where: P j differece i reveues from customer j; K k (j) cost of the so-called trade credit with regard to customer j. The cost of the trade credit was calculated from the formula: T r ts Kk Ns s * AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MINING AND GEOENGINEERING, DEPARTMENT OF ECONOMICS AND MANAGEMENT IN INDUSTRY, AL. A. MICKIEWICZA 30, KRAKOW, POLAND. t-zak@agh.edu.pl ** AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF APPLIED MATHEMATICS, DEPARTMENT OF DISCRETE MATHEMATICS, AL. A. MICKIEWICZA 30, KRAKOW, POLAND. zakadrz@agh.edu.pl

2 942 where r iterest rate o the bak loa, % t s collectio time for the receivable i days (e.g. t 1 = 30, t 2 = 45,, t 12 = 360); N s value of the receivable at collectio date t s. This paper presets the geeral model of liear biary programmig for maagig receivables by gratig rebates. The model, i its geeral form, aims at: miimisig the objective fuctio: with the restrictios: j1 K x k mi cr j j j1 ( N x N ) qn tji j ji i i = 1,..., m ad: x j (0,1) where: N t ji value of the timely paymets of a customer j i a i th moth of the period aalysed; N ji value of the overdue receivables of a customer j i a i th moth of the period aalysed; q the assumed miimum percetage of timely paymets collected; N i summarised value of all receivables i the moth i; m the umber of moths i the period aalysed. The geeral model was used for applicatio to the example of the operatig Mie X. Furthermore, the study has bee exteded through the presetatio of a biary model of liear programmig, i which the objective fuctio should miimise the aticipated value of the cost of rebates: EK ( ) p xk cr j j j j1 where: p j deotes the probability of use of the rebate by customer j. The paper presets two mathematical models. Oe is a determiist model which ca be used uder certaity coditios, whereas the other cosiders the risk of the rebates ot beig used by the customers. The paper also describes some radom experimets with the Mote Carlo method. Keywords: liear programmig method, risk ad ucertaity, rebate, discout, receivables maagemet W artykule zastosowao metodę biarego programowaia liiowego w celu wyboru odbiorców, którym zostaie zapropooway rabat. W zamia za propooway rabat, day odbiorca zobowiązuje się spłacać w założoym termiie ależość kopali. W te sposób przedsiębiorstwo ma osiągąć odpowiedi poziom ściągalości ależości termiowych. Staie się to oczywiście kosztem zmiejszeia przychodów, które moża zrekompesować poprzez zwiększeie sprzedaży. Wybór odbiorców dokoay został w taki sposób aby sumaryczy koszt zapropoowaego rabatu był dla kopali jak ajmiejszy, czyli: K x k mi cr j j j1 gdzie: K cr całkowity koszt udzieloych rabatów przez kopalię k j koszt udzieleia rabatu j-temu odbiorcy, x j zmiee decyzyje, j = 1,, poszczególi odbiorcy.

3 943 Do obliczeń wykorzystao arzędzie Solver (program Exel). Koszty rabatów wyliczoo ze wzoru: (j) k j = P j K k gdzie P j różica w przychodach od odbiorcy j, (j) K k koszt tak zwaego kredytu kupieckiego pooszoego względem odbiorcy j. Koszt kredytu kupieckiego wyzaczoo ze wzoru: T r ts Kk Ns s gdzie r stopa procetowa zaciągiętego kredytu bakowego, %, t s okres ściągaia ależości, di (p. t 1 = 30, t 2 = 45,, t 12 = 360), N s wartość ależości o termiie ściągalości t s. W artykule zaprezetowao ogóly model biarego programowaia liiowego do zarządzaia ależościami, z wykorzystaiem rabatu. Model te w ogólej postaci ma: zmiimalizować fukcję celu: przy ograiczeiach: j1 K x k mi cr j j j1 ( N x N ) qn tji j ji i i = 1,..., m oraz: x j (0,1) gdzie: N t ji wartość ależości termiowych odbiorcy j w i-tym miesiącu aalizowaego okresu, N ji wysokość ależości ietermiowych odbiorcy j w i-tym miesiącu okresu, q założoy, miimaly odsetek ściągalości ależości termiowych, N i sumarycza wartość wszystkich ależości w miesiącu i, m liczba miesięcy w aalizowaym okresie. Ogóly model posłużył do jego aplikacji a przykładzie daych fukcjoującej kopali X. Poadto rozważaia te rozszerzoo przedstawiając biary model programowaia liiowego, w którym fukcja celu powia miimalizować oczekiwaą wartość kosztów rabatów, czyli: EK ( ) p xk cr j j j j1 gdzie p j ozacza prawdopodobieństwo skorzystaia z rabatu przez odbiorcę j. W artykule zostały sformułowae dwa modele matematycze. Pierwszy z ich to model determiistyczy, który może być stosoway w warukach pewości, zaś drugi uwzględia ryzyko ieskorzystaia z rabatów przez odbiorców. W artykule przeprowadzoo rówież pewe losowe eksperymety za pomocą metody Mote Carlo. Słowa kluczowe: metoda programowaia liiowego, ryzyko i iepewość, rabat, zarządzaie ależościami

4 Itroductio Liear programmig has bee applied to a umber of optimisatio problems (Czopek, 2001; Osiadacz, 2000; Jaśkowski, 1998). It ca be also applied i the maagemet of receivables which result from trade credits grated i the course of busiess, icludig various forms of rebates (Cooke, 2001; Jasiukiewicz at.al., 2001; Kloowska, 2005; Lewadowska, 2000). Covetioal receivables-maagemet methods are based o the so-called icremetal method, i which the icrease i reveue after the rebate compesates for the cost of gratig the rebate. However, liear programmig eables methods of receivables maagemet to be optimised, for example, as follows, through: maximisig reveue from sales, ad thus profit; optimisig the periods of rebates grated; miimisig the cost of trade credit grated; miimisig the risk of overdue or lost receivables; restructurig timely paymets ad overdue receivables. The liear programmig problem ivolves fidig a optimal solutio to the liear objective fuctio f(x) with argumets x 1, x 2,..., x, give certai restrictios icludig the variables x j (j = 1, 2,..., ). This ca be preseted i geeral form, as follows: to optimise the objective fuctio: f ( x1, x2,..., x ) c x optimum(max,mi) (1) j j j1 with the restrictios: or ij j i j1 a x b (2) ij j i j1 a x b, for i = 1, 2,..., m (3) x j 0, for j = 1, 2,..., (4) m < (5) where: f(x 1, x 2,..., x ) objective fuctio; c j coefficiets of objective fuctio; a ij, b j directio coefficiets, real umbers; m umber of restrictios. I the group of liear programmig problems uder cosideratio, it is aturally ecessary to limit the rage of acceptable solutios (Gass, 1980; Jasiukiewicz et. al., 2001; Radzikowski, 1979), sice two further coditios limitig the solutio usually have to be satisfied. I the case of a mie, such restrictios might be, for example, limited resources or limited miig output.

5 945 I receivables maagemet, restrictios (2) ad (3), depedig o the rebate scheme chose, may ivolve (Lewadowska, 2000; Pluta & Michalski, 2005; Portalska & Koratowicz, 2003; Sierpińska, 2005; Szyszko, 2000) the followig: the acceptable value of receivables for the etire mie or a idividual customer; the acceptable value of the overall cost of the trade credit grated, or grated idividually to each of the customers; the expected reveue of the mie after applyig the rebate; miimum prices eeded to guaratee required reveue; miimum rate of collectio of timely paymets, etc. I the case of maagemet of a miig compay, geerally speakig, the decisio-maker, eve i the case of receivables maagemet, is ofte faced with a yes or o choice. If this is the case, biary problems are ivolved, usually decisio-related problems, with the atural iterpretatio x j = 0, whe a decisio is made ot to execute a project or x j = 1 whe decided otherwise. This paper addresses this very issue, which depeds o the selectio of customers to whom the mie will ad will ot grat the rebate. The paper formulates two mathematical models. Oe ca be applied uder determiistic coditios, whereas the other cosiders the risk of the rebates ot used by the customers. Some completely radom experimets were also coducted (usig the Mote Carlo method). Therefore, the followig aalysis of the problem ca be uderstood as how to make decisios uder certaity, risk or ucertaity coditios. It is worth metioig that strategies for the above-metioed situatios have also bee ivestigated (Cyrul, 2004) i a paper presetig differet methods. 2. Formulatio of the geeral biary model of the liear programmig Let us suppose that the mie accurately moitors its customers ad is able to specify (at least approximately) the followig values: k j cost of rebate grated to customer j, j = 1, 2,...,. N t ji value of the timely paymets of customer j i the i th moth of the period aalysed; N ji value of the overdue receivables of a customer j i the i th moth of the period aalysed; umber of customers. Let N ti deote the overall value of all timely paymets from all customers i the moth i, N ti N j1 t ji, i = 1, 2,..., m (6) ad N i deote the overall value of all overdue receivables from all customers i the moth i, N i N (7) j1 ad N i deote the overall value of all receivables i the moth i: ji N i = N ti + N i (8)

6 946 It is further assumed that a decisio o gratig a rebate to a give customer applies to the etire period aalysed. The customer, i retur for the rebate, udertakes to pay all receivables i timely fashio every moth. The objective of the mie is to grat rebates to some of the customers, so that the overall percetage of timely paymets from all customers will icrease to the assumed miimum level i every moth. Such a miimum percetage will be deoted by q. I additio, the overall cost of rebates should be as low as possible. To create a mathematical model for the above problem, the followig decisive variables will be itroduced: x j 1, customer j grated the discout (9) 0, otherwise. Let us aalyse how timely paymets chage after rebates are applied. While gratig a rebate to the customer j (i.e., whe x j = 1), the timely paymets of that customer will icrease by N ji i each moth i, ad thus yield N t ji + N ji, which equals N t ji + x j N ji. If o rebate is grated to the customer j, i.e. whe x j = 0, the timely paymets of that customer will ot chage ad will, therefore, remai at the level N t ji, which, agai, equals N t ji + x j N ji. Therefore, the coditio uder which the timely paymets i each moth i achieve the assumed miimum percetage of the overall receivables of that moth will be fulfilled by the restrictio: NtjixjNji qni (10) j1 Assumig that the cost of the rebate grated to customer j is deoted by k j, the the actual cost of the rebate grated to customer j is x j k j, which meas zero cost whe the rebate is ot grated (because the x j = 0); if the rebate is grated, it is simply k j (because the x j = 1). Hece, the overall cost of the rebates give is: K x k (11) cr j j j1 To sum up, the mathematical model of biary liear programmig of the problem is as follows: miimise ad: with the restrictios K x k (12) cr j j j1 NtjixjNji qn (13) i j1 x j {0,1} j = 1, 2,..., (14) It is kow that gratig the rebate to customer j meas, i practice, reducig the sales price for that customer. If o additioal terms were added to the cotract save the requiremet to reduce

7 the term of paymet for customer j, the the reveue of the mie would be reduced. To prevet that, extra provisios should be added to the cotract. Therefore, let the followig deote: A oj the reveue of the mie received so far from customer j; A j the expected reveue from customer j, give the rebate. The the additioal provisio will take the followig form: for x j = 1 for x j = A j x j A oj (15) A j = A oj (16) Note, however, that coditios (15) ad (16) do ot costitute part of the model (12)-(14), but are icluded oly i a modified cotract with a give customer. 3. Applicatio of the biary model of the liear programmig Let us preset the above model i a hypothetical example. Table 1 shows the data regardig the paymet of receivables by te customers withi a period of 3 moths i the past year. The mie aticipates the same values i the first quarter of the year aalysed. Paymet of mie receivables by 10 selected customers, PLN TABLE 1 Customer Moth of the 1 st quarter year A Receivables paid o time Overall overdue receivables Overdue receivables, moths , ,597 92,597 20, customer , ,032 53,008 80, ,749 23,445 66,280 67, ,859 39,031 19,420 19, customer ,821 35,745 7,613 18,475 9, ,858 42,478 17,719 9,714 15, , customer ,656 34,945 33,303 1, ,470 63,179 48,180 13,357 1, ,733 30,405 30, customer ,245 23,174 23, ,780 27,797 21,345 6, customer , ,785 6,278 6,

8 948 TABLE ,603 16,279 16, customer ,702 16, , , ,100 5,926 5, customer ,581 7,725 7, , ,750 64,287 15,210 13,455 16,928 18, customer ,647 61,288 29,750 12,274 4,381 14, ,108 81,106 29,818 19,750 12,274 19, ,204 40, ,222 0 customer ,330 40,303 13, , ,444 76,578 36,275 13, , , customer ,392 29,054 29, ,725 29,211 15,965 13, TOTAL 1,109,986 1,173,668 Percetage, % As show above, timely paymets costitute oly 49% of all receivables. Assumig that the goal of the compay s maagemet is to pla a strategy which would result i a receivable collectio rate of at least 80% each moth, that goal ca be achieved by givig rebates i retur for timely paymet of the receivables. Cosiderig, for example, customer 9 ad the third moth, assumig that the aual iterest rate of a bak loa is r = 9.5% ad the rate of the rebate is u = 9%, the, the profit achieved through payig off the trade credit would be: Kk PLN 12 6 The loss i the third moth due to the rebate would be: P 0.09 ( ) PLN Therefore, i the third moth, the cost of the rebate grated to customer 9 would be: P K k PLN The value k 9 would be achieved after addig up the cost of the rebate i all 3 moths. Table 2 below presets the cost calculated. Cost of rebate to idividual customers i three aalysed moths i PLN TABLE 2 k 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 k 10 41, , , , , , , , , , The curret status of overdue receivables ad timely paymets is show i tables 3 ad 4.

9 949 Value of overdue receivables paid by idividual customers, PLN TABLE 3 Customer Overall overdue receiveables Moth 1 112,597 39, , ,279 5,926 64,287 40, ,747 Moth 2 133,032 35,745 34,945 23, ,279 7,725 61,288 40,303 29, ,545 Moth 3 156,749 42,478 63,179 27,797 6, ,106 76,578 29, ,376 Value of timely paymets paid by idividual customers, PLN TABLE 4 Customer Overall timely paymets Moth 1 80,744 17,859 6,598 29, ,603 43,100 99,750 23,204 35, ,611 Moth 2 36,718 19,821 50,656 26,245 33,063 50,702 37,581 64,647 49,330 19, ,155 Moth ,858 10,470 17,780 26,785 88,168 43,882 76,108 25,444 26, ,220 As show above, the overall performig ad overdue receivables are respectively PLN 693,358, 769,700 ad 820,596 i moths 1, 2 ad 3, ad 80% (the assumed miimum percetage q) of that sum is respectively PLN 554, i moth 1, PLN 615,760 i moth 2 ad PLN 656, i moth 3. After substitutig the data i the model (12)-(14) ad solvig it with Excel s Solver tool, the followig solutio was obtaied: x1 x3 x4 x9 1, x2 x5 x6 x7 x8 x10 0 ad the total cost of the rebate grated was K cr = PLN 83, Therefore, the best choice would be to grat rebates to customers 1, 3, 4 ad 9. The the etire cost of the rebate would be PLN 83, For example, achievig the required level of timely paymets would be possible if the compay grated rebates to customers 1, 3, 8 ad 10. The, however, the overall cost of rebates would be PLN 102, Biary liear programmig method i gratig rebates to mie customers, cosiderig risk I practice, the requiremet that those customers give rebates pay their debt o time might ot be met. Therefore, a additioal radom elemet has bee itroduced. Let p j be the approximate kow (as a result of moitorig) probability that customer j will use the rebate. The objective is to idetify those customers to whom rebates will be grated, so that the expected value of timely paymets is at least q 100% of all receivables (ote that q is the miimum assumed percetage of collected timely paymets) ad the cost of the rebates give is as low as possible.

10 950 N t ji deotes a radom variable which idicates a icrease i the collectio of timely paymets from customer j i moth i, whe that customer was grated the rebate. Therefore: N t ji = N ji with the probability p j N t ji = 0 with the probability 1 p j (the above equatios result from the fact that i cases where the rebate is used, the customer is obliged to pay overdue receivables o time). Therefore, the expected value of the icrease i timely paymets from customer j i moth i equals: E( N ) N p (17) tji ji j Fially, the expected value of the icrease i timely paymets from customer j i moth i equals: EN ( N ) N E( N ) N p N (18) tji tji tji tji tji j ji owig to the liearity of the expected values (ote that N t ji is ot a radom variable but a kow value). O the other had, whe customer j is ot grated the rebate, the performig receivables will ot chage ad will remai N t ji. I geeral, both situatios ca be described i a commo formula: EN ( N ) N xp N (19) tji tji tji j j ji usig agai the iterpretatio of the decisio variables x j. Note that the overall cost is also a radom variable with the expected value programmig problem will be ivolved: miimise with the restrictios cr j j j j1 EK ( ) p x k cr j j j j1. Therefore, the followig biary EK ( ) p xk (20) Ntjixj pjnji qni i = 1,..., m (21) j1 x j {0,1} j = 1,..., (22) The above model will be described agai with the data from table 1. Moreover, all probabilities p j were determied as 0.8. After pluggig the data ito the model (15)-(17) ad solvig it with Excel s Solver tool, the followig solutio was obtaied: x1 x3 x8 x9 x10 1 x2 x4 x5 x6 x7 0 ad the expected cost of the rebate grated was K cr = PLN 94,

11 Summary Note that the cost of gratig rebates k j, j = 1,..., ca be uderstood i the broadest sese as the cost of debt collectio resultig from other methods of receivables maagemet (e.g. court fees, hirig a debt collectio compay, or the cost of credit isurace). Therefore, model (12)-(14) would also be applicable i such aalysis (this refers also to model (20)-(22)). The above model (12)-(14) was also applied to aalysis of the receivables of a real mie ( X ) ad its 110 selected customers. The simulated level of the compay s timely paymets i Year 1 of the aalysis was aroud 41%. The required share of timely paymets was fixed, as i the example, at the level of at least 80% i each moth of the year aalysed. To solve the biary problem (12)-(14), the Solver tool was used agai. However, due to the great umber of variables (110 customers), Excel did ot fid a optimum solutio withi a reasoable time. Istead of optimisig the origial biary problem, it was decided to solve its liear relaxatio, i.e. the problem of liear programmig resultig from (12)-(14), by replacig the restrictios (14) with 0 x j 1 The solutio of the relaxatio is the value of PLN 2,227,114 which is the lower limit for K cr, i.e. K cr PLN 2,227,114. Such a solutio, however, is uacceptable, sice the variables assume fractioal values ad have o practical iterpretatio. The solutio was the rouded to the closest biary oe. Oe drawback of such roudig is that it may ot fulfil all the required restrictios. Also, i this case, after roudig, the percetage of timely paymets was maitaied above the level of 80% i all moths except oe, i which it was slightly below 79%. Sice that level oly slightly differed from the assumed oe, such roudig was cosidered acceptable. The cost of such a solutio was PLN 2,354,076, so the relative error does ot exceed 5.7%. The value of relative error was derived from the formula: * * opt Z Z Z Z Z where: Z * the accepted solutio, Z opt ukow value of the optimum solutio. opt opt The iequality i the formula results from the fact that Z opt 2,227,114 PLN. It is merely a estimatio of the relative error of approximatio. I practice, a much smaller error should be expected. The solutio was, therefore, cosidered satisfactory. To compesate for the resultat cost of the rebate by icreasig sales, the mie would have to sell 48, Mg more product at a average price of PLN 48.08/Mg. I the mie i questio, the average mothly sales i the year aalysed amouted to 130, Mg, so the plaed icrease i sales would be 37.5% of the mothly average. Some radom experimets were also coducted, usig the Mote Carlo method, i which rebates were grated to all customers, ad the probability of usig the rebate was set at 0.7 for each of the customers. The expected value of timely paymets slightly exceeded 80% i each moth. The expected value of the total cost of the rebates grated was PLN 3,020, As ca be see, careful selectio (simplex method) of customers to whom rebates should be grated

12 952 works, i this respect, better tha radom choice (Mote Carlo method). However, model (12)- (14) will produce good results oly whe all customers are carefully moitored ad whe most of the predictios prove to be correct. Calculatios with the Mote Carlo method are less sesitive to uexpected chages i some values. The model described by formulas (20)-(22) is a sort of combiatio of the optimum choice method with a certai degree of radomess i allowig the customers ot to use the rebate. I cotrast to the Mote Carlo method, however, a group of customers to whom we are iclied to grat the rebate is first idetified. The, radom methods are applied to the selected group, whereas i the Mote Carlo method, radom methods were applied to all customers. Model (20)-(22) is a kid of compromise (limited radomess) betwee optimum choice ad radom choice of the recipiets of the rebates. Model (20)-(22) was also applied to the aalysed Mie X. A procedure was adopted similar to the case of model (12)-(14). The probability of usig the rebate was set at 0.7. A solutio meetig all coditios ad with a expected value of the total cost of PLN 2,855,210 was obtaied. Thus it ca be stated that the attempt to safeguard agaist certai radom occurreces resulted i icreasig the overall expected cost of the operatio. Note, however, that complete radomess i gratig rebates resulted i a expected overall cost of PLN 3,020, (see the summary of model (12)-(14)). The paper was writte i 2013 withi the framework of statutory research registered with the AGH Uiversity of Sciece ad Techology i Krakow, uder the umber Refereces Cooke K.A., Positive Cash Flow. Career Press. New York. Cyrul T., Risk ad decisios - selected theories. Arch. Mi. Sci., Vol. 49, Spec. Iss., p Czopek K.,2001. Budgetig of the miig output costs by meas of dual programmig. Arch. Mi. Sci., Vol. 46, No. 1, p Gass S.I., Programowaie liiowe: metody i zastosowaia. PWN, Warszawa. Jasiukiewicz B., Jokiel E.M., Koźma Z., Ożóg M., Skrzycka-Pilch B., Widykacja ależości w obrocie gospodarczym. ODDK, Gdańsk. Jaśkowski A., Multi-dimesioal optimizatio of quatitative-qualitative productio structure of a group of coal mies (holdig compaies) i the light of basic requiremets of coal cosumers. Arch. Mi. Sci., Vol. 43, No. 4, p Kloowska S., Należości, wycea, ewidecja księgowa, ujęcie podatkowe. ODDK, Gdańsk. Lewadowska L., Niekowecjoale formy fiasowaia przedsiębiorczości. ODDK, Gdańsk Osiadacz A., Optimizatio i atural gas trasmissio etwork. Arch. Mi. Sci., Vol. 45, No. 3, p Pluta W., Michalski G., Krótkotermiowe zarządzaie kapitałem. C. H. Beck, Warszawa. Portalska A., Koratowicz O., Odzyskiwaie ależości pieiężyh. C. H. Beck, Warszawa. Radzikowski W., Programowaie liiowe i ieliiowe w orgaizacji i zarządzaiu. Wydawictwo Uiwersytetu Warszawskiego. Sierpińska M., Nowoczese arzędzia zarządzaia fiasami w przedsiębiorstwie góriczym-cz. 3. Wiadomości Góricze, Katowice, R. 56, r 3, s Szyszko L., Fiase przedsiębiorstwa. PWE, Warszawa. Received: 23 November 2012