Interpretation of Different Approaches to Sensitivity Analysis in Cash Flow Problem

Transcription

1 MATEMATIKA, 01, Volume 9, Number 1b, 1-9 Department of Mathematcal Scences, UTM Interpretaton of Dfferent Approaches to Senstvt Analss n Cash Flow Problem Alreza Ghaffar-Hadgheh Department of Mathematcs, Azarbajan Unverst of Tarbat Moallem Klometere, Tabrz / Azarshahr Road Tabrz, Iran e-mal: hadgheha@azarunv.edu Abstract Cash s the drvng power of all busness and cash-flow statement s one of the major ssues of nsttutons, especall n crss. Optmal cash-flow plan of a compan could be one of the most mportant ndcators of that busness's fnancal health and can be consdered as ts fnancal analsts' ablt and skll. Lnear optmzaton (LO) s one of the mathematcal tools n modelng the cash-flow problem and ts rch lterature helps analsts to devce the optmal one when the stuaton satsfes the requrements of the LO model. However, the stuaton s alwas due to varaton and the optmal soluton arsen from the LO model have to be analzed accordng to measurable varaton of nput data. Senstvt analss and parametrc programmng s the tool to ths analss. Usng the Smple method to fnd a basc optmal soluton and havng multple optmal solutons s one of the reasons that dfferent solvers lead to dfferent optmal solutons. In these stuatons, senstvt analss ma produce confusng results. Moreover, there are dfferent ponts of vews to senstvt analss such as optmal bass nvaranc, optmal partton nvaranc, support set nvaranc to name some eamples. Here, we brefl revew dfferent approaches to senstvt analss n LO and a short term cash-flow problem of a dumm nsttuton s modeled as an LO problem. It s shown that the problem has multple optmal solutons whch are degenerate, the stuaton that usuall occurs n practce and causes of ambguous and unclear results. The confusng results n analzng the senstvt of these solutons are hghlghted n ths eample. Then, a strctl complementar optmal soluton s provded and ts useful nterpretaton n senstvt analss s mentoned n a nutshell. In the sequel, the concept of the results arsng n dfferent ponts of vews to senstvt analss s analzed. Kewords Cashflow Problem, Lnear Optmzaton, Senstvt Analss. 010 Mathematcs Subject Classfcaton 6P0 1 Introducton Optmzaton plas a major role n fnance nowadas. Approprate decson makng s of the most mportant crteron n survvng of all the fnance and economc nsttutes and ndustral organzatons specall n crses. Man problems n quanttatve fnance and rsk management such as asset allocaton, dervatve prcng, value at rsk modelng and model fttng, are now effcentl solved usng state-of-the-art optmzaton technques. Two of the closel related fnancal problems are the cash-flow and the asset prcng. In ths stud, we consder the cash-flow problem [1] and revew dfferent models, emphaszng the rch n theor and mature n methodolog and mplementaton, the LO Model. Senstvt analss of obtaned solutons based on ever-changng atmosphere of the world s not an opton but s an oblgaton. Dfferent aspects of senstvt analss and parametrc programmng have dvert nterpretatons. Havng ratonal understandng of the results leads to choose rght practcal and sustanable fnancal polces. In ths stud, we consder dfferent aspects of senstvt analss and parametrc programmng appled to a cash-flow problem that s formulated as an LO problem. The paper goes as follows. The net secton ntroduce the cash-flow problem n a nutshell and revew dfferent possble mathematcal models. In Secton, LO problem s mentoned and the concern turns to dfferent aspects of senstvt analss and parametrc programmng n LO. The LO model of cash-flow problem s 1

2 devsed n Secton, and prvleges and drawbacks of the LO model are mentoned then. Interpretaton of senstvt analss n ths model s the man am Secton. Concludng remarks mght help the researchers for further studes. Cash Flow Problem and Optmzaton Models In ts smplest form, owng the actual cash n and out of the busness, and beng dentfed both ther sources and uses to recognze cash-flow varaton over a perod s the subject of cash-flow analss. Cash management,.e. controllng the cash-flow, s vtal to busnesses of all szes. Small busnesses are especall vulnerable to cash-flow problems snce the tend to operate wth nadequate cash reserves or none at all, and worse, tend to mss the mplcatons of a negatve cash-flow untl t's too late. Cash-flow analss could be descrbed n several steps, whch allow one to model t as an optmzaton problem. Frst lst cash nflows (sources). The sources of cash ma be lmted to: new nvestment, new debt, sale of fed assets and operatng profts. Then, record cash outflows (uses) and dentf when (b date) cash flows n or out. Tmng s the net step; that s, cash nflows mnus cash outflows. One of the man task s to dentf the major consequences of cash as t currentl flows and ndcate the constrants; nflows or outflows, whch cannot be changed. Consderng these steps could lead to an approprate optmzaton model. Fnall establshng a plan for postve cash-flow s the goal of the problem. B solvng the problem and dong senstvt analss, the nflows and outflows, whch can be changed (rescheduled) wthout consderabl nfluencng the optmalt of the plan can be recognzed. Observe that postve cash-flow can be consdered as a measure of a compan's fnancal health and much postve cash-flow the better. There are some assumptons nherentl accompaned wth LO and ts success n modellng refers to how closel relatvel matches up wth these assumptons. Lneart of the objectve functon and constrants are the two mportant ones. The other s proportonalt assumpton, whch means that the contrbuton to the objectve of an decson varable s proportonal to the value of the decson varable. Smlarl, the contrbuton of each varable to the left-hand sde of each constrant s proportonal to the value of the varable. Moreover, the addtvt assumpton asserts that the contrbuton of a varable to the objectve and constrants s ndependent of the values of the other varables. The other assumpton s the dvsblt assumpton, meanng that takng an fracton of an varable s permtted. The fnal assumpton s the certant assumpton. In LO, no uncertant s permtted on the nput parameters. It s obvous that for a cash-flow problem, satsfng all these assumpton ma not happen smultaneousl. Losng the addtvt or proportonalt assumptons leads to a nonlnear programmng model. If the dvsblt assumpton does not hold, then a technque called nteger programmng rather than LO s requred. Ths technque takes orders of magntude more tme to fnd solutons but ma be necessar to create realstc solutons. Problems wth uncertan parameters can be addressed usng stochastc programmng or robust optmzaton approaches. Though these weak ponts tempts one to avod usng the LO model, however the rch theor, estence of man effcent solvers and over all these, possblt of senstvt analss ma persuade us to use t effcentl at least for a short term cash-flow problem. In ths paper, we consder an eample devsed n [1], to descrbe dfferent ponts of vew to senstvt analss on ths problem. LO and Dfferent Aspects of Senstvt Analss Let us consder the LO problem n standard form as mn { c T A b, 0} =, where A s a matr of dmenson m n and c,, b are of approprate dmensons. Its dual can be defned as ma { T T b A s c, s 0} optmal solutons s guaranteed. A prmal-dual soluton s denoted b (,, ) + =. Havng feasble soluton of prmal and dual problems, estence of s that satsfes the

3 complementart propert ( ) T { 1,,..., n} s parttoned to two sets ( B, N) s = 0. When the gven optmal soluton s basc, the nde set π = where B and N are correspondng to the nde set B of the gven basc and non-basc varables, respectvel. Ths partton s referred to as bass optmal partton. Observe that degenerac ests n almost ever LO problem and when the problem has multple (basc) optmal solutons, the basc optmal partton s not unque. For a vector 0 σ ( ) = j j > 0. If we are gven an, the support set of s denoted b { } arbtrar prmal optmal soluton, there s another partton dentfed. For a gven prmal optmal soluton, let a partton be defned and denoted b π = ( σ, ς ), where σ = σ ( ) and { n} σ ς = 1,,..., \ ( ) and be referred to as prmal support set partton. Analogous partton could be defned when a dual optmal soluton { n} ς s (, ) P s s gven and ς σ ( s ) { j s j 0} = = >. In ths case σ = 1,,..., \ ( ). Ths partton mght be dented b π = ( σ, ς ) and referred to as dual support set partton. Observe that these two recent parttons are dfferent wth each other when the prmal and dual optmal solutons are degenerate smultaneousl. In ths case, t s dfferent from the basc optmal soluton as well. Analogous to the basc optmal partton, these parttons are depends to the gven optmal solutons and, consequentl are not unque. On the other hand, the prmal-dual optmal soluton (,, s ) ma satsf the strctl complementar propert, when + s > 0. B the Goldman-Tucker Theorem [6], the estence of strctl complementar optmal solutons s guaranteed f the prmal and dual problems are feasble. Ths leads to a partton of the nde set { } = s > 0 for an arbtrar prmal-dual optmal soluton D 1,,...,n nto two sets B = { > } and { } (,, s ). Observe that the nde belongs to B, when the correspondng varable s postve n an optmal soluton. Analogousl, s ncluded n, when the correspondng dual slack varable s s postve n a dual optmal soluton. Identfng ths partton s possble f the problem s solved b an nteror pont method [1]. Ths partton s smpl referred to as optmal partton and denoted as π =(B, ). Unlke other parttons, ths partton s unque because of the convet of the optmal soluton sets. All the aforementoned parttons are dentcal wth the optmal partton when the gven prmal and dual optmal solutons are non-degenerate. There are dfferent ponts of vew towards senstvt analss and parametrc programmng dependng on the tpe of the n-hand optmal soluton. Let us consder the perturbed LO problem as mn ( c+ α c) T A= b+ β b, for arbtrar perturbaton vectors c and b of approprate { } dmenson and real parameters α, β. We are nterested to fnd the regon for these parameters where specal characterstcs holds for a current optmal soluton. There are dfferent ntervals correspondng and dependng the gven optmal soluton. Here we revew some of them. We ma have a (degenerate or non-degenerate) prmal basc optmal soluton. Fndng the regon for parameters where the assocate bass optmal partton remans optmal. As mentoned above, havng dfferent basc optmal solutons leads to dfferent confusng ntervals (e.g. [1]). A good reference to fnd detals n ths pont of vew for a un-parametrc case s [11]. A strctl complementar optmal soluton s n hand that leads to dentf the optmal partton π. Fndng the regon where ths partton remans nvarant s the am of ths case. Recall that ths nterval s ndependent of the tpe of prmal and dual optmal solutons. Fndng ths nterval onl needs to solve the followng two problems: =mn : =, B =0, =ma : =, B =0,

4 where =(B, ) s the known optmal partton. A good reference for a un-parametrc case could be [1]. The goal s to dentf the regon for parameters, where onl postve varables of the gven optmal soluton remans postve after an change on the parameters n ths regon. When we are nterested to the prmal optmal soluton, dentfng the nvaranc nterval of the prmal support set partton π P s ntended [] and the correspondng nterval (, ) can be dentfed b the followng two allar problems: =mn : =, 0 =ma : =, 0 where P= σ ( ) and s the gven optmal soluton. When the dual optmal soluton s of the nterest, fndng the nvaranc nterval of the support set partton π D s the am. For more detals n ths case, we refer the nterested reader to []. We restate that these ntervals are dentcal when the problem has unque nondegenerate prmal-dual optmal soluton. However, the are dfferent when the soluton s not a basc one, or when t s a degenerate basc optmal soluton. Ignorng ths fact ma lead to confusng result (e.g. [7]). In ths stud, we hghlght ths msunderstandng n cash-flow problem, and to keep t eas to track, we onl consder the un-parametrc case, when ether α or β s nonzero. Lnear Model for Cash-Flow Problem To llustrate the man dea of the paper we consder an eample from [1]. Eample: Consder a compan has the followng short-term fnancng problem: Month Jan Feb Mar Apr Ma Jun Net cash-flow Net cash-flow requrements are gven n thousands of dollars. The compan has the followng sources of funds: a lne of credt of up to $100k at an nterest rate of 1% per month; n an one of the frst three months, t can ssue 90-da commercal paper bearng a total nterest of % for the three-month perod; ecess funds can be nvested at an nterest rate of 0.% per month. Followng [1], we use the followng decson varables: the amount drawn from the lne of credt n month, the amount of commercal paper ssued n month, the ecess funds z n month and the companes wealth v n June. Here we have three tpes of constrants: () cash nflow = cash outflow for each month, () upper bounds on, and () nonnegatvt of the decson varables, and z. We remnd that s the balance on the credt lne n month, not the ncremental borrowng n month. Smlarl, z represents the overall ecess funds n month. For the detal of formulaton we refer to [1]. The LO model of ths problem n standard form s as follows; ma v + z1 = z1 z = 100

5 + z z 1.01 z z 1.0 z z 1.0 z v = = = = w1 = w = w = w = w = 100,, z, w 0. Replacng 1,, wth 6, 7, 8, respectvel; z1,..., z wth 9,..., 1 ; v wth 1 ; and w1,..., w wth 1 19,...,, we onl deal wth the varable vector 19 R. Moreover, wthout loss of generalt we can add nonnegatvt of v to the problem. To have the standard form, we replace the objectve functon wthe mnmzaton of the negatve of v= 1. In ths wa the matr A s of dmenson and the nde set s {1,,19}. Solvng ths problem wth the EXCEL solver leads to the followng basc soluton; 1 = = = = = 0.98, 1= 1 = 9. = 0. (1) z = z = z = z = z = 1.9, v= w = w = w = w = w = It should be restated that there s another basc optmal soluton as 1 = = = = =, 1= 1 = 10 = 11.9 () z = z = z = z = z = 1.9, v= w = w = w = w = w = 8, 1 10 and consequentl strctl optmal soluton ests, sa 1 = = = = 18.07, =.7 1= 1 = 81.9, = () z = z = z = z = z = 1.9, v= w1 = w = w = 10 w = 81.9, w = 66.. Havng multple optmal solutons means that the problem s dual degenerate and a strctl optmal soluton reveals the optmal partton =(B, ), where B = {,, 6, 7, 8, 11, 1, 1, 16, 17, 18, 19} and = {1,,, 9, 1 1, 1}. The nterpretaton of the soluton s eas. In all optmal solutons, the companes wealth v n June wll be $9,00. To acheve ths goal, n the basc optmal soluton (1) for eample, the compan wll

6 ssue $1000 n commercal paper n Januar, $9,00 n Februar and $0 n March. In addton, t wll draw $980 from ts lne of credt n Februar. Ecess cash of $1,90 n March wll be nvested for just one month. Analogous nterpretaton can be consdered for other optmal solutons. If we denote the dual varable b t j, j = 1,,11, the unque dual optmal soluton s, t 1= t = t = t = t = t 6 = 1 t7 =... = t11 = 0 These values are known as shadow prces. The nonzero dual slack varables are s 1= 0.001, s = , s = 0.001, s 9 = , s 10 = , s 1 = and s 1 = 0.007, whch are n complementart wth (all) prmal optmal solutons. These values are known as reduced costs. Shadow prces and reduced costs pla mportant rules n senstvt analss and the nterpretatons. Most of solvers have tremendous nformaton on senstvt analss and the have useful eplanaton on the problem n queston such as allowable ncrease and decrease for the Rght-Hand- Sde (RHS) of each ndvdual constrant and objectve functon coeffcent of ndvdual decson varable (e.g. [1]). However, there are a lttle publshed eplanatons when the varaton of nput data, sa the RHS of the constrants occurs smultaneousl. Ths can be categorzed as parametrc programmng. In the net secton we consder such cases and menton some fnancal descrpton of the parametrc programmng. Descrpton of Parametrc Programmng n LO Model To realze a concrete case, consder there s an opton offered to the compan as follows: Opton 1. Reduce the net cash-flows n months Januar, Februar and Aprl b the rate of, 1 and respectvel, and pa back them equvalentl n months March, Ma and June wth the fed nterest rates % 1.7 percent n these months. In ths wa the perturbaton vector of the RHS of constrants could be b= (,1, 1.017,, 1.017, 1.017,0) T, and we are nterested to dentf the range for the parameter value β, n the followng cases. (1) The basc optmal soluton (1) s gven and fndng the bass nvaranc nterval for ths soluton s amed. Smple calculaton reveals that ths range s [-9.1,.996] and the range of varaton for objectve value s from 19.1 to 0. It means that less negatve cash-flow n months Januar, Februar and Aprl s allowed,.e., acceptng ths opton ncreases the fnal compans wealth $ 19.1 at the end of ths 6-month perod. (1) When () s the known basc optmal soluton, the bass nvaranc nterval s [-0.7,.996]. It has analogous nterpretaton as the other basc optmal soluton. Observe that ths nterval s dfferent from the obtaned nterval for the basc optmal soluton (1). In ths case, the objectve functon value starts from $ 196,1 and decreases to $ 0 when the parameter goes to the rght end of the nterval. () When a strctl complementar prmal optmal soluton sa () s gven (and consequentl the optmal partton π s known), and we are nterested n fndng the optmal partton nvaranc nterval. In ths case, the nterval s (-6.91,.996) and the objectve functon value decreases from 1,88 to 0 when the parameter value ncreases. The mportant ssue here s that, the nvaranc nterval of the optmal partton s greater than of the bass 6

7 nvaranc nterval n both basc optmal solutons (1) and (). Moreover, ths nterval s an open one [1], whle the two others are closed. Remark: For all parameter values n these nterval specall the largest one, the dual optmal soluton (.e., the shadow prces) s vald. There are useful nterpretatons are mentoned n [1]. One of the nterpretaton of the results of senstvt analss s the nformaton arses from shadow prces. We menton one of ths nformaton from [1] and make the analss deeper. Opton. Assume that the negatve net cash-flow n Januar s due to the purchase of a machne worth $1000. The vendor allows the pament to be made n June at an nterest rate of % for the fve-month perod. Would the compans wealth ncrease or decrease b usng ths opton? What f the nterest rate for the -month perod was %? Snce the shadow prce of the Januar constrant s 1.07, reducng cash requrements n Januar b $1 ncreases the wealth n June b $1.07. In other words, the break-even nterest rate for the fve-month perod s.7%. So, f the vendor charges less than ths amount, we should accept, but f he/she charges more, we should not. For the eact nterest value.7%, acceptance or rejecton of the proposal has dentcal result. We restate that ths analss s vald when the amount of change n the RHS of the correspondng constrant s wthn the allowable decrease. Observe that for ths analss the perturbed vector s b= (1,0) T. For ths perturbaton vector, f the gven optmal soluton s (1) the bass nvaranc nterval s [-89.17,10]. However, f the n hand optmal soluton s (), ths nterval reduces to [-89.17,19.11]. It seems that the aforementoned analss s not vald n the later case. However, the optmal partton nvaranc nterval for ths perturbaton s dentcal wth nteror of the bass partton nvaranc nterval (-89.17,10). Thus we can ensure the management on hs decson. Eample s revsted: We saw that n ths eample the prmal problem has multple optmal solutons those are not degenerate. Let us change the model modestl as follows. The credt lne up to $0k at an nterest rate of 1% per month s avalable onl at Februar. Wth ths opton, onl the constrants of upper bound for varables changes accordngl. In ths wa the problem has degenerate optmal soluton: = = = = = 0 1 1= 1 = 0 = 0.8 () z = z = z = z =.98, z = w = w = w = w = w = v= Observe that n ths optmal soluton, we onl have 7 postve varables, whle a basc soluton needs to have 11 n ths eample. Therefore, other varables among zero ones must be chosen to etend these postve varables to a basc one. Ths means that there are man fntel correspondng bass'. Let us agan consder the Opton 1. In ths case, there mght be dfferent bass optmal partton nvaranc ntervals. For eample, when one let correspondng bass' to basc optmal solutons (1) and (), strangel the optmal bass nvaranc for these two ones s (-, ). However, the prmal support set nvaranc nterval of ths soluton s (-9.99), that s clearl an open nterval []. Moreover, there s a prmal strctl complementar optmal soluton, sa, 1 = = = = = 0 1= 10.6, = 9.6, = 0.8, () z = z = z 1= 0.6 z =.98, z = 0.6, w = w = w = w = w =, v= It s clear the the optmal partton for ths problem s: =(B, ), where 7

8 B = {, 6, 7, 8, 9, 11, 1, 1} and = {1,,,, 1 1, 1, 16, 17, 18, 19}. Observe that ths optmal soluton s not basc agan and not surprsngl, the optmal partton nvaranc nterval s eactl dentcal wth the prmal support set nvaranc nterval. There s an nterestng nterpretaton for the support set nvaranc nterval. Recall that n prmal support set nvaranc, we want to keep postvt of postve varables n the gven optmal soluton, whle the parameter vares n the nterval. In the optmal soluton (), the compan wll ssue $1000 n commercal paper n Januar, $000 n Februar and $8 n March. In addton, t wll draw $000 (the mamum possble) from ts lne of credt n Februar. Ecess cash of $,98 and $997 n March and Aprl wll be nvested, respectvel. If the compan wants to keep ths polc along wth Opton 1,.e., nvestng n two months March and Aprl whatever possble, ssung commercal paper n Januar, Februar and March n addton to drawng from the credt lne n Februar, allowable change of the parameter s onl the nterval (- 9.99) but not the whole real lne as obtaned for the two correspondng basc optmal solutons. There s another fndng. For the varaton of the parameter n ths nterval, there s no change n reduce costs and shadow prces. Because the dual optmal soluton s nvarant n the optmal partton nvaranc nterval [1]. Thus smlar analss to the Opton can be carred out agan. 6 Concluson It s obvous that the nherent uncertant s not the thng that can be answered onl b the LO. However, ts rch theor n senstvt analss and parametrc programmng have man thngs to sa to economsts. In ths stud, we onl nterpreted the parametrc programmng results when the RHS s perturbed. Analogous analss can be carred out for the case when the objectve functon s perturbed [1]. Un-parametrc case can be consdered when the parameter presents n both the objectve functon and the RHS of constrants []. Moreover Mult-parametrc programmng n LO has been studed b man authors (e.g., [, 8, 9]) wth dfferent ponts of vews. Developng of multparametrc analss of LO could be a tool to tackle some of the dffcultes n man fnancal problems whose can be modelled as an LO problem. Interpretaton of the result ma clear some facts n cashflow problem as well as other fnancal problems whch can be formulate as an LO. References [1] Cornuéjols, G. and Tütüncü, R. Optmzaton methods n fnance, Mathematcs, fnance, and rsk, no. v. 1. Cambrdge Unverst Press [] Ghaffar-Hadgheh, A., Romanko, O. and Terlak, T. Senstvt analss n conve quadratc optmzaton: Smultaneous perturbaton of the objectve and rght-hand-sde vectors. Algorthmc Operatons Research (): [] Ghaffar-Hadgheh, A., Romanko, O. and Terlak, T. B-parametrc conve quadratc optmzaton. Optmzaton Methods and Software : 9-. [] Ghaffar-Hadgheh, A., Mrna, K. and Terlak, T. Senstvt analss n lnear and conve Quadratc optmzaton: Invarant actve constrant set and nvarant set ntervals. INFOR: Informaton Sstems and Operatonal Research (): [] Ghaffar-Hadgheh, A. and Terlak, T. Senstvt analss n lnear optmzaton: Invarant support set Intervals. European Journal of Operatonal Research (): [6] Goldman, A.J. and Tucker, A.W. Theor of lnear programmng, n: H.W. Kuhn and A.W. Tucker (Eds.), Lnear Inequaltes and Related Sstems. Annals of Mathematcal Studes : [7] Jansen, B., de Jong, J.J., Roos, C. and Terlak, T. Senstvt analss n lnear programmng: just be careful!, European Journal of Operatonal Research : 1-8. [8] Kherfam, B. Mult-parametrc lnear optmzaton: nvarant actve constrants set. Pacfc Journal of Optmzaton (1):

9 [9] Kherfam, B. and Mrna, K. Quaternon parametrc optmal partton nvaranc senstvt analss n lnear optmzaton. Advanced Modelng and Optmzaton (1): 9-0. [10] Kolta, T., Terlak, T. The dfference between manageral and mathematcal nterpretaton of senstvt analss results n lnear programmng. Internatonal Journal of Producton Economcs : 7-7. [11] Murt, K.G. Lnear Programmng. New York, USA: John Wle & Sons [1] Roos, C., Terlak, T., Val, J.-Ph., Roos, C., Terlak, T. and Val, J. Interor Pont Methods for Lnear Optmzaton. Hedelberg/Boston: Sprnger Scence