APPLICATION OF GOAL PROGRAMMING METHODS IN MANAGING THERMAL ENERGY PROJECTS - CASE STUDY CHANGING RADIATORS IN DISTRICT HEATING BUILDINGS

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1 Internatonal Journal of Mechancal Engneerng and Technology (IJMET) Volume 9, Issue 6, June 2018, pp , Artcle ID: IJMET_09_06_010 Avalable onlne at ISSN Prnt: and ISSN Onlne: IAEME Publcaton Scopus Indexed APPLICATION OF GOAL PROGRAMMING METHODS IN MANAGING THERMAL ENERGY PROJECTS - CASE STUDY CHANGING RADIATORS IN DISTRICT HEATING BUILDINGS Fsnk Osman, Ftm Zeqr* Faculty of Mechancal Engneerng and Computer, Unversty of Mtrovca, Mtrovca, Kosova. *Correspondng author: ftm.zeqr@umb.net. ABSTRACT Contemporary economc and engneerng studes tend to focus on applyng multcrtera (mult-crtera) decson makers as polcy makers contnue to face a range of ncreasngly complex management polcy ssues. One of the analytcal tools that can provde further knowledge, for mult-purpose problems, s known as the "goal program" (GP). Goal Programmng (GP) s an mportant class of wdely-used decson-makng models to analyze and solve appled problems nvolvng conflctng goals. Multple, contradctory and qualtatve goals of manageral and engneerng performance requre the use of mult-crtera decson-makng. The paper dscusses a sustanable example for use n the feld of management scences. Specfcally, we dscuss ths example to make decsons to partcpate n a "plot project" for the replacement of radators from the two basc mantenance unts of the "Mountng" Unt and the Mantenance Unt that can temporarly transfer staff to help wthn Dstrct Heatng "TERMOKOS" whch n dsposal they have a lmted number of workers. Termokos can hre temporary staff from local Aterm, for temporary assstance, whch s specalzed n techncal support on radator replacement. Therefore, fndng a compromse soluton n the engagement of some staff for help from the Mantenance and Alterm unt s requred to gan an effectve goal-settng soluton. The analyss wll be done usng the programmng of goals wth the help of PHPSmplex and Excel. Key words: Goal Programng, Multrctera Decson Makng, Management, Analyss, Dstrct Heatng Cte ths Artcle: Fsnk Osman and Ftm Zeqr, Applcaton of Goal Programmng Methods n Managng Thermal Energy Projects - Case Study Changng Radators n Dstrct Heatng Buldngs, Internatonal Journal of Mechancal Engneerng and Technology 9(6), 2018, pp edtor@aeme.com

2 Applcaton of Goal Programmng Methods n Managng Thermal Energy Projects - Case Study Changng Radators n Dstrct Heatng Buldngs 1. INTRODUCTION Cty heaters are specalzed companes for the supply of thermal energy - manly extend to the densest parts of ctes and publc buldngs, such as hosptals, schools and admnstratve buldngs. The central heatng company "Termokos sh.a" supples about 10,000 customers n the servce and economy sector n Prstna. Heatng servces are provded only durng the heatng season. Dstrbuton networks (prmary network) supply wth heat from the heat plants, heat substatons, from where heat s dsspated through nternal nstallatons of buldngs (secondary network). The sector has been challenged by the old technology and a low level of bllng and collecton of generated energy payments. Another challenge s the poor management of secondary network regulaton and the ntroducton of consumer electrcty meters nto ndvdual households n apartment buldngs. Now there are varous projects n the phase of "plot projects" from varous donors that should be mplemented, whch serve as a model for broader, smlar nterventons. Several prorty nterventons have been dentfed and plot projects have been mplemented or are about to be mplemented n the near future. One of them s the connecton of several buldngs n parts of the cty, e old buldngs that have not been locked - so the equpment nstalled many years ago need to be changed, among other thngs, the heatng bodes. It s about a radator older than forty years that has passed the deprecaton perod. The project wll be mplemented by DH "Termokos" respectvely the "Mantenance" unt and the "Producton" Unt operatng wthn ths company, whle avalable are a lmted number of employees and are a conflct over partcpaton n ths project. Therefore, n ths case, the Multcrtera Decson Makng Respectvely Goal Programng (GP) may be used [1]. GP seems to be an approprate, robust and flexble technque for analyzng the decsonmakng of the troubled modern decson-maker, whch s n charge of achevng numerous contradctory objectves under complex constrants. Decson-makng models and multcrtera goals (GP) modelng are mportant tools for scentfc research management wth extensve applcatons n engneerng and socal scence. Complexty n most of the real-world problems s due to dffcultes n modelng and solvng a sngle purpose. GP models are a remote-based approach that optmzes multple goals by mnmzng target devatons from aspratonal levels or decson maker (DM) goals. When devatons are shfted to zero, the set goals of the model can be acheved, n addton, devatons can be postve and negatve, mplyng overestmaton or underestmaton of goals that are subject to some lmtatons. As noted by Romero (1991), GP s the most wdely used way of decson-makng wth many crtera. The GP modelng structure s easy to understand and apply and can be solved usng commercal software for mathematcal programmng. For detaled mathematcal treatment and soluton we refer to readers for some nterestng works on GP models [2] from Tamz, Jones and Romero (1997) then Jones and Tamz, (2010). 2. GOAL PROGRAMMING Goal Programmng (GP) s now an mportant feld of optmzaton of multple crtera [3]. The dea of goal programmng s to create a level of achevement goal for each crteron. The goal programmng method requres the decson-maker to set goals for each objectve that he / she wants to take. A preferred soluton s then defned as mnmzng devatons from defned goals. The GP can then be formulated as the next achevement functon [4] edtor@aeme.com

3 Fsnk Osman and Ftm Zeqr k Mn ( d d ) s.t. f ( x) d d b, 1,..., k, 1 x X d d 0, 1,..., k, d, d 0, 1,..., k, DMs for ther purposes set some acceptable aspraton levels, b ( = 1,..., k) for these purposes and try to reach a range of goals as near as possble. The goal of the GP s to mnmze devatons between achevng goals, f (x) and these acceptable aspraton levels, b ( = 1,..., k). Also, know d + are respectvely sub - and over - reachng the goal. As the global crteron and the mnmum devaton crteron represent strateges that seek to establsh a soluton as close as possble to a targeted soluton n whch target values of all goals are ther optmal values. Ths makes such an deal soluton, but mpossble. In many decson-makng stuatons, management s able to create more practcal levels of achevement (goals). Optmum solutons are requred where all goals wth total mnmum devaton are acheved [5]. What s common for these purposes s that they are gudelnes, not sold restrants. In ths sense, settng the goal offers a soluton of the knd of compromse, because t allows devatons from the full achevement of dfferent goals [6] 3. FORMULATION OF THE MATHEMATICAL PROBLEM Termokos [7] has asked to start her job to be the man contractor to carry out the "Plot Project" for replacng old radators n a buldng complex. As a result, the project s expected to need addtonal temporary support from Termokos Substaton Mantenance Unt (Dvson) to handle a sgnfcant ncrease n radator assembly to be completed untl the project s completed. Termokos can hre temporary staff from Aterm, a local temporary ad company that specalzes n techncal support n changng radators, or can temporarly transfer staff from other dvsons wthn the company. Gven the qualfcatons and results n performng radator swtchng, t s known that a Termokos employee who moved could have changed around the NRT radators per workng day. Such a job wll requre around two hours per day of techncal assstance from supervsory staff. Because Aterm's workshop s less famlar wth the envronment than Termokos employees, t s estmated that the Atermt employee could only change NRA radators per workng day, meanng NRT> NRA. However, due to the hgher level of Aterm's techncal tranng, each employee only seeks around NOA (one) hour of techncal assstance from supervsory staff, e. NOT> NOA. Termokos has allocated up to NS work statons to swtch radators for addtonal support so that the dvson can be used by most NP employees. He has further allocated full-tme staff for techncal assstance, provdng NOAT hours of techncal support tme avalable each day. After a ferce negotaton between the Unt Manager Unt Manager and the Mantenance Manager of the Central Heatng Substatons Unt n Termokos, t was agreed that two goals should be used n determnng the employment plan: a) Number of radators mounted per day (a target called "RAD"). b) The dfference between the number of workers employed and the transferred workers (a purpose called "DIFF") edtor@aeme.com

4 Applcaton of Goal Programmng Methods n Managng Thermal Energy Projects - Case Study Changng Radators n Dstrct Heatng Buldngs To solve ths problem we should start from the assumptons that: 1) We have an analyst who works for management n the Assembly Unt. For ths case a multfaceted model s desgned wth the above goals. Workng for the Fttng Unt, t should be stpulated that for each employee employed by Aterm, at least two employees wll be transferred from the Mantenance Unt wthn Termokos, X2 = 2 X1. Then buld an effcent bound for ths model. Then should the answer be gven f the Unt Dvson Mantenance Manager s ready to agree to P1 (25%) less than the best possble DIFF value, whch s the optmal employment soluton t wll suggest? 2) We have an analyst who works for management n the Central Heatng Substaton Mantenance Unt (dvson) wthn Termokos. He set a goal-modelng model for ths problem. The purpose of "RAD" s the number of radators mounted per day NRAD and the rato of employment s (X1 / X2) = 1/2 (1: 2) two workers are transferred from the Mantenance Unt wthn Termokos to each employee employed by Aterm (ths s just one purpose = gude). We need to approach ths goal as near as possble, the relevant model should be resolved. 3) We need to make sure that goal resoluton s effcent. If there s no management ad, need to fnd an optmal optmal soluton of goal programmng. 4) Challenge: Repeat part (1), but ths tme Unt Manager (Dvson) Mountng s ready to fx the problem for P2 (28.6%) less than the best possble value of X1 / X2 ( Unt Unts Mountng s nterested n the smallest possble report). What s the optmal employment soluton that wll be suggested (proposed) for ths problem? The table 1 below shows the data for problem solvng. Table 1 Data for the purpose programng problem Input data for the purpose programmng problem Companes TERMOKOS ATERM Number of radators mounted per day when employees are transferred from N RT 3.5 N RA 3 Number of hours for techncal assstance by supervsory staff N OT 2 N OA 1 Number of work statons n mountng radators for addtonal support N S 12 Maxmum number of employees N P 24 The number of hours of techncal support tme avalable each day N OAT 36 Number of Employees Employed by X2 X1 Number of radators mounted per day (target called "RAD") RAD The dfference between employee employees and employees transferred (target called "DIFF") DIFF For every employee employed by ATERM, at least two employees N wll be transferred from the Termokos Mantenance Unt AT 2 N PA 1 1 The Mantenance Manager at Termokos s ready to agree for 30% P less than the best possble value of DIFF 2 The purpose of "RAD" s N RAD 74 Postve very small constants (d) for ganng an effcent 3 programmng of the goal programmng The Mantenance Manager at Termokos s ready to settle for 28.6% 4 less than the best possble value of the X1 / X2 P edtor@aeme.com

5 Fsnk Osman and Ftm Zeqr 4. PROBLEM-SOLVING APPROACH 4.1. Desgn a Mult-Purpose Model Intally, we determne the number of employees of ATERM (X1) and the number of employees from TERMOKOS Mantenance Unts (X2). The proposed approach can be formulated by a mathematcal model of lnear programmng wth the followng two functons of purpose: [8] Max DIFF Max RAD X 2 X 1 (The Mantenance Dvson wants to be maxmzed DIFF!) 3.5X 3 S.T. X 24, 1 X 2 X1 2 2X1 X2 36, 2X1 X 2 0. In optmzng for each purpose wth the help of the PHP Smplex [9] program (applyng two tabular and graphcal methods) Maxmze "RAD ": RAD = 76, [X 1 = 8, X 2 = 16], DIFF = 8. Maxmze "DIFF": DIFF = 24, RAD = 72, [X 1 = 0, X 2 = 24]. The respectve optmal values are gven n Table 2 and Table 3 and graphcally n Fgure 1 and Fgure 2. Fgure 1. Optmal Values by graphcal method Table 2. Value of the objectve functon Fgure 2 Optmal Values by graphcal method Table 3 Value of the objectve functon 78 edtor@aeme.com

6 DIFF DIFF Applcaton of Goal Programmng Methods n Managng Thermal Energy Projects - Case Study Changng Radators n Dstrct Heatng Buldngs Applyng the above-dscussed procedure to dentfy the effcency lmt, we draw a straght lne lnkng the two ponts n the RAD - DIFF plan as seen n the graph shown n fg RAD Fgure 3 RAD DIFF dagram 4.2. Fndng a Compromse Soluton The management unt management s ready to gve up 30% of the best DIFF level. The best value s DIFF = 24 and the worst s DIFF = 8 ganed when the "RAD" s maxmzed. Thus, n the nterval between 8 and 24, the Mantenance Unt Management s wllng to agree to DIFF = 20 (25% below 24 at ntervals between 8 and 24). From the effcent boundary chart we can get an approxmate RAD value when DIFF = 20. But an accurate result can be obtaned algebracally. There s a lne that s the effectve lmt. Therefore, the followng equaton should be true (see also graphc demonstraton) 24 8 DIFF 24 (RAD 72) 4RAD y = 4x R² = RAD Fgure 4 RAD DIFF dagram By solvng ths equaton we have RAD = 73. Now we can solve for X1 and X2 usng the defntons of DIFF and RAD: Usng DIFF: 20 X 2 X1 Usng RAD: X1 3X 2. When solvng these two two-varable equatons, we have: X1 = 2; X2 = 22. A short settlement procedure: Use the lnear programmng model where the objectve s to maxmze the RAD, but set a lmt on the DIFF value to be X2 - X1 = 20. You must get a soluton at the effectve boundary and snce DIFF = 20 we wll gan the soluton we have obtaned wth the algebrac method above. Max RAD 3.5X 3 S.T. X 24, 1 X 2 X edtor@aeme.com

7 Fsnk Osman and Ftm Zeqr 4.3. Model of Goal Programmng Mn U 1 E1 U 2 X X X1 X2 36, 2X1 X 2 0. S.T. X 24, X1 2 2X1 X2 36, 2X1 X2 U1 E X 3X 2 U 2 E 2 1 The optmal soluton of the target programmng model s: X1 = , X2 = Whle U1 = E1 = U2 = E2 = 0. Ths means that all goals are fully acheved! Snce the nstructon s that 2X1 = X2, or 2X1 - X2 = 0, the scope lmt reflects the devaton from the value '0'. 74 Fgure. 5 Optmal Values by graphcal method Table 4. Value of the objectve functon For the purpose of the employment relatonshp, the Management Unt's management s nterested n mnmzng any devaton from the '0' goal, whch explans why both U1 and E1 are ncluded n the goal functon. In ths case the queston arses as to whether the Mantenance Management unt should be very pleased, or not? No. Ths soluton s neffectve because all strong constrants are welcomed as nequaltes (both strong lmtatons have a crawl n the optmal soluton). Specfcally, t results: X 2 1 X and 2X1 X2 2( ) So ths "optmal" soluton s reached at an nner pont of the potental regon! Such a soluton can always be mproved wth respect to both objectves, and ths makes t neffectve. A goal programmng model has a goal functon that expresses the amount of devatons from goals (whch should be mnmzed). Regardng the lmtatons, two types of restrctons can be dentfed. 1) Sold (strong) constrants, the same as n the regular lnear programmng model. These restrctons can not be volated edtor@aeme.com

8 Applcaton of Goal Programmng Methods n Managng Thermal Energy Projects - Case Study Changng Radators n Dstrct Heatng Buldngs 2) Soft (goal lmtatons, representng the desred level of achevement for each purpose). These lmtatons of purpose allow for underestmaton and overcomng goals Acqurng an Effcent Programmng Goal Soluton Let's change the current functon of the goal to provde an effcent soluton that does not change the overall level of achevement of the goal. The modfed goal s bult where the orgnal devatons are ncluded as before then the maxmzed goal s added as RATIO, whle the mnmzed goal s added as RAD. Mn U1 + E1 + U RATIO RAD That s, two new varables that get the values RATIO 2X1 X 2 of goal functons are defned (Rato and Radators). (Goal values). RAD 3.5X 3X X 1 0; X 2 0; U 1 0; E 1 0; U 2 0; E (Non - negatve) Non-negatvty constrants are needed because the RATIO varable s allowed to receve negatve values. (In the Soluton, under "Optons", we should leave "suppose all varables are non-negatve", but then we need to specfy whch varables are non-negatve). You do not - negatve. The optmum soluton modfed under PHP Smplex s: The strong restrcton on the total number of employees s satsfed as an equalty (the total number of employees s 24). Consequently, the current optmal soluton les n the boundary of the potental regon, where the soluton must be effcent, because both goals can not be mproved smultaneously. The prce of ths compromse s that E2 = 2 and not '0' Harmonzng contradctory goals Unt Management (Dvson) Mantenance s ready to gve up 30% of the best rato rato. The best rato for the unt (dvson) Mantenance s 'zero' (X1 / X2 = 0/24) taken when maxmzng "DIFF". The worst rato for unt (dvson) Mantenance s '0.5' that s obtaned when maxmzng "RAD" (X1 / X2 = 8/16 = 0.5). Thus, n the nterval between 0 and 0.5 the syndcaton s wllng to agree on a rato of (28.6% on raton) of 0 n the nterval between 0 and 0.5.Ths s X1 / X2 = Ths can be wrtten as X1 = 0.143X2. When replacng 0.143X2 for X1 n the mult-objectve model stated above both goals are actually as follows: () X 2 X1 X X X 2 and () 3.5X 1 3X 2 3.5(0.143X 2 ) 3X X 2. Snce both objectve values should be maxmzed, X2 should be maxmzed, subject to the followng lmtatons expressed only n terms X2: X 1 + X 2 24 s 0.143X 2 + X 2 24 ose 1.143X 2 24 whch s done X 2 (24 / 1.143) = X 1 + X 2 36 s 0.286X 2 + X 2 36 ose 1.286X 2 36 whch s done X 2 36 / = X 1 X 2 0 and 0.286X 2 X 2 0 whch s true of any non-negatve value of X 2. So, hghest values of X 2 completes the constrants s X 2 = So, X 1 = ( ) = edtor@aeme.com

9 Fsnk Osman and Ftm Zeqr 5. CONCLUSIONS Ths procedure dealt wth n ths paper guarantees an effcent soluton and works only when there are multple optmal solutons. If not all functons have multple optmum solutons, the process wll stop the frst tme n an optmzed purpose that has a sngle optmal soluton. Settng goals approprately s an art of ts knd. The set goals that are too hgh result n dsappontment for not beng achevable. Goal settng approach for mult-purpose decson-makng can end wth neffcent solutons. Such solutons fully meet all goals but can not be perceved as optmal. Ths s so because all goals can be mproved smultaneously from the optmal soluton of goal programmng to another possble soluton. Snce an neffcent soluton occurs at an nternal pont n the potental regon, when all restrctons are satsfed as nequaltes (all unstable and / or excessve varables are postve), the soluton s neffcent. It may be optmal on the bass of goal satsfacton, but not optmal on the ground very objectve due to neffcency (the problem s not wth the settlement procedure but wth the levels of ntent). To fx the problem of neffcent solutons, we change a bt of goal functon. Reachng a new goal wll not be affected much, but the soluton wll be effcent. REFERENCES [1] G. O. Odu, and O. E. Charles-Owaba. Revew of Mult-crtera Optmzaton Methods Theory and Applcatons, IOSR Journal of Engneerng (IOSRJEN, Vol. 3, Issue 10 (October. 2013). [2] Carlos Romero, Dylan Jones, Mehrdad Tamz. Goal programmng for decson makng: An overvew of the current state-of-the-art, European Journal of Operatonal Research 111 (1998) 569±581. [3] G. O. Odu, and O. E. Charles- Owaba. Revew of Mult-crtera Optmzaton Methods Theory and Applcatons, IOSR Journal of Engneerng (IOSRJEN) e-issn: , p-issn: Vol. 3, Issue 10 (October. 2013), V2 PP [4] Pradeep Tomar, Jagdeep Kaur. Mult Objectve Optmzaton Model usng Preemptve Goal Programmng for Software Component Selecton; I.J. Informaton Technology and Computer Scence, 2015, 09, Publshed Onlne August 2015 n MECS ( DOI: /jtcs [5] Konstanπnos Ρ. Soutsas. Goal programmng - nput-output analyss ινforest management; ΕΠΙθ Ε Ω ΡΗ Η ΟΙΚ ΟΝΟΜΙΚΩΝ Ε ΠΙΣΗΜΩΝ Σεύχ ο ς 8 (2005) [6] Andrew Mankas, Mchael Godfrey. Integratng sustanablty nto a goal programmng exercse; busness educaton & accredtaton, Volume 6, Number 1, [7] Dstrct heatng Termokos [8] Davd G. Luenberger, Ynyu Ye. Lnear and Nonlnear Programmng, Fourth Edton, Stanford Unversty Page 72. [9] Smplex method software edtor@aeme.com