A special case of the generalized pooling problem arising in the mining industry

Transcription

1 A pecal cae of the generalzed poolng problem arng n the mnng ndutry Nataha Boland Thoma Kalnowk Faban Rgternk Martn Savelbergh Aprl 26, 216 Abtract Iron ore and coal are ubtantal contrbutor to Autrala export economy. Both are blended product that are made-to-order accordng to cutomer dered product qualte. Mnng compane have a great nteret n meetng thee target qualte nce devaton generally reult n contractually agreed penalte. Th paper tude a varaton of the generalzed poolng problem (GPP) arng n th context. The GPP a mnmum cot network flow problem wth addtonal blnear contrant to capture the blendng of raw materal. In the varaton we tudy, cot are not aocated wth network flow but wth devaton from target qualte. We propoe a blnear program (BLP) that we olve locally ung nonlnear programmng olver to obtan upper bound. We lnearly relax the BLP ung McCormck relaxaton and olve the reultng lnear program (LP) to obtan lower bound. A computatonal tudy on 26 ntance, repreentng a real-lfe ndutry ettng and havng quarterly, half-yearly, annual and trannual plannng horzon, how that even for large-cale BLP, thee bound can be calculated effcently. Keyword Blendng, generalzed poolng problem, blnear programmng, nonlnear programmng 1 Introducton and problem decrpton Mne extract Stockple runoff mne tock Proce cruh & wah Stockple raw materal tock Tranport mne to port Stockple & aemble fnhed product tock Export load & hp Fgure 1: Schematc of a mnng upply chan The am of th paper to model the raw materal tock blendng and fnhed product tock blendng of a mnng upply chan, hown n Fgure 1. The notaton that we ue throughout th paper gven n Table 1. The key feature of the mnng upply chan blendng model are a follow. After the raw materal mned and proceed, t tored on tockple for ntermedate torage. We refer to the tockple a upply pont, or nventore, denoted by S := {1,..., S}. Important properte of the materal to be blended are repreented a a et of qualte Q := {1,..., Q}. In the cae of coal thee may be ah, moture, ulfur and volatle matter. We aume that for all qualte, cutomer prefer mall qualty value. For all S, upply come n at known pont n tme t T := {t 1,..., t T }, where t T >... > t 1 >. The quantty and qualty of ncomng upply, t and q kt for all k Q, repectvely, alo known a pror. When a raw materal tored on a tockple, the nventory quantty and qualty change. For Georga Inttute of Technology, Atlanta, GA, USA The Unverty of Newcatle, Callaghan, Autrala 1

2 all S and t T, let I t capture the quantty tored on nventory at tme t, and for all k Q, let x kt capture the value of qualty k of that nventory at that tme. We aume lnear blendng,.e., the nventory new qualty a weghted lnear combnaton of the nventory old qualty and the qualty of the ncomng upply. I t and x kt not only ncorporate the precedng ncomng upply quantte, t, and qualte, q kt, for all t T, t t, but alo the ucceedng outgong demand. It wll become apparent n Secton 3 why we chooe th notaton. Demand goe out at known pont n tme t T d := {t d 1,..., t d T }, where t d d T >... > t d d 1 >. We ue the raw materal tored on the tockple to meet the demand d t. Let y t be the flow from upply pont S to the demand pont at tme t T d. Note that S y t = d t mut hold. For all k Q, let x d kt be the qualty after blendng the y t flow,.e. the qualty of d t. Agan, we aume lnear blendng. If demand met wth product that doe not atfy precrbed qualty pecfcaton, a penalty ncurred. The penalty thu a functon of x d. The partcular penalty functon we ue n th paper decrbed n the next ecton. We aume that every raw materal flow (may t be a ncomng upply, outgong demand or a a flow between the upply pont and the demand pont) happen ntantaneouly. The quantte and qualte are alo updated ntantaneouly. We do not conder any tranfer tme or cot between the upply pont and the demand pont. We aume that, at any tme, the pat, aggregated, ncomng upply larger than or equal to the pat, aggregated outgong demand: demand can alway be met. The problem to decde how much from each upply hould be ued to meet the demand at each demand tme pont, o a to mnmze the total penalty. A we hall how n Secton 3, th problem can be vewed a a poolng problem; we call t the mnng poolng problem (MPP). The poolng problem wa frt propoed by Haverly [12] n Snce then, an extenve lterature ha been publhed. Revew on the poolng problem and t varaton are found n [4, 8, 16]. We alo refer the reader to two PhD thee on th ubject [1, 15]. There are a number of oluton technque to olve poolng problem uch a ucceve lnear programmng, the reformulaton-lnearzaton technque and lnear relaxaton. In th paper, we tudy the latter. Applcaton are found n chemcal engneerng, e.g. n petroleum refnng [16]. A decrpton of a computatonal tool that globally optmze poolng problem, APOGEE, ha been publhed recently [17]. The lterature on poolng problem arng n the mnng ndutry, however, pare [6, 9, 19]. 2 Blnear programmng model Recall that the objectve to mnmze the total penalty that ha to be pad by the mnng company a a conequence of meetng demand wth product havng qualte that do not atfy precrbed product pecfcaton. Here, we ue a penalty functon that cont of two lnear pece for each qualty k Q and for each demand tme pont t T d. Let u kt be the contractually agreed oft upper bound on qualty k for demand tme pont t. The left pece repreent qualty value x d kt < u kt for whch no penalty ha to be pad. The rght pece repreent qualty value x d kt u kt, for whch the mnng company ha to pay a nonnegatve per unt penalty p kt. We defne the exce qualty functon e kt (x d kt ) := max{, xd kt u kt} to be the devaton of x d kt from u kt f x d kt u kt, and otherwe. The penalty ncurred from qualty k n meetng demand d then gven by the per unt penalty p kt multpled by e kt (x d kt ), the devaton of the qualty value xd kt from t oft upper bound u kt, multpled by the demand d t (hence a per unt penalty). Th yeld the pecewe affne convex objectve functon g(x d ) = p kt e kt (x d kt)d t. k Q t T d Penalty contrant: The objectve functon above modelled lnearly by the ntroducton of new varable, δ u kt, for each k Q and t T d, that are requred to atfy the followng contrant: δ u kt, k Q, t T d, (1a) δ u kt x d kt u kt, k Q, t T d. (1b) 2

3 Set and ndce Table 1: Overvew of the notaton S := {1,..., S} Supply pont k Q := {1,..., Q} Qualte t T := {t 1,..., t T } Tme pont at whch upply come n at S t T d := {t d 1,..., t d T } Tme pont at whch demand goe out d Varable Inventory and flow varable I t y t Inventory at S at t T Flow from S to the demand pont at t T d Qualty varable x kt Qualty value of k Q of I t x d kt Qualty value of k Q of d t δkt u, Devaton of xd kt from u kt f x d kt u kt, ele Parameter Inventory and flow parameter I t d t Intal nventory at S Incomng upply at S at t T Outgong demand at t T d Qualty parameter x k u kt p kt q kt Functon P (, t) := {{t T : t < t} max{p (, t)} f P (, t) p(, t) := otherwe S(, t) := {t T : t > t} (, t) := mn{s(, t)} P T (t) := {t T : t t} p T (t) := max{p T (t)} S T d(, t) := {t T d : t t < (, t)} f S(, t) {t T d : t t } otherwe T d(, t) := mn{s T d(, t)} Qualty value of k Q of I Soft upper bound on x d kt Per unt penalty to be pad f x d kt u kt Qualty value of k Q of t Predeceor() wthn T upply tme pont n T the mmedate predeceor to t., S: P (, t) T the et of precedng t T and p(, t) T Succeor() wthn T, S: S(, t) T the et of upply tme pont n T ucceedng t T and (, t) T the mmedate ucceor to t (provded that t max{t } S(, t) ). Cro-predeceor(): P T (t) T tme pont n T and p T (t) T the et of upply precedng the demand tme pont t T d the mmedate predeceor to t T d. Cro-ucceor(): S T d(, t) T d the et of demand tme pont n T d ucceedng the upply tme pont t T and T d(, t) T d the mmedate ucceor to t T (provded that t max{t d } S T d(, t) ). 3

4 The optmzaton model requre the functon f(δ u ) = p kt δktd u t k Q t T d to be mnmzed. Th objectve, together wth contrant (1a) and (1b), and nonnegatvty of all p kt, enure that, n any optmal oluton, δ u kt = max{, xd kt u kt}, for each k Q and t T d. Supply de nventory contrant: When upply come n, the nventory quantty and qualty change. The followng contrant capture the change n quantty: I t = I t + t y t, S, t T, t = p(, t). (2) t S T d (,t) Supply de nventory blendng contrant: The followng contrant capture the change n qualty. The nventory new qualty a weghted lnear combnaton of the nventory old qualty and the qualty of the ncomng upply: x kt = x kt I t + q kt t I t + t, S, k Q, t T, t = p(, t). (3) Snce each tockple aumed to be ntantaneouly, perfectly blended at each upply tme pont, for tme t between conecutve upply tme pont, the qualty value of a tockple reman contant: all materal taken off the tockple to meet demand at uch tme have precely the qualty value at the tme the lat upply wa added, (x kp T (t) ) k Q, and thee are dentcal to the qualty value of the materal remanng n the tockple after materal ha been take off to meet demand. Thu qualty value need only be calculated at the upply tme pont a hown above, and thee can be ued to determne the qualty value of materal taken off to meet demand, a hown n the next contrant. Demand de blendng contrant: To calculate the qualty after blendng the y t flow,.e. the qualty of d t, we lnearly blend the precedng nventory qualte weghted by the y t flow (where we ue S y t = d t ): x d ktd t = S x kt y t, k Q, t T d. (4) t =p T (t) Boundng contrant: Both I t and y t are nonnegatve. I t bounded above by It u, whch deduced from ung a lttle raw materal a poble from upply pont to meet demand. y t bounded above by d t, whch deduced from ung only raw materal from upply pont to meet demand. x kt and xd kt are at leat a good a the wort qualty and at mot a good a the bet qualty of all ncomng upply. Thu, we have I t I u t, S, t T, (5a) y t d t, S, t T d, (5b) mn{q kt t T, t t} x kt max{q kt t T, t t}, mn{q kt S, t T, t t} x d kt where I u t := t T t t t max S, k Q, t T, (5c) max{q kt S, t T, t t}, k Q, t T d, (5d) {, t T d t t d t S t T t t We are now ready to formulate the mnng poolng problem (MPP): t }, S, t T. mn f(δ u ).t. (1) (5). (MPP) δ u,i,y,x,x d (3) and (4) are blnear functon wth blnear term of the form Ix and yx, repectvely. Such form are not convex, thu MPP a nonconvex, nonlnear program. It can be vewed a a pecal cae of the generalzed poolng problem, whch we defne next. 4

5 Input 1 Input 2 Input 3 Pool 1 Pool 2 Output 1 Output 2 Fgure 2: Example of a GPP graph 3 A pecal cae of the generalzed poolng problem We conder an acyclc drected graph G = (N, A), where N the et of node and A the et of arc. N parttoned nto three nonempty ubet I, L, J N: I the et of nput, L the et of pool and J the et of output. Flow are blended n pool and output. We aume that A (I L) (L L) (L J) (I J),.e., there are no arc between two nput ((I I) = ) or two output ((J J = )) and no arc from pool to nput ((L I) = ) or output to pool ((J L) = ) or output to nput ((J I) = ). We further aume that every pool ha n-degree and out-degree of at leat 1. Smlarly, every nput (output) ha out-degree (n-degree) of at leat 1. Problem ntance wth A (L L) = are referred to a tandard poolng problem (SPP) and a generalzed poolng problem (GPP), otherwe. An example of a GPP graph hown n Fgure 2. Both SPP and GPP can be modelled a blnear program. Problem ntance wth L = are referred to a blendng problem (BP). BP can be modelled a lnear program. It worth notng that (2) (4) conttute the man contrant of what commonly referred to n the lterature a the p-formulaton (concentraton model) of the GPP [1]. There are alternatve formulaton uch a the q-formulaton (proporton model), the pq-formulaton, the hybrd formulaton, and mult-commodty flow formulaton [2, 3, 4, 7]. The GPP tated a a tatc problem, to be olved at one pont n tme, wherea MPP a dynamc problem, decdng how to blend at multple tme pont over a plannng horzon. Neverthele, the followng contructon enable u to repreent the MPP ung a GPP graph, and hence how that the MPP can be vewed a a pecal cae of the GPP. 1. Node (a) I node: For all S, t T, create an nput node t. (b) L node: For all S, t T, create a pool node t. (c) J node: For all t T d, create an output node t. 2. Arc (a) (I L) arc: For all S, t T, create an arc from nput node t to pool node t. Flow on thee arc repreent t. (b) (L L) arc: For all S, t T, t = p(, t), create an arc from pool node t to pool node t. Flow on thee arc repreent I t. (c) (L J) arc: For all S, t T d, t = p T (t), create an arc from pool node t to output node t. Flow on thee arc repreent y t. 4 Lnear relaxaton of the blnear program On the one hand, we can olve MPP a t wth a nonlnear programmng olver able to handle nonconvex problem. However, tate-of-the-art global olver uch a BARON [18] and Couenne [5] are relatvely low 5

6 and truggle to olve large-cale nonconvex problem. On the other hand, local olver uch a Ipopt [2] are fat, but only fnd locally optmal oluton, whch provde upper bound on the MPP. In the GPP lterature, a tandard approach to fndng lower bound to ubttute the Ix and yx term n contrant (3) and (4) by an auxlary varable, z, o that (3) and (4) become lnear, and the problem aume the form of a blnear program (BLP): mn x,y,z f(x, y, z).t. g(x, y, z), h(x, y, z) =, z j = x y j, (, j) B, x L x x U, y L y y U, (BLP) where x (y) a vector of I (J) contnuou varable, z j the blnear term of x and y j ( {1,..., I}, j {1,..., J}), B = {(, j) z j = x y j } the et of blnear term, f(x, y, z) a lnear functon and g(x, y, z) and h(x, y, z) are lnear vector functon [11]. Then one relaxe the blnear term z j = x y j for all (, j) B ndvdually ung o-called McCormck relaxaton. Let xy be a blnear term and let x L, x U, y L and y U be the lower and upper bound on x and y, repectvely. In [14], McCormck ntroduce a lnear relaxaton of xy ung the followng four nequalte: z xy L + x L y x L y L, z xy U + x U y x U y U, z xy L + x U y x U y L, z xy U + x L y x L y U. Al-Khayyal and Falk prove n [1] that the former two of thee four nequalte provde the convex envelope whle the latter two provde the concave envelope of xy. In other word, the four nequalte form the convex hull of xy. The convex and concave envelope (or under- and overetmator) of xy on [ 1, 1] [ 1, 1] are hown n Fgure 3. We wll refer to the lnear relaxaton of MPP obtaned n th way a MPP-L x y 1.5 y 1 x.5 1 Fgure 3: Under- (red) and overetmator (blue) of xy on [ 1, 1] [ 1, 1] 5 Computatonal tudy Our ndutry partner provded u wth an example data et repreentng upply and demand data (ncludng qualty pecfcaton and contract penalte) of a real-lfe mnng company for the tme horzon We plt the data nto problem ntance of year, half-year and quarter. All problem ntance hare the followng charactertc: There are two upply pont,.e. S = 2. There are four qualte: ah, moture, ulfur and volatle matter,.e. Q = 4. In the orgnal data et, there are T 1 +T 2 = 4132 upple and T d = 53 demand. However, we pre-proceed all problem ntance to enure feablty and to decreae ther ze a follow: 6

7 Conder a upply pont S and two tme pont at whch upply come n, t, t T, t < t. If there ext no tme pont at whch demand goe out n between t and t,.e. there ext no t T d, t t < t, then t and t (and ther repectve qualty value q kt and q kt ) can be equvalently repreented a a ngle ncomng upply at tme pont t. The qualty value of new t = t + t are then calculated a qkt new = q kt t + q kt t, k Q. t + t Th true for any number of ncomng upple for whch there ext no outgong demand n between. Applyng th pre-proceng tep reduce T 1 to 234 and T 2 to 244. If at any tme the cumulatve ncomng upply maller than the cumulatve outgong demand (.e. demand cannot be met), we adjut the demand data. More precely, we teratvely delete any demand d t, t T d, where t < d t. Th reduce T d to 349. S t T t t Fgure 4 how the quantty of ncomng (aggregated) upply and outgong (aggregated) demand, Fgure 5 the qualty of ncomng upply and oft upper bound on outgong demand over tme. t T d t t 2 1t 2t d t t dt 12.5 t, dt (kt) t, dt (Mt) Fgure 4: Quantty of ncomng (aggregated) upply and outgong (aggregated) demand over tme The lnear problem MPP-L wa olved wth CPLEX [13], and the nonlnear problem MPP wth Ipopt [2]. Table 2 how the tet reult. Our computatonal tudy mple that, even for large-cale BLP, both locally optmal upper bound oluton and McCormck relaxaton lower bound can be calculated effcently. However, the gap between the lower and upper bound n ome cae qute large, rangng from 1.7% to 41.9% on the ntance teted, wth an average of 19.4% (accurate to one decmal place). To cloe the gap, one may now conder parttonng I, y and x to tghten the McCormck relaxaton, and combne oluton of MPP-L wth a branch-and-bound algorthm; th the ubject of future reearch. Acknowledgement Th reearch wa upported by the ARC Lnkage Grant no. LP112524, Hunter Valley Coal Chan Coordnator (hvccc.com.au) and Trple Pont Technology (tpt.com). 7

8 q1t, d1t (%) q 11t q 21t u 1t Ah (k = 1) q2t, d2t (%) q 12t q 22t u 2t Moture (k = 2) q3t, d3t (%) q 13t q 23t u 3t q4t, d4t (%) q 14t q 24t u 4t Sulfur (k = 3) Volatle matter (k = 4) Fgure 5: Qualty of ncomng upply and oft upper bound on outgong demand over tme 8

9 Table 2: Number of varable (#Var), number of contrant (#Con), objectve value (Obj) and olve tme n econd for all problem ntance, MPP-L and MPP 9 MPP-L Intance T 1 T 2 T d #Var #Con Obj Tme #Var #Con Obj Tme Gap All (29 212) ,48 31,931 35,381, ,319,1.2 25% 29 Year ,69, ,141, % MPP Half-year ,69, ,141, % Quarter ,611, ,281, % Quarter ,22, ,757, % 21 Year ,32 8,747, ,822, % Half-year ,86, ,192, % Half-year ,612, ,242, % Quarter ,378, ,963,57. 2% Quarter ,938, ,999, % Quarter ,956, ,456,252. 2% Quarter , , % 211 Year ,672, ,659, % Half-year ,563, ,517, % Half-year ,57, ,935, % Quarter ,68, ,348,32. 11% Quarter ,57, ,185,97.1 4% Quarter ,37, ,264, % Quarter ,219, ,28, % 212 Year ,392, ,32, % Half-year ,25, ,63, % Half-year ,4, ,385, % Quarter ,4, ,626, % Quarter ,917, ,967,36. 2% Quarter ,254, ,395,868. 6% Quarter , , %

10 Reference [1] F. A. Al-Khayyal and J. E. Falk. Jontly Contraned Bconvex Programmng. Mathematc of Operaton Reearch, 8(2): , [2] M. Alfak and D. Haugland. A mult-commodty flow formulaton for the generalzed poolng problem. Journal of Global Optmzaton, 56(3): , 213. [3] M. Alfak and D. Haugland. Strong formulaton for the poolng problem. Journal of Global Optmzaton, 56(3): , 213. [4] C. Audet, J. Brmberg, P. Hanen, S. Le Dgabel, and N. Mladenovć. Poolng Problem: Alternate Formulaton and Soluton Method. Management Scence, 5(6): , 24. [5] P. Belott. COUENNE: a uer manual. con-or.org/couenne/couenne-uer-manual.pdf. [6] A. Bley, N. Boland, G. Froyland, and M. Zuckerberg. Solvng mxed nteger nonlnear programmng problem for mne producton plannng wth tockplng. Optmzaton Onlne e-prnt, November 212. optmzaton-onlne.org/db HTML/212/11/3674.html. [7] N. Boland, T. Kalnowk, and F. Rgternk. New mult-commodty flow formulaton for the poolng problem. Journal of Global Optmzaton, 216. Advance onlne publcaton, 42 page. DOI: 1.17/ x. [8] S. S. Dey and A. Gupte. Analy of MILP Technque for the Poolng Problem. Operaton Reearch, 63(2): , 215. [9] J. E. Everett. Iron ore producton chedulng to mprove product qualty. European Journal of Operatonal Reearch, 129(2): , 21. [1] A. Gupte. Mxed nteger blnear programmng wth applcaton to the poolng problem. PhD the, Georga Inttute of Technology, 212. hdl.handle.net/1853/ [11] M. M. F. Haan and I. A. Karm. Pecewe Lnear Relaxaton of Blnear Program Ung Bvarate Parttonng. AIChE Journal, 56(7): , 21. [12] C. A. Haverly. Stude of the Behavor of Recuron for the Poolng Problem. SIGMAP Bulletn, 25:19 28, [13] IBM. IBM ILOG CPLEX Optmzaton Studo: CPLEX Uer Manual, Veron 12 Releae 6, 215. [14] G. P. McCormck. Computablty of global oluton to factorable nonconvex program: Part I Convex underetmatng problem. Mathematcal Programmng, 1(1): , [15] R. Mener. Novel Global Optmzaton Method: Theoretcal and Computatonal Stude on Poolng Problem wth Envronmental Contrant. PhD the, Prnceton Unverty, 212. ark.prnceton.edu/ark:/88435/dp15q47rn787. [16] R. Mener and C. A. Flouda. Advance for the poolng problem: Modelng, global optmzaton, and computatonal tude. Appled and Computatonal Mathematc, 8(1):3 22, 29. [17] R. Mener, J. P. Thompon, and C. A. Flouda. APOGEE: Global optmzaton of tandard, generalzed, and extended poolng problem va lnear and logarthmc parttonng cheme. Computer & Chemcal Engneerng, 35(5): , 211. [18] N. V. Sahnd. BARON: A General Purpoe Global Optmzaton Software Package. Journal of Global Optmzaton, 8(2):21 25, [19] G. Sngh, R. García-Flore, A. T. Ernt, P. Welgama, M. Zhang, and K. Munday. Medum-Term Ral Schedulng for an Iron Ore Mnng Company. Interface, 44(2):222 24, 214. [2] A. Wächter and L. T. Begler. On the mplementaton of an nteror-pont flter lne-earch algorthm for large-cale nonlnear programmng. Mathematcal Programmng, 16(1):25 57, 25. 1