mPAL. F. .• KUCl-ITA. M .• and NEWMAN. A. E.x.lensiOlls 10 an elfici",,( opI imizatioD DIOdeI for long-term proo.Iu~.;tion 1)larmillg III LKAB's Kirnna MillC. A.ppU,·alk", u/CompuICI7UJlW Opu ul;flI'" RQ ('(1rch in the Minerals ItUlus/riu, Soulh Arrican InstilUtc of Mining and Met.:lllu f8Y. 2003 .

. Extensions to an efficient optimization model for long-term production planning at .LKAB's Kiruna Mine

E. TOPAL, M. KUCHTA, and A. NEWMAN C()I()radQ School of Mjnes. Golden, CO, USA

LKAB's Kiruna Mine, located above the Arctic Circle in nortbem Sweden. is the second-largest underground mine in thc world today. The orebody is a world-class high-grade magnetite deposit, from which three raw ore products are extracted. We use mixed integer programming to de\emlinc a-production-scheduie,-i:e;-; whieh production'blockno'mine, 'and-whelno"Stmt mining' them, SjTa~ ­to minimize deviation from planned production quantities while adhering to the operational constraints of the mi ne. The resulting model contains thousands of binary variables and a commensurate number of constraints, precluding us fro m obtaining a schedu le in a reasonable amount of time. Wc describe a method that build~ Utt prior, related work, to expedite solution time.

Keywords: underground milling, production planning. optimi7.ation, integer programming

Introduction

LKAB's Kiruna Mine is located above the Arctic Circlc in northern Sweden, and produces ab(lut 24 million tons or iron ore per year, making it thl! second ta(gest underground mine in the world loday. The orebody, a world-clas!'> bigh­grade nUlgnetite deposit., is approx imalely 4 km long and 80 m wide on average, and lies roughl y in the nOrlh-south direction, with a dip of some 60 degrees. Surface mining

commenced around the turn of the last century, and half a cellltlry later, operations began underground. Since 1962, mining has been done exclusively via large-scale slIblevel caving. The mine is divided iOlO ten main production areas, aboul400 m to 500 m in length, each with its own group of ore pa!'>ses, also known as a shaft grQUp, located at the centre of Cl.lch production area and extending down to the current main 1045 m level (Figure I). One la two 25-lon­capaciry electric Load Haul Dump Units (LHDs) operating

Hoisting

fr'igure 1. Schematk of the orebody beillg mined at its current level (LKAB, Intenlal promotiOnAl Dlalel'ial, 2001)

EXTENSIONS TO AN EFACIENT OPTIML7.ATlON MODEL FOR LONG-TERM PRODUCTION PLANNING 289

on a sublevel within each production area transport thc orc from the crosscuts to the ore passes. The site on which each LHD operates is also referred to as a machine placement. Each machine placement is usually 200 to SOU m in length and contains from one t(f three million tons of ore, equating to between onc and five 100 m production blocks; the machine placements possess thc same height as the mining sublevel and extend from tile hangingwall to the footwall. Once started, mining restriction~ requirc continuous production of the blocks within a machine placement until all available ore has been removed.

From the two main in situ ore types, the mine deliver~ three raw orc prl)ducts used to supply four ore post­processing plants, or mills. Phosphorus is the main ore contaminant. The B 1 product contains the least amount of phosphorus and is used to produce high-quality fines. The other ore types, B2 ore and ])3 ore, are processed into pellets and PHSSOSS medium-and·high-phosphorous ·eontent,· respectively.

Scheduling model Optimization models are commonly used. not only in the mining sector, to makc dccisions, represented with decision variables, so as to attain a goal, specified as an ohjective function, while meeting operational limits, called constraints. We employ such an optimization model, specifically, a multi-pcriod mixed integer program, to provide mine planners with a production schedule that details when to start mining each production block within each machine placement in order to satisfy demand for each ore type at the mills as closely as possible while not violating any mine sequeneing comtraints. An optimal solution would give the minimum deviation from planned production quantities and the means by which to attain this minimum, i.e., for our application, the time at which a given production block should be mined; a sub-optimal solution would give a value for the deviation that is higher than what could theoretically be realized. In eitller case, the solution would satisfy all operational constraints. A sub­optimal solutipll may still be usable, if its deviation from optimality is relatively low, e.g., 1-5%.

For ease of presentalion, we assume that each production block requires exactly one time period to mine. The model follows:

Indices a machine placement b,b' production block k ore type, i.e., Bl, B2. D3

time period (month) )! shaft group, i.e., 1 ... 10

Sets Db "" set of eligible time pcriods in which

production block b can· be mined (rcstrictcd hy production block location and the start time of other relevant production blocks)

All = sct of production blocks within machine placement a

BlockVh = set of production blocks who~e access is restricted vertically by production block h

BlOCkRb = set of production blocks whose access is forced by right adjacency to production block h

Blockh set of production blocks whose access is forced by left adjacency to production block b

290

S.. set of machine placements contained in shaft group v

Parameters rbk

d" EarlYb Late/) T UlD"

amount of ore type k in block b (tons) demand for ore type k in time period I (tons) earliest start time for production block b latest start time for production block b

= length of the planning horizon = the maximum number of simultancously­

operational LHDs in each shaft group v

P _{I, If block b of machine placement a is in shaft grOllp v nbv - O. otherwise

Decision variables

_{I. If start mining production block h in time peliod I) Ybl- O. otherwise

dUll amount mined above thc desired demand of ore type k in time period t (tons)

ddkl amount mined below the desired demand of ore type k in time period t (tons)

Objective function

Min '2:,( du" + dd,, ) ' ,'

COl1straint.~

Lrbk *' Y/rt -dU,t + dd'k = dk' 'rfk,t e nb " I,Yht :?: Yb,,' 'rfb,b' E BlockYv,t' E n'b' ,~n,"

LYbf ~ Y/!'t' 'rfb,b' E BlOckRb,t' E nb' ,~n .

L Yb'~Yb't' Vb,b'eBlock4,t' eQb'

1~f1,

LL L ~,",'y" ~LHD"V, ~GS. b~A" "GQh~,5'

Yb, = I 'rfb I EarlYb = Lateb

dU,p dd'k

2::: 0 \it,k Y/it binary Vh,t

III

[2]

[3]

[4J

[51

[6]

[71

[8]

[9]

The objective function minimizes the deviation from the production targets for each ore type ~o that the mills can meet their respective production demands. Constraints LIJ calculate the tons of each ore type mined per time period and thc corresponding deviations from the specificd production levcls. Constraints [2], the vertical scqucncing constraints between mining sublevcls, preclude mining a production block under a given production block until operationally feasible. Constraints (3J and [4J enforce horizontal sequencing constraints between adjacent production blocks. Constraints [5 J limit the numbcr of LHDs active within a shaft group to the maximum allowable number. Constraints [61 place activc production blocks into the production schedule. Constraints [7/J

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I

I

I

j

I

..

preclude a production block fm m filarling to be mined more than once during the lime horiwn if its late start date occurs beyond the maximum time hmizon. Constraints [81 require a production block \() fi larl being mined ul some point. during the time hori7.01l if ils late start date falls within tIle lime horinm. Conslnl int~ (9] enforce non-negativity and integrality oHhe variables. as appropriate.

Literature review Open pit mining problems1,2 are early applications of optimization to problems in thc mining sector. Some of the~e models possess special ~lrLlctures that a llow the opti.miz.'l.lion problems 10 be soLvlXl despi te the primitive state of hard ware and software at the time. Subsequently, a nothe r s peciaJ class of optimi za ti on model s, linear programs, is applied in underground mines. e.g., copper or co31 mines 3-7. UnJortunatelY:,Jhese models lack the ability (0 capture discrete decisions, e.g., whether or not to minc a given production block. In an crfort 10 capture tb e~e di scretc dec is ions, some modeJ,~ co mbine li near progr.tmming with simulation or m~nua l interventions-I!. fo r example , a I)toch:'lstic dynamic program determines long-term optimal generalion levels for u coaJ- and gas­fired power station under a set of scenarios; the authors then use simulation to determine corresponding mining and stockpiling strategies i3• Researchers recogni 7..c the need to incollJorate discrete dec i sion~ in their models l 4-16, but none of these models provides detailed optimal multi -period mine sc hedules_ O ne suc h mulLi -period lnode ll1 uses discrete decision variables to de termine the location o f processing facilities and whether a mine produces or nOl, but doe~ not pro vide a detailed produclioll schedu le. Several prior models for production planning at Kiruna Mine do not yield adequate multi-period schedules in an operationally acceptable amount of time18-2t. TheSt: models therefore either provide ::>ub-optimaJ solutions or production sc hedules over a s horte r timcframe than requ ired. An integer-programming model provides a production schedule for a s ublcvel stoping operation at Stillwater Mining Company21. Thi s model is the closest to ours. and yields l1crtr-optimal soLutions to maximize reVcnue from mining p latinum and palhHlium: however, Ihe authors do not describe any speci:'lllcehniques to expedi lc solution time.

Increasing the efficiency of the optimization model

The size and struCture of our model motivates our current work. MIP solution time increases expollentially with the numher of in~ger variables. To d t; tcrmine a three-year prod uction schedule req ui res 36 (time periods)* 11 73 ( produc lion bl ocks) = 42,228 va riables. Gi ven th e mathe matica l Hructure of OUT mode l, es peciaJl y Ih e complicaling scquencing constraints, lhe corresponding solu tion time can be hours, or even days. In a scparate research endeavour12 , we reduce the number of integer variables by preprocessing Ule production d31a., carefully fo rmu lating the model , a nd impt e mcntin g several algori thms to elimi nate unnecessary vari:'lbles. Relevant It)

our c urrent di ficu~sion is the formulation of the mode] to make decisions for machine placements. ratJler than fo r production blocks. Becausc all production bl{lCks within a machine placeme nt mu~L be mined contin uously in a spec ific o rder, it suffices to define a binary variabl e indicating whcthel' nuu:hine placement a starts to be mined

in time period f , i.e .. y,,,. rmher than dellning a valiable for whether or not each individual production block is mined. This change of variables requires nontrivial accounting lo c ()n~ider the amo unt or time re quired to min e eaeh production bloc k, and, given the numher of production blocks ill each machine placement , the amount of time 10 mine (some portion at) the machine placement. Huwever, because the re :'l re far fewer mac hine place ments tha n individual production blocks, using y"" rather than Yrn, as the binary variable reduces the number of binary v:lriables in our model by an order or magnitude. Tn the ens uing discLl~sion , we use thesc new variables. Yah rather than the variables that appear in Ule original formulation.

Despite prior work: to expedite so lution times, intcger programs a re notorious for their lack o f robustness with re:.:pcet to tr:lctrtbility, and even these advanced techniques fail to produce a threc-year schedule in a timely fashion for all the_mining_scenarios we hnve encountered at Kirun.a Mine. Motivated by new data sets that nOl only incn::a~e the size of the model, but also introduce ~ymmetry (making various feasible solutions less di ~lingu ishable from oue another), we explore additional lechniques to reduce the search ~ce, i.e., Ihe set of fcasible solutions from which we derive the optimal.

Specifically. we add redundant, but valid, constrainls 10 our modeL In llt hcr words . these constraints arc not necessary to e liminate operationally infeasible solutions (because the eon~lraints are redundant with those already in the model) , but Ihey are va lid, i.e., they are satisfied by every fc:.ls ible solutioll in the original mode l. Intuitively, th is may appear to ex acc rbale problem trac trtbiJity by (unnecessarily) increasing the size of the model. However, unlike the addition of variables. whi ch provides more options (feasib le ~olutions) the 8o lution algorithm mu st explore, adding constraints reduces the number of fcasible pos.~ib ilities, there hy reducin g the search space and expediting solution lime. III fact. constrainL .. [7J and nn in the existing model already serve this purpose. bul wc can also add redunda nt vertical a nd horizontal sequcncing constraints.

We add conso:aints between pairs of machine placements if both th e early and the late st ar! of each ma c hin e placement lie within the time horizon. Specifically, if we define a as a mac hine placemen t Hnd a' as a mac hine placement whose mining stan d.lte is affected by a, ES(a) and 1..S(a) as the early and late SIMt dales, respecti vely, for mach ine placement G, rtnd assume that ES(a) ;:.: ES (cl) . we can add a (redundant) vertical sequenciug constraint of the form:

L~(a ) !.S(II')

L" y. - L "'y,;,>ES(a)-ES(a') l - £S(a) t·"£.~,,,) [10]

Va,a' affected by a

In other words, given machine placement a starts to be mined in some lime period f (where ( lies between th e earliest and lalest possible times that a can start /() be mincd), machine phl.Cement u' can start to be mined no earlier than ,11 somc li me period (where t' lies between the earliest and la test possible ti me.\( th rt t u' can stuI1 to be mined) pl us the ditl"erence between the early slart times betwecn the lWO production blocks, a and 0', i.e .• the requircd lag time after a starts being mined and before a' .~tm1s to be mined,

We can Cllllstr uct similar (red undant) hori zonta l sequencing constraints as fol lows, where the first set of

EXTENSIONS TO AN EFFICIENT OPTIMIZATION MODEL FOR LONG-TERM PRODUCTION PLANNING 291

constraints corresponds 10 )e n adjacency constraints :md the ~"Qnd se! to right adjacency constraints;

IS( .. ) ~')

Lt:+: Yn/ -2,.1':+: Y .. " ~ lag,(a) - l '_ES( .. } ,'_ES(,,' ) [11]

Tia ,a' affected by a

LS(,,) ~'l

L,1 *Ym - l...t'*Y,,·,$iagr(a) - 1 ,=&"( ... ) ,'",t::l(II')

[1 2J

"la, a' affected by a

where lag;(a) and lag,(a) represent the maximllm amount of time between when macnille placement a slarts to be mined <!Dd when machine placement a' must corresponding ly start 10 be mined for machine placements 10 the left and to the right of machine placement a, respectivel y. Analogous 10 the firsL set of constraints, these two constraints require TilacJiille- pIa-cement (i' Ri bemiriedWfihfii 'tnc J~eq-U:jred amount of lime after machine placement (J starls to be mined.

All three of these sets of constraints are met by using the original horizontal and vertical sequcncing constraints, i.e., constraints [2]- f41 o f the formulation s hown earl ie r. However, the addition of constrain ts flO]- (l2] to this original formulation helps expedite solution time.

Numerical results We illu~trate the e ffect on the solution lime of imposing conslrninl~ [IO}-[1 2] of our model by providing results for three separalc scenarios. The first scenario uses the original model presented earlier. The second scenario replaces the scquenclng constraints ([2)-[4]) in the original model with cons traints [l01- [12]. The third scenari o combines the odgiual constraints wit h our redundant constraints , fIO}-[1 2]. Wc use both acmal data from LKAB's Kiruna Mine, and other d11a sets for which we modify demand data and the available numbe r o f LHDs to reflect realistic changes given the :\vll i1 abili ty of each ore type in each machine placement and the availability of load haul dump units, respectively. The first four data sets possess a three­year time horizon and the fifth data set a five-year hori7.on. As is common with integer programming algorithms, our algorithm provides both the best objective function value i t has found al any point in its search, and the best bound on the optimal objecti ve function value, Le., an objective function value at least as good as the optimal solution (and perhaps better). In our numerical study here, we attempt to solve all instances within a 4-hour time limit to within 5% o f optirnality, i.e., to willtin 5% of what is guaranteed to be the 'bc~t possible' solution.

We imple me nt our mi xed integer program using the AMPL programming language2.J,24, and the CPLEX solver, Version 7.025. We run our model instances on a Sun Ultra 10 machine with 256 MB RAM . We report results in Table 1.

Results are inconclusive as to whether adding constraints f lO)-[12J or replacing [2}-14] ill the original model with cons truints I IOJ-[l 21 produces fa sle r so lution [ime~. Ho wever , the run time o r lhe original model is clea rl y dominated by the introduction (If the new constraints. While the run times for the model with the first two data sets exhibit modest improvements wit h the addition of the redundant constraints, the solution times for the model with the third data set arc significantly reduced by adding the redundant consrrainl'l. We are unable 10 obtain a solution provably within 5% of optimality for the scenario with the fourth data set. However, the optimality gap is much lower with ... Jhan without , the redundant conll.lraintll.. Th e mo:-;t significant improvement occurs with the fifth data seL Without the redundant constraints, the algorithm fail s to find a solution provably within 5% of optimality within the 4-honr time limit we impose. By COll trast, nsjng the Dew constraints. lhe algori thm find!l a SHlutiou wilhin 5% of optimality in 1C.'I.'1 lh 'lIl 30 minutes.

Conclusions and extensions We use integer programmin g to generate production schedules at LKAB's Kiruna Minc; however, the size of our mode l requires that we develop techniques 10 e,;ped ite solution time. By reformulating tbe model to reduce the number of integer variables, and by employing lightening constraints, ill all b ut one case we test, we generate ncar­optimal three- to five-year schedules in less than an hour. Extensions to our research include continuing to evaluate our current solucion methodology, and generating new Icchniques to even further reduce solution times for other types of data sets. We note that Kinuta Mine has adopted schedules that we have produced with our optimization model, and is now llSing optimization 10 aid in its schedule generation. Wc present a complelC comparison of CW'l'ent practice and production scheduling at Kiruna Mine prior to the introduction of this optimization model elsewherc26.

Acknowledgements The authors would like to thank LKAB for the opportunity Lo work on this challenging project and for permission to publish these results. The authors would also specifically like to than k LKAB employees Hans Engberg, Anders Lindholm. and Jun-01Qv Nilsson for providing information, assistance, and suppon in this endeavour.

Table I Solution times for five sets of data using the three models: (i) the original model, (ii) r~p'ncing ronstraints 12]-[4] with IIO]-I12], and

(tiI) the oriKinal modellO'lth cQllstraints [10K121

Scelloriv Original modo:l MOtkI leplacing 12H4J wi lh I IOH I21 Model wilh !'2}-14] and [ tOH121

011(1\ Sel I tl OO SC:COOOIi 1 200~ood~ 950 second. Dill:! Set 2 1200 ~Ildl; 150[) secomls '750 &ecoods DalaSd3 820 second~ 350 seconds 200 seconds Data Set 4 .14400 seconds (7.5% gap) 14400 stx:onds (5.7% gap) 14400 ~econd~ (5.(1% gap) Oata Set 5 14400 r;eco)}(il; (5.2% gap) 730 .\eCoo~ 130') ~cconds

292 APC:::OM 2003

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