Integrating advanced demand models within the framework of mixed integer linear problems: A Lagrangian relaxation method for the uncapacitated

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1 Integratng advanced demand modes wthn the framework of mxed nteger near probems: A Lagrangan reaxaton method for the uncapactated case Mertxe Pacheco Paneque Shad Sharf Azadeh Mche Berare Bernard Gendron Transport and Mobty Laboratory (EPFL) STRC 17th Swss Transport Research Conference Monte Vertà / Ascona, May 17 19, 2017

2 Transport and Mobty Laboratory (EPFL) Integratng advanced demand modes wthn the framework of mxed nteger near probems: Mertxe Pacheco Paneque, Mche Berare Transport and Mobty Laboratory Écoe Poytechnque Fédérae de Lausanne EPFL ENAC IIC TRANSP-OR, Staton 18, CH-1015 Lausanne phone: fax: {mertxe.pacheco, Shad Sharf Azadeh Operatons Research & Logstcs group, Econometrc department, Schoo of Economcs Erasmus Unversty Rotterdam H11-27 Tnbergen Budng, Rotterdam, Netherands phone: fax: Bernard Gendron Département dnformatque et de recherche opératonnee, CIRRELT Unversté de Montréa, Canada C.P. 6128, Succursae Centre-Ve phone: fax: Abstract The ntegraton of customer behavora modes n optmzaton provdes a better understandng of the preferences of cents (the demand) to operators whe pannng for ther systems (the suppy). These preferences are formazed wth dscrete choce modes, whch are the state-of-the-art for the mathematca modeng of demand. However, ther compexty eads to mathematca formuatons that are hghy nonnear and nonconvex n the varabes of nterest, and are therefore dffcut to be ncuded n (mxed) nteger near probems (MILP). These probems correspond to the optmzaton modes that are consdered to desgn and confgure a system. In ths work, we present a genera framework that ntegrates advanced dscrete choce modes n MILP. Nevertheess, a near formuaton comes wth a hgh dmenson of the probem. To address ths ssue, and gven the underyng structure of the mode, decomposton technques such as Lagrangan decomposton can be apped. Two subprobems wth common varabes have been dentfed: one regardng the user and one regardng the operator. In the former, the user has to perform a decson based on what the operator s offerng, whereas n the atter, the operator needs to decde about the features of the suppy to attract the users. Keywords dscrete choce modes, combnatora optmzaton, smuaton, decomposton technques

3 1 Introducton Even though demand and suppy cosey nteract n many appcatons, such as arnes or hgh speed raway, these two research feds have evoved ndependenty, wthout payng too much attenton to the exstng nterdependences between the two. Indeed, ncorporatng the preferences and tastes of customers, whch are usuay characterzed wth dscrete choce modes, aows for a better pannng of the systems for the operators. The desgn and confguraton of such systems are typcay addressed wth optmzaton modes, beng MILP a reevant porton of the modes reported n the terature. In Pacheco et a. (2016), we propose a genera framework that aows to ntegrate n MILP dscrete choce modes for whch t s possbe to draw from ts assocated probabty dstrbuton. Ths feature s mportant because we rey on smuaton to crcumvent the nonnearty ntroduced by the demand mode. To ustrate how the framework can be empoyed, an appcaton on revenue maxmzaton s consdered. The performed experments exhbt that the resutng formuaton s a powerfu too to pan systems based on the heterogeneous behavor of customers. However, the dsaggregate representaton of the cents behavor, together wth the nearty of the formuaton, eads to a computatonay expensve probem (see Secton 3.1). To address ths ssue, we can beneft from the structure of the probem and expore decomposton technques to sove t n an effcent way. These technques are used n optmzaton probems that have the approprate structure. They aow to speed up and factate the souton approach and/or to obtan good bounds for the optma vaue of the objectve functon (Conejo et a., 2006). In practce, there are two dfferent decomposabe structures: one characterzed by compcatng constrants, wth Lagrangan reaxaton as the man assocated decomposton technque, and another one characterzed by compcatng varabes, wth Benders decomposton as the prncpa technque. In our case, even f both coud be apped, we focus on the former, snce some constrants n the MILP can be seen as compcatng because they are nvoved n both the demand and the suppy mode. The remander of the paper s organzed as foows. In Secton 2, the modeng framework s summarzed, and the mathematca formuaton for the appcaton on revenue maxmzaton s provded. Secton 3 motvates the use of Lagrangan reaxaton and characterzes the dfferent probems comprsed wthn ths technque. Fnay, n Secton 4 we descrbe the future avenues of research assocated wth ths nvestgaton. 1

4 2 Modeng framework In ths approach, we embed a dscrete choce mode, whch modes the choces of customers, nsde a MILP, whch modes the decsons of the operator. In ths secton, a summary of the mathematca formuaton s provded. 2.1 Genera context Regardng the demand mode, we denote by C the choce set, whch contans a potenta servces pus an opt-out opton to capture the customers eavng the market (.e., customers choosng a servce from a compettor or not choosng anythng at a). We consder a popuaton of N customers, and we assume that the choce set of each customer may be dfferent (e.g., due to dfferent tastes, restrctons, etc.). The choce set of customer n s denoted by C n. The common specfcaton of the utty assocated wth servce by customer n s U n = V n + ε n, (1) where V n s the determnstc part and ε n the error term, whch s assumed to foow a probabty dstrbuton. We note that for the ntegraton of the choce mode n a MILP, the endogenous varabes (present n both the demand and the suppy modes) need to appear neary n the utty functon. Thus, we assume V n has the form V n = β nk x e nk + qd (x d ), (2) k where x e nk are the endogenous varabes for servce and customer n (ndexed by k), and qd (x d ) s a term that depends on other exogenous demand varabes x d (ony present n the demand mode), n a possby nonnear way defned by the functon q d. The behavora assumpton s that customer n chooses servce f the assocated utty s the argest wthn the choce set C n. The probabty that customer n chooses servce s P n ( C n ) = Pr(U n U jn, j C n ). (3) Ths formuaton s typcay nonnear as a functon of the endogenous varabes. For the sake of the ntegraton, we work drecty wth the utty functons, and not wth the correspondng probabtes. To dea wth the random nature of the utty functon, we rey on smuaton. 2

5 For each error term ε n, we generate R draws ξ n1,...,ξ nr based on ts dstrbutona assumpton. Each draw corresponds to a behavora scenaro. For the specfcaton (1), we have U nr = V n + ξ nr = β nk x e nk + qd (x d ) + ξ nr. (4) k A reevant appcaton to ustrate the descrbed methodoogy s the maxmzaton of revenue. We use ths exampe to characterze a concrete MILP. The objectve s to fnd the best strategy n terms of prcng and capacty aocaton n order to maxmze the revenue of the operator, whch ses servces at a certan prce (to be decded) and wth a gven capacty. In ths formuaton, the prce s the ony endogenous varabe, appearng n both the utty functon (demand) and the objectve functon (suppy). We defne p n R as the prce that customer n must pay to access servce. We assume t can take a fnte number of vaues, caed prce eves, between a ower bound n and an upper bound m n (bounds on the nteger representaton of the prce). For nearzaton ssues, t s characterzed as foows: p n = 1 ( L n 1 10 k n + 2 λ n ), (5) where the term n brackets corresponds to the bnary representaton of the prce as a nteger vaue (λ n are the assocated bnary varabes, and there are L n of such varabes), and 1 10 k transforms the nteger vaue nto the actua contnuous vaue of the prce (wth a precson of k decmas). The decson varabes are therefore λ n, and not p n. =0 2.2 Mathematca formuaton The mathematca formuaton for the appcaton descrbed above s depcted n Fg. 1. We provde a bref descrpton of ts sets, nput parameters, varabes, objectve functon and constrants. Sets I = C : number of servces, ndexed by ( = 0 denotes the opt-out opton) L n : number of bnary varabes characterzng the prce eves, ndexed by N: number of customers, ndexed by n R: number of draws, ndexed by r 3

6 Input parameters c : capacty of servce n, m n : nteger bounds on the prce (set by the operator) nr U nr m nr : bounds on U nr (p n s bounded and q d (x d ) s gven) nr = mn j Cn jnr : smaest ower bound for customer n and scenaro r m nr = max j Cn m jnr : argest upper bound for customer n and scenaro r M nr = m nr nr, M nr = m nr nr (bg-m constrants) q d (x d ): term dependng on exogenous demand varabes (gven) β n : parameters assocated wth the prce varabes p n (for the sake of compactness of the formuaton, we mpose β n = 0 for = 0, so that the prce term vanshes) ξ nr : r-th draw from the error term for servce and customer n Varabes U nr : maxmum dscounted utty, U nr = max C z nr y n {0, 1}: avaabty at operator eve, 1 f servce s avaabe to customer n and 0 otherwse (expct decson of the operator) y nr {0, 1}: avaabty at scenaro eve, 1 f servce s avaabe to customer n n scenaro r and 0 otherwse (resut of the choces of other customers when servce capacty s nsuffcent to satsfy the tota demand) z nr : dscounted utty, U nr when the servce s avaabe and nr otherwse w nr {0, 1}: choce, 1 f servce s chosen by customer n n scenaro r and 0 otherwse α nr {0, 1}: nearzaton of the product w nr λ n, 1 f w nr λ n = 1 and 0 otherwse λ n {0, 1}: bnary representaton of the prce (see (5)) Objectve functon Maxmzaton of the tota expected revenue (6), whch s obtaned by addng the revenues from a servces but the opt-out. The revenue of each servce s cacuated wth the assocated prce and the expected demand (obtaned from the choce varabes). Constrants Utty: (7) defnes the utty assocated by customer n wth servce n the r-th scenaro (obtaned from (4) wth the prce as the ony endogenous varabe) Avaabty: (8) makes servce unavaabe f C n and (9) reates both avaabtes (a servce cannot be avaabe at scenaro eve f t s not made avaabe by the operator) Dscounted utty: (10) (13) correspond to the near formuaton of z nr 4

7 Choce: (14) (17) characterze the choce (.e., servce s chosen by customer n n scenaro r f s the argument of the maxmum dscounted utty) Capacty aocaton: (18) (20) hande the capacty mtatons Prcng: (21) bounds the prce from above and (22) (24) are the nearzng constrants assocated wth α nr Fgure 1: MILP for the demand-base revenue maxmzaton appcaton subject to max 1 1 [ R 10 k n >0, C n U nr = β n 1 10 k ( n + r ( n w nr + )] 2 α nr ) 2 λ n + q d (x d ) + ξ nr, C n, n, r, (7) y n = 0, C n, n, r, (8) y nr y n,, n, r, (9) nr z nr,, n, r, (10) z nr nr + M nr y nr,, n, r, (11) U nr M nr (1 y nr ) z nr,, n, r, (12) z nr U nr,, n, r, (13) w nr = 1, n, r, (14) w nr y nr,, n, r, (15) z nr U nr,, n, r, (16) U nr z nr + M nr (1 w nr ),, n, r, (17) y nr y (n+1)r, > 0, n < N, r, (18) n 1 w mr (c 1)y nr + (n 1)(1 y nr ), > 0, C n, n > c, r, (19) m=1 n 1 c (y n y nr ) w mr, > 0, n, r, (20) m=1 n + 2 λ n m n, > 0, C n, n, (21) λ n + w nr 1 + α nr, > 0, C n, n, r,, (22) α nr λ n, > 0, C n, n, r,, (23) α nr w nr, > 0, C n, n, r,. (24) (6) 5

8 3 A Lagrangan reaxaton method for the uncapactated case 3.1 Motvaton The framework descrbed n Secton 2 has been tested on a case study from the recent terature. We have consdered a parkng servces operator, whose dsaggregate choce mode s characterzed n Ibeas et a. (2014). More precsey, they estmate a mxtures of a ogt mode (.e., aowng for dfferent coeffcents among customers) to descrbe the behavor of potenta car park users when choosng a parkng pace. For the experments performed n Pacheco et a. (forthcomng), we consder a sampe of N = 50 customers to avod sovng huge optmzaton probems. The choce set s composed of I = 3 servces (ncudng the opt-out opton), and L = 4 bnary varabes are used to characterze the prce, gvng rse to 16 possbe prce eves (L n = L,, n). Then, dfferent vaues of R have been empoyed to evauate the compexty. For R = 250 (the argest number of draws consdered), the uncapactated case (when servces are assumed to have unmted capacty,.e., constrants (18) (20) are gnored) takes 2.5 hours, whereas the capactated case (the fu mode n Fgure 1) takes amost 42 hours. These resuts confrm that the probem s computatonay expensve, even for nstances of moderate sze. In practce, popuatons are way arger and a hgh number of draws s desrabe to be as cose as possbe to the true vaue, so t s mportant to deveop methodooges n order to overcome ths mtaton. As mentoned n Secton 1, we consder Lagrangan reaxaton to speed up the souton approach. In ths technque, the constrants that are consdered hard are reaxed by brngng them to the objectve functon wth assocated parameters, caed Lagrangan mutpers. The resutng optmzaton probem s caed Lagrangan subprobem. It hods that the optma vaue of the Lagrangan subprobem s a ower (upper) bound on the optma vaue of the orgna mnmzaton (maxmzaton) probem. In order to obtan the tghtest possbe ower (upper) bound, an optmzaton probem on the Lagrangan mutpers, caed the Lagrangan dua probem assocated wth the orgna optmzaton probem, s soved. The remander of ths secton s organzed as foows. In Secton 3.2, we defne a smper optmzaton probem to start wth by makng some assumptons on the probem modeed n Fgure 1, and we characterze the correspondng Lagrangan subprobem (Fgure 3). We note that 6

9 t can be spt nto two subprobems wth common varabes: the choce subprobem, concernng the choce made by each customer (Fgure 4), and the prce subprobem, concernng the seecton of the prce eve for each servce (Fgure 6). We provde an agorthm to sove each subprobem (Fgure 5 and Fgure 8, respectvey). In Secton 3.3, we deta the assocated Lagrangan dua (63) and we approxmate t wth a subgradent method (Fgure 9). 3.2 Lagrangan subprobem Gven the compexty ntroduced by the capacty constrants (18) (20), we consder at frst the uncapactated case to characterze the Lagrangan subprobem. Indeed, as soon as we forget about capacty, the probem s reduced to that of assgnng the servce wth the hghest utty to each customer (among the avaabe ones) and of computng the prce for each servce. Other consderatons that are taken nto account are temzed next. The resutng formuaton s presented n Fgure 2. Avaabty Snce the capacty of the servces s unmted, the varabes for the avaabty at scenaro eve (y nr ) are not needed (they are a equa to 1). Furthermore, the avaabty at operator eve s characterzed by means of the subsets C n, and not wth the varabes y n. Ths s a smpfcaton wth respect to the mode n Fgure 1, snce t does not enabe the operator to cose and open servces dependng on the arrvng customers. However, t can ft some contexts where t s not possbe to decde on the avaabty of the servces n an onne way (e.g., parkng servces). Consequenty, we get rd of constrants (8) (9) and (15). Dscounted utty Gven that there s no avaabty at scenaro eve, the concept of dscounted utty s not necessary, and the vaue of the utty can be used drecty (.e., z nr = U nr, C n, n, r). Then, constrants (10) (11) nvove the bounds on the utty nr U nr m nr, (25) and constrants (12) (13) are redundant, and can therefore be gnored. Choce foows: In ths approach, we remove the varabes U nr and we defne the choce varabe as w nr = 1 f = arg max n, Cn {U nr } 0 otherwse,, n, r. (26) 7

10 Wth ths specfcaton the mode becomes nonnear (we refer to t as pseudo-milp). Fgure 2: Pseudo-MILP for the uncapactated case max s.t. 1 1 R 10 k >0, C n [ ( n w nr + n r )] 2 α nr (27) 1 U nr = β n 2 λ 10 k n + b nr, C n, n, r, (28) nr U nr m nr, C n, n, r, (29) 1 f = arg max w nr = n, Cn {U nr },, n, r, (30) 0 otherwse, w nr = 1, n, r, (31) n + 2 λ n m n, > 0, C n, n, (32) α nr λ n, > 0, C n, n, r,, (33) α nr w nr, > 0, C n, n, r,, (34) where b nr = β n 1 10 k n + q d (x d ) + ξ nr C n, n, r. (35) Capacty We gnore constrants (18) (20) to account for unmted capacty of servces. Prcng Constrant (22) s not needed because of constrants (23) (24) and the objectve functon (6). Ths can be verfed by consderng two cases: λ n = 1 w nr = 1 α nr = 1: constrants (23) (24) are aways verfed, but do not force α nr to be equa to 1. However, snce the objectve functon s to be maxmzed and these varabes appear postvey n the objectve functon, they are pushed to the upper bound, that s 1, so that constrant (22) can be gnored. λ n = 0 w nr = 0 α nr = 0: constrants (23) (24) aready force α nr = 0 (regardess of the objectve functon), and therefore constrant (22) can aso be gnored. We note that constrant (22) coud have aso been removed n Fgure 1, but as mentoned n Secton 2.1, we are amng at keepng a genera framework, and the MILP n Fgure 1 s just a concrete characterzaton (.e., wth a partcuar objectve functon). 8

11 The constrants assocated wth the choce performed by the customers are (28) (31), whereas the ones assocated wth the prce seecton are (32) (34). We note that these sets of constrants have a common varabe: w nr. In order to separate the Lagrangan subprobem nto the choce and the prce subprobems, we ntroduce copy varabes v nr and the correspondng copy constrants (whch are reaxed n a Lagrangan way): v nr = w nr,, n, r. (36) We take advantage of the structure of the probem and we wrte constrant (36) as foows: v nr w nr,, n, r, (37) v nr = 1, n, r. (38) To prove the equvaence between (36) and (37) (38), assume that for a certan customer n and scenaro r, there exsts a servce j such that v jnr > w jnr. Ths mpes v jnr = 1 (snce v nr = 1) and w jnr = 0. So there must exst j such that w j nr = 1 (snce w nr = 1) and v j nr = 0 (snce v jnr = 1). But ths mpes v j nr < w j nr, whch s a contradcton to our orgna assumpton. Hence, we necessary have v nr = w nr, n, r. The advantage of constrants (37) (38) s twofod. On the one hand, the repacement of the equaty n (36) by the nequaty n (37) ntroduces non-negatve Lagrange mutpers (nstead of unconstraned mutpers), whch factates the correspondng optmzaton. On the other hand, we ntroduce redundant assgnment constrants (38), whch strengths the Lagrangan subprobem. Constrant (37) s the one reaxed n a Lagrangan way. In addton to constrant (37), constrant (28) defnng the utty varabe U nr s aso transferred to the objectve functon wth unconstraned Lagrange mutpers. Ths aows us to decoupe the choce subprobem from the prce subprobem. The resutng Lagrangan subprobem s depcted n Fgure 3. We denote by θ nr R the Lagrangan mutpers assocated wth constrant (28), and by γ nr 0 the ones assocated wth constrant (37). The choce subprobem nvoves the varabes U nr and w nr, and comprses constrants (40) (42) (Fgure 4), whereas the prce subprobem nvoves λ n, α nr and v nr, and comprses constrants (43) (46) (Fgure 6). In both cases, the objectve functon s composed by the terms dependng on the correspondng varabes. Furthermore, two decomposton sources are dentfed n the mode n Fgure 1. On the one hand, each draw r represents an ndependent scenaro, and a scenaros are couped ony n the objectve functon (6). On the other hand, each customer n ams at choosng the servce 9

12 among the avaabe ones maxmzng her utty, and a customers are couped n the capacty constrants (18) (20), to ensure that the capacty of each aternatve s not exceeded, and n the objectve functon (6), to cacuate the tota demand. For the Lagrangan subprobem defned n Fgure 3, the choce subprobem decomposes by n and r (.e., a subprobem s soved for each customer and draw), and the prce subprobem decomposes by n (.e., a subprobem s soved for each customer). We denote the subprobems by Znr(θ, c γ) and Zn p (θ, γ), respectvey. Fgure 3: Lagrangan subprobem Z(θ, γ) = max 1 1 [ ( )] R 10 k n w nr + 2 α nr n >0, C n r ( 1 ) + θ nr U nr β n 2 λ 10 k n b nr n C n r + γ nr (v nr w nr ) s.t. n r (39) nr U nr m nr > 0, C n, n, r (40) 1 f = arg max w nr = n, Cn {U nr },, n, r, (41) 0 otherwse, w nr = 1, n, r, (42) n + 2 λ n m n, > 0, C n, n, (43) α nr λ n, > 0, C n, n, r,, (44) α nr v nr, > 0, C n, n, r,, (45) v nr = 1, n, r. (46) Choce subprobem The choce subprobem for each customer n and draw r s defned n Fgure 4. The souton of ths probem s easy to compute by takng nto account the foowng observatons: The vaue of U nr s ether m nr or nr. Indeed, f θ nr > 0, then U nr = m nr, snce the objectve functon s to be maxmzed and the term θ nr U nr contrbutes postvey. If θ nr < 0, then U nr = nr, snce the term θ nr U nr contrbutes negatvey. If θ nr = 0, U nr can take any vaue, so for the sake of smpcty we merge ths case wthn θ nr > 0. 10

13 Constrant (49) states that the servce wthn C n wth the hghest vaue of U nr (.e., U nr ) s chosen, and constrant (50) mpes that ony one servce can be seected. Thus, the servce C n such that max θnr 0{m nr } and max θnr <0{ nr } s the one chosen. If severa C n acheve U nr, we must choose the one wth the hghest contrbuton to the objectve functon,.e., wth the argest { n + θ nr U nr γ nr }. Fgure 4: Choce subprobem Znr(θ, c γ) = max 1 1 R 10 k n w nr + θ nr U nr γ nr w nr (47) >0, C n C n s.t. nr U nr m nr, C n, (48) 1 f = arg max w nr = n, Cn {U nr },, (49) 0 otherwse, w nr = 1. (50) The agorthm to compute Z C nr(θ, γ), based on the above observatons, s descrbed n Fgure 5. Fgure 5: Agorthm to sove the choce subprobem Input: θ nr, γ nr m nr f θ nr 0 Set U nr = C n nr otherwse Obtan where U nr = max Cn U nr s acheved Compute = arg max Cn { n + θ nr U nr γ nr } 1 1 Output: Znr(θ, c R 10 γ) = k n + C n θ nr U nr γ nr, f 0, C n θ nr U nr γ nr, f = Prce subprobem The prce subprobem for each customer n s defned n Fgure 6. For each draw r, the genera dea s to cacuate the contrbuton to the objectve functon for each servce when t s chosen (v nr = 1) and when t s not (v nr = 0). Then, the servce wth the hghest contrbuton for that draw s chosen, and the assocated prce s computed. 11

14 Fgure 6: Prce subprobem Z p n (θ, γ) = max s.t α R 10 k nr (51) >0, C n r 1 θ 10 k nr β n 2 λ n + γ nr v nr C n r n + 2 λ n m n, > 0, C n, (52) α nr λ n, > 0, C n, r,, (53) α nr v nr, > 0, C n, r,, (54) v nr = 1, r. (55) Unchosen servce If v nr = 0, then α nr = 0 (due to constrant (54)). The contrbuton to the objectve functon for servce and draw r s ζ 0 1 nr (θ, γ, λ) = 10 θ nrβ k n 2 λ n. (56) We can consder three cases to characterze ζ 0 nr based on the vaues of θ nrβ n : 1. If θ nr β n < 0, then ζ 0 nr s aways postve, and snce the objectve functon s to be maxmzed, we obtan λ n = 1 such that 2 λ n = m n n (the remanng λ n are equa to 0). 2. If θ nr β n > 0, then λ n = 0, snce ζ 0 nr s aways negatve and we are maxmzng the objectve functon. 3. If θ nr β n = 0, then ζ 0 nr = 0, so λ n can take any vaue. We merge ths case wthn case 2, so that λ n = 0. We note that for = 0, ζ 0 nr (θ, γ, λ) s not defned, snce the prce varabes are ony defned for > 0. However, as we have set β n = 0 for = 0, we can consder a contrbuton of 0 for = 0, and merge ths possbty under case 3. By takng a possbtes nto account, we defne ζ 0 nr (θ, γ) as foows: ζ 0 nr (θ, γ) = 1 θ 10 k nr β n (m n n ), f θ nr β n < 0, 0, f θ nr β n 0 = 0, C n, r. (57) 12

15 Chosen servce If v nr = 1, then α nr = λ n (due to constrant (53)). The contrbuton to the objectve functon for servce and draw r s ζnr(θ, 1 γ, λ) = 1 ( 1 ) 10 k R θ nrβ n 2 λ n + γ nr. (58) We can consder three cases to characterze ζ 1 nr based on the vaues of { 1 R θ nrβ n }: 1. If 1 θ R nrβ n > 0, then ζnr 1 s aways postve (γ nr 0), and snce the objectve functon s to be maxmzed, we obtan λ n = 1 such that 2 λ n = m n n. 2. If 1 θ R nrβ n < 0, then λ n = 0 so that ζnr 1 takes the hghest possbe vaue (.e., γ nr). 3. If 1 θ R nrβ n = 0, then ζnr 1 = γ nr and λ n can take any vaue. We merge ths case wthn case 2, so that λ n = 0. We note that for = 0, ζ 1 nr (θ, γ) = γ nr (the other term s not defned). By takng a possbtes nto account, we defne ζnr 1 (θ, γ) as foows: ζ 1 nr(θ, γ) = ( k R θ nrβ n )(m n n ) + γ nr, f 1 R θ nrβ n > 0, γ nr, f 1 R θ nrβ n 0 = 0, C n, r. (59) We can wrte the prce subprobem n Fgure 6 n terms of ζ 0 nr and ζ1 nr. The resutng formuaton s defned n Fgure 7. Ths probem s easy to sove. For each draw r, the chosen servce s the one wth the hghest contrbuton to the objectve functon,.e., r = arg max ζnr 1 (θ, γ). The compete agorthm s descrbed n Fgure 8. Fgure 7: Prce subprobem n terms of ζ 0 nr and ζ1 nr Z p n (θ, γ) = max s.t. ζ 0 nr (1 v nr) + ζnrv 1 nr (60) C n r v nr = 1, r. (61) C n 13

16 Fgure 8: Agorthm for the prce subprobem Input: θ nr, γ nr C n, r Compute ζ 0 nr and ζ1 nr Sove the probem n Fgure 7 by computng r = arg max Cn ζ nr 1 r 1 f = r Set v nr =, r. 0 otherwse Output: Zn p (θ, γ) = r ζ 1 rnr + r r ζ 0 nr 3.3 Lagrangan dua As mentoned n Secton 3.1, the souton of the Lagrangan reaxaton provdes a ower (upper) bound of the nta mnmzaton (maxmzaton) probem. Furthermore, n the case of (convex) near programs, the optma souton of the Lagrangan subprobem concdes wth the optma souton of the nta probem. In ths case, the Lagrangan subprobem presented n Fgure 3 s wrtten as Z(θ, γ) = Znr(θ, c γ) + Zn p (θ, γ) θ nr b nr. (62) n r n n C n r Snce the orgna probem s a maxmzaton probem, the Lagrangan dua s defned as mn Z(θ, γ). (63) θ,γ The Lagrangan dua can be approxmated by an teratve method that updates the vaues of the Lagrangan mutpers by sovng the so-caed reaxed prma probem (RPP), whch s the Lagrangan subprobem for the gven vaues of the mutpers. The number of teratons depends on the desred accuracy of the resut, and t can be set by the anayst or can obey a concrete stoppng crteron. There are dfferent procedures for updatng the Lagrangan mutpers. Here we consder the subgradent method, because t s smpe to mpement and ts computatona burden s sma. Fgure 9 shows the agorthm apped to our case. In step 1, we ntaze the vaues of the Lagrangan mutpers. In step 2, the Lagrangan subprobem (62) s soved for the gven vaues of the mutpers, obtanng vaues for the varabes of the probem. The mutpers are updated n step 3 by cacuatng a subgradent of the functon Z(θ, γ) at the mutpers of the current 14

17 teraton (k). The subgradents are obtaned as foows: g k nr = Uk nr β 1 n 2 λ k 10 k n b nr C n, n, r, (64) h k nr = vk nr wk nr, n, r. (65) If they are equa to 0, the optma souton s Z(θ k, γ k ). If not, the mutpers are updated. To do so, the step sze needs to be computed. Many dfferent types of step sze rues are used. For the sake of smpcty, we consder a constant step sze (δ). As the mutpers assocated wth equaty constrants are free, the update s not restrcted (.e., θnr k+1 = θnr k + δ gk nr, C n, n, r, k), whereas for the mutpers assocated wth ess or equa constrants, we have to ensure that they are postve. Thus, the mutpers take the maxmum vaue between 0 and the vaue of the update (.e., γ k+1 nr = max{0, γnr k + δ hk nr },, n, r, k). Fgure 9: Subgradent method 1. Intazaton: set k = 0 and choose θ 0 nr C n, n, r and γ 0 nr, n, r; 2. Souton of the RPP: sove Z(θ k, γ k ) and obtan vaues for the varabes Unr k, wk nr, λk n, αk nr and v k nr where t s acheved; 3. Update the mutpers: choose subgradents g k, h k of the functon Z(θ, γ) at θ k and γ k (g k assocated wth the reaxaton of the utty constrant and h k assocated wth the copy constrant). The subgradents g k, h k are obtaned wth (64) (65) ; f (g k, h k ) T = 0 then ese end the optma souton s Z(θ k, γ k ), stop; compute θnr k+1 = θnr k + δ gk nr, C n, n, r; compute γnr k+1 = max{0, γnr k + δ hk nr },, n, r; ncrement k and go to step 2; The subgradent method provdes an upper bound of the orgna probem Fgure 2. In order to obtan a ower bound, we can foow these steps: 1. Fx the prces of the prce subprobem. 2. Compute the choce of each ndvdua based on the fxed prces. 3. Compute the orgna objectve functon vaue based on the prces from step 1 and the choces from step 2. 15

18 4 Concusons and future work The Lagrangan reaxaton technque descrbed n Secton 3 has to be evauated and compared wth the resuts obtaned wth the exact method. We note that n the case study consdered for the exact method, the servces were assumed to be accessbe to a customers at an operator eve. Thus, both approaches can be safey compared, snce the feature enabng the operator to open or cose a servce dependng on the upcomng customer s deactvated. Despte the smpcty of the descrbed technque, the upper bound provded by the Lagrangan subprobem s restrcted, n the sense that the prce varabes p n and the utty varabes U nr can ony take the assocated extreme vaues. In practce, however, ths s not the case and ntermedate prce eves and vaues of the utty are acheved. Hence, the proposed technque can be modfed n order to defne more compex subprobems, so that ess mportance s paced n the subgradent method. One possbty to be expored s to dupcate the prce varabes nstead of the choce varabes, snce as soon as the former are fxed, the uty vaues can be easy computed, and so the choces performed by the customers. The extenson of the method to the capactated case s smpe. Indeed, we notce that ony the choce subprobem has to be adapted, as the capacty constrants determne the set of avaabe aternatves to choose from. Snce the prorty st determnng the order n whch customers are processed s assumed to be known, t s possbe to terate over the customers n the gven order (the subprobem now ony decomposes by r), and track the remanng capacty of each aternatve. As soon as an aternatve has reached ts capacty, t s not offered anymore to the upcomng ndvduas. Ths s done by updatng the assocated ndvdua choce sets C n. As a future avenue of research, n case the dfference between the upper and ower bound for the orgna probem s sgnfcant, coumn generaton can be used to defne an exact method to sove the Lagrangan dua. The attractveness of coumn generaton s to work ony wth a suffcenty meanngfu subset of varabes, and to add more varabes ony when needed. 16

19 5 References Conejo, A. J., E. Casto, R. Mnguez and R. Garca-Bertrand (2006) Decomposton technques n mathematca programmng: engneerng and scence appcatons, Sprnger Scence & Busness Meda. Ibeas, A., L. de Oo, M. Bordagaray and J. Ortúzar (2014) Modeng parkng choces consderng user heterogenety, Transportaton Research Part A: Pocy and Practce, 70, 41 49, ISSN Pacheco, M., S. S. Azadeh and M. Berare (2016) A new mathematca representaton of demand usng choce-based optmzaton method, paper presented at the Proceedngs of the 16th Swss Transport Research Conference, Ascona, Swtzerand. Pacheco, M., S. S. Azadeh, M. Berare and B. Gendron (forthcomng) Integratng advanced dscrete choce modes n mxed nteger near programmng, Technca Report, Transport and Mobty Laboratory, Ecoe Poytechnque Fédérae de Lausanne. 17