Capacitated Assortment Optimization with Pricing under the Paired Combinatorial Logit Model

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1 Capactated Assortmet Optmzato wth Prcg uder the Pared Combatoral Logt Model Daha Zhag 1, Zheghe Zhog 1, Chug Gao 1, Ru Che 2, 1 Sparzoe Isttute, Beg, 10084, Cha 2 Departmet of Idustral Egeerg, Tsghua Uversty, Beg , Cha E-mal Address: jacdaha.zhag@outloo.com (Daha Zhag) @qq.com (Zheghe Zhog) @163.com (Chug Gao) cher15@mals.tsghua.edu.c (Ru Che) Abstract - I ths paper, we vestgate the capactated assortmet optmzato problem wth prcg uder the pared combatoral logt model, whose goal s to detfy the reveue-maxmzg subset of products as well as ther sellg prces subject to a ow capacty lmt. We model customers purchase behavor usg the pared combatoral logt model, whch allows for covarace amog ay par of products. We formulate ths problem as a o-lear mxed teger program. The, we propose a two-step approach to obta the optmal soluto based o solvg a mxed teger program ad Lambert-W fucto. To further mprove ts performace, we desg a greedy heurstc algorthm ad a greedy radomzed adaptve search procedure to obta hghqualty solutos so as to balace the tradeoff betwee accuracy ad computatoal effcecy. A seres of umercal expermets are coducted to gauge the effcecy ad qualty of our proposed approaches. Keywords - assortmet optmzato, prce optmzato, pared combatoral logt, reveue maagemet I. INTRODUCTION I the feld of reveue maagemet, retalers who sell several categores of products, usually have flexblty o determg the subset of products to offer as well as ther sellg prces to some extet. Therefore, gve a ow capacty costrat to lmt the total shelf space cosumpto or the total costs of troducg products to the store, these retalers face the capactated assortmet optmzato problem wth prcg, referred to as the CAOPP, to maxmze the expected reveue obtaed from customers. Tae a home applace store as a example, the retaler has to choose a reveue-maxmzg subset of televsos, refrgerators, ad other products wth ther correspodg prces of dfferet szes to dsplay as the total shelf space s lmted. To tacle ths problem, researchers maly troduce varats of dscrete choce models, especally the multomal logt ad the ested logt models [see, e.g., 1-3], to capture customers purchase behavor as well as the substtuto amog products. Uder the multomal logt model, several effcet algorthms are proposed to solve the correspodg CAOPP wth dfferet sde costrats [1-2, 4]. Referece [5] characterzes the equlbrum a compettve evromet wth multple retalers. Although aalytcally coveet, the multomal logt model suffers from the depedece of rrelevat alteratve X property, whch s capable of hadlg wth the substtuto amog products [6]. Researchers, therefore, resort to the ested logt model developed by [7]. A samplg of these research o the CAOPP cludes [3] ad [8-10]. We refer terested readers to [11] for study wth ra-based oparametrc model. I ths paper, we vestgate the CAOPP uder the pared combatoral logt (PCL) model, whch has bee prove to be a effectve model capturg the decso mag process. There are several beefts of the PCL model: (a) The PCL model s cosstet wth the radom utlty maxmzato prcple; (b) The PCL model allows for covarace amog ay par of products, whch leads to a more accurate represetato of choce settg wthout specfyg a structural sequece; ad (c) Most, f ot all, of extat emprcal studes pot out that the PCL model outperforms the multomal logt ad the ested logt models predctg users route choce [see, e.g., 12-15]. However, there are very scarce applcatos volvg the PCL model the feld of reveue maagemet. Referece [16] s the frst to study the prcg problem uder the PCL model. The authors show that the uqueess of the optmal prces ca be acheved uder certa codtos based o the cocept of P-matrx ad develop the correspodg soluto algorthms. Recetly, Referece [17] shows that the assortmet optmzato problem uder the PCL model s NP-hard eve whe there are o capacty costrats ad develop a geeral approxmato framewor based o teratve varable fxg ad coupled radomzed roudg. Later, Referece [18] proposes aother approxmate algorthm based o approxmatos to the apsac problem whe there s a ow capacty costrat. As far as the authors have cocered, o lterature has ever cotrbuted to the CAOPP uder the PCL model, ad our research flls ths bla. We frst formulate ths problem as a mxed teger program volvg a o-lear objectve fucto. The, we propose a two-step approach to obta the optmal soluto based o the observato that the optmal prces for all offered products are detcal. I the frst step, we solve a o-lear auxlary problem ad provde ts equvalet teger program, whose lear programmg relaxato provde a umercally tght upper boud. I the secod step, we obta the optmal prces for all products based o Lambert-W fucto ad the output the frst step. To further mprove ts emprcal performace, we apply a greedy heurstc algorthm ad a radomzed

2 adaptve search procedure (GRASP) to obta hgh-qualty solutos. Numercal expermets dcate that our greedy heurstc algorthm s qute effcet. The optmalty gap of the solutos obtaed by the heurstc algorthm s less tha 0.18% o average ad 0.46% at the worst case over 300 radomly geerated problem staces. We summarze the cotrbutos of ths research as follows: (a) To the best of authors owledge, we are the frst to vestgate the capactated assortmet optmzato problem wth prcg uder the pared combatoral logt model; (b) We obta the optmal assortmet as well as the optmal prces through solvg a mxed teger program ad a Lambert-W fucto; (c) To further mprove ts performace, we proceed to develop a greedy heurstc algorthm ad a GRASP to obta hgh-qualty solutos wth reasoable tme cosumed; ad (d) We demostrate the effcecy ad qualty of our approaches through a seres of umercal expermets. The remader of ths paper s orgazed as follows. I Secto II, we formulate the CAOPP uder the PCL model as a mxed teger program. I Secto III, we develop a two-step approach based o a teger program to obta the optmal soluto, the upper boud of the orgal optmzato problem, ad to develop a greedy heurstc algorthm as well as a GRASP to obta hgh-qualty solutos qucly. I Secto IV, we coduct a seres of umercal expermets to gauge the performace of our approaches. Fally, we coclude ad outle potetal future research drectos Secto V. II. MODELING FRAMEWORK I ths secto, we formulate our jot capactated assortmet ad prce optmzato problem uder the PCL model. Cosder a retaler, who has eough flexblty o determg the subset of products as well as ther sellg prces out wth lmted dsplay space for offered products. Wthout loss of geeralty, we assume that the set of all possble products to offer s dexed by N = {1,2,, }. For each product N, let p be ts sellg prce ad w be ts weght, or requred dsplay space. Moreover, we use x = 1 to dcate that product s offered; otherwse, x = 0. Gve a capacty lmt, deoted by C R +, to lmt the total space cosumpto of the offered products, the retaler wshes to fd the reveue-maxmzg subset of products as well as ther sellg prces. I ths paper, we assume that customers choce behavor s captured by the PCL model, whch ca accurately capture the substtuto effects amog products [16-18]. Throughout ths paper, the determstc utlty of product s captured through α βp, where α s a ow costat to measure the qualty of ths product, ad β > 0 s the prce-sestvty parameter to capture the utlty varato whe the sellg prce chages. Uder the PCL model, products are parttoed to several ests whch cota exact two products, ad we use, j ( < j) to deote the est wth oly products ad j. Let p = (p 1, p 2,, p ) ad x = (x 1, x 2, x ). We use γ (0,1] to deote the dssmlarty parameter assocated wth est, j, the choce probablty of purchasg product ca be calculated through q = v 1/γ (p )x /V (p, x), where v = exp(α βp ) s the preferece weght of product 1/γ 1/γ ad V (p, x) = v (p )x + v j (p j )x j s the total preferece weghts of est, j. Moreover, the coveto that q = 0 s adopted f x = x j = 0. The, the expected reveue that we obta from ths customer s R (p, x) = p q + p j q j. Furthermore, the choce probablty q that a customer wll mae a purchase uder est, j s q = γ 1 γ V (p, x)/(1 + =1 j=+1 V (p, x)). Fally, the choce probablty of product, deoted by q (p, x) wth gve x ad p, s q (p, x) = q j q, ad the opurchase opto, whch meas the customer leaves wthout 1 purchasg aythg, s q 0 = 1 =1 j=+1 q. As a result, the CAOPP uder the PCL model ca be formulated as the followg o-lear mxed teger program: 1 * z max R( p,x)= p q p q q 1 1 j1 s.t. 1 w x 1 1 j1 1 1 j1 V ( p,x)r ( p,x) 1 V ( p,x) (1) C, (2) p 0, x {0,1}, N. (3) The objectve fucto maxmzes the expected reveue over all products. Costrat (2) esures that the total weght cosumpto of offered products wll ot exceed the capacty lmt. Costrats (3) restrct the decso varables. III. SOLUTION APPROACH I ths secto, we preset our soluto approaches to obta the optmal soluto as well as hgh-qualty solutos to the problem defed (1)-(3). The startg pot of our soluto approaches s a result from the observato that wth ay gve feasble assortmet x that satsfes Costrat (2), the jot optmzato problem reduces to the mult-prce optmzato problem uder the PCL model, whch s well studed [16]. I the prce optmzato problem, [16] proves that the optmal prces are detcal for all products. That s, lettg p (x) be the optmal prce for product, we have that p (x) = p (x) holds for ay N. Furthermore, the optmal expected reveue wth gve x ca be characterzed usg p (x), whch leads to the followg lemma. Lemma 1. Wth gve feasble assortmet x, the optmal prce to maxmze R(p (x), x) ca be represeted by * 1 W( A( x) / e) p ( x), (4)

3 where A(x) s calculated trough 1 A( x ) (exp( / ) x exp( / ) x ). (5) 1 j1 j j Ad the optmal expected reveue s R(p (x), x) = W(A(x)/e)/β, where W(y) s the Lambert-W fucto to solve we w = y for w. The proof of Lemma 1 s omtted here to save space sce t s a drect geeralzato of Theorem 1 [16] accoutg for gve assortmet x. Due to the fact that R(p (x), x) = p (x) 1/β, maxmzg the expected reveue s equvalet to maxmzg p (x). So, for ay gve assortmet x, the optmal prces for all products are detcal ad ca be calculated by (4), volvg Lambert- W fucto. Sce Lambert-W fucto s a strctly creasg fucto (0, + ), the optmal soluto to maxmze A(x) subject to a capacty costrat must be the optmal assortmet to the CAOPP. That s, we ca further propose a two-step procedure to obta the optmal soluto to (1)-(3): I the frst step, we fd the feasble assortmet x wth the largest value of A(x ) subject to the capacty lmt; I the secod step, we calculate p (x ) based o Lemma 1. As a result, (p (x ), x ) must be the optmal soluto to the CAOPP. So, all that remas s to fd x, whch ca be obtaed by solvg the followg o-lear teger program: 1 max A( x) (exp( / ) x exp( / ) x ) (6) 1 j1 s.t. w x C; x {0,1}, N. 1 j j To solve the above problem, we frst learze the objectve fucto by usg the characterstc of the PCL model that there are exact two products uder each est. To do so, we follow the smlar approach used [17-18]. Lettg ρ = (exp(α /γ ) + exp (α j /γ )) γ ad θ = exp(α ), A(x) ca be re-wrtte as A(x) = 1 =1 j=+1 ρ x x j + θ x (1 x j ) + θ j (1 x )x j = 1 =1 j=+1 μ x x j + θ x + θ j x j, where μ = ρ θ θ j < 0. Replacg the product x x j wth y 0, we ca esure that y = 1 f ad oly f x = x j = 1 by troducg addtoal costrats 1 + y x + x j, x y, ad x j y. Usg the fact that μ < 0, t s show [17] that the latter two costrats are redudat. Now, we ca formulate our mxed teger program wth lear objectve fucto to solve (6) as follows: 1 max A( x) y x x (7) 1 j1 s.t. w x C; 1 j j 1+ y x x, j N; j y 0, x {0,1}, N. The advatage of ths formulato s that ay MIP software pacage ca be mplemeted to obta the optmal assortmet x wth gve capacty lmt. However, solvg a MIP to optmalty ca be qute tme-cosumg whe there are thousads of products to cosder sce t s NPhard. Ths complexty result prompts us to develop effcet heurstc algorthms to obta hgh-qualty solutos to (7) qucly. We frst use the lear relaxato of the MIP (7), whose optmal objectve fucto s deoted as A(x), to boud the value of A(x ), whch s used to gauge the qualty of the solutos obtaed by our heurstcs. Sce the optmal expected reveue s a o-decreasg wth respect to A(x), we ca have p (x) = (1 + W(A(x)/ e)))/β > p (x ) ad R = p (x) 1/β > R(p (x ), x ). That s, we ca use the lear relaxato of the MIP (7) ad Lemma 1 to obta a upper boud of CAOPP. I our basc heurstc algorthm, a feasble soluto s costructed as follows: (a) Sort the caddate products decreasg order accordg to the values of { exp(α ) : N}; (b) Greedly add the feasble products, satsfyg the capacty lmts, to the curret assortmet utl there s o feasble products that ca be added to the assortmet. As a result, we ca obta a feasble soluto, deoted as x H, wth O(log + ) = O(log) tme. To further mprove the qualty of the obtaed heurstc soluto, we troduce a meta-heurstc algorthm based o a Greedy Radomzed Adaptve Search Procedure, referred to as GRASP. I our GRASP, a feasble soluto s radomzed costructed based o our greedy heurstc ad local perturbatos are followed to get a local optmal soluto. To do so, we troduce a parameter α [1, Alpha] the secod procedure (b) of our heurstc. That s, we radomly add the feasble product the lst composed of the α best dces accordg to the values of exp(α ) /w for ay product, whch has ot bee selected the curret assortmet, utl o feasble products exst. The, the local search phase, we radomly choose two varables x, x j (x x j ) ad flp ther values to zero or oe. If the ew soluto s feasble ad mproves the objectve fucto, we store the chage ad eep t as the curret soluto. Otherwse, we cotue the radom perturbato utl the prescrbed maxmum umber of terato s reached. For ease of readg, the pseudo-code for our GRASP s show Procedure GRASP(Alpha, MaxIter). As a result, we ca obta a feasble soluto, deoted as x GRASP ad we ca have A(x GRASP ) A(x H ) because whe α = 1, the costructo phase s the same wth the heurstc algorthm. Procedure GRASP(Alpha, MaxIter) For α = 1,2,, Alpha do Obta a feasble soluto wth heurstc algorthm Wth parameter α For terato=1,2,, MaxIter do Local search for better solutos by flppg the values of x, x j Ed For Ed For w

4 To sum up, we ca ow solve the MIP (7) to obta the optmal assortmet, apply the greedy heurstc algorthm as well as the GRASP to obta hgh-qualty assortmet wth reasoable tme cosumed, ad solve the lear relaxato of (7) to get a upper boud of A(x). The, for these four approaches, we ca further use Lemma 1 to obta the correspodg optmal prce for all products as well as the correspodg expected reveue obtaed from each customer. Therefore, we ca have R(p (x), x) R(p (x ), x ) R(p (x GRASP ), x GRASP ) R(p (x H ), x H ) based o the dscussos ths secto. Now, we try to test the umercal performace of our approaches the ext secto. IV. RESULTS I ths secto, we compare the umercal performace of proposed algorthms Secto III based o the qualty of the solutos obtaed ad the rug tme cosumed. Throughout our umercal expermets, we use the followg approach for geeratg problem staces. We vary the umber of total products over {400, 600, 800, 1000}. We set the capacty lmt C = κ =1 w, where κ s a fracto of the total space cosumpto of all products ad vares over κ {0.02, 0.04, 0.06}. Ths setup provdes 12 parameter combatos for (, κ). For each combato, we geerate 25 radom problem stace, each of whch we sample the parameters as follows: we sample {exp(α ) : N} from the uform dstrbuto U(0, 5]; the weght w of product s geerated from the uform dstrbuto U[1,10]; ad the scale parameter γ for est, j s geerated from the uform dstrbuto U[0.1, 1] ad γ = γ j. Moreover, we set prce sestvty β = 0.1. The value of Alpha ad the umber of maxmum terato local search used our GRASP are set to 5 ad 80, respectvely. The, for each problem stace, we obta the optmal soluto through solvg MIP ad Lambert-W fucto. We also apply the greedy heurstc algorthm ad the GRASP to obta hgh-qualty solutos. Our umercal expermets are coducted o a Wdows server wth CPU of 2.00 GHz ad RAM of 512 GB. The algorthms are all coded Matlab led to the CPLEX 12.6 optmzato routes. The umercal results for solvg the CAOPP uder the PCL model s summarzed Table I. Colum 1 shows the parameter combato for staces the form of (, κ). Let Opt obj, UB obj, He obj, ad GRASP obj be the expected reveue obtaed by the MIP (7), the lear relaxato of the MIP, the heurstc algorthm, ad the GRASP, respectvely, for problem stace {1,2,, 25} wth gve parameter combato. Colums 2-3 report the average ad maxmum rug tme 1 secod to obta the optmal soluto based o solvg the MIP ad Lemma 1, respectvely. Smlarly, Colums 4-9, respectvely, report the average ad maxmum rug tme 1 secod of obtag a upper boud through lear relaxato of the MIP, a feasble soluto through our greedy heurstc algorthm, ad a soluto through the GRASP. Colums report the average ad maxmum optmalty gap wth MIP, defed as (1 Opt obj ) 100%, UB obj respectvely. The average ad maxmum optmalty gaps wth heurstc algorthm ad GRASP, defed as (1 He obj UB obj ) 100% ad (1 GRASPobj UB obj ) 100%, respectvely, are reported Colums correspodgly. Show Table I, the rug tme for our optmal approach based o MIP creases dramatcally as the umber of products ad κ crease. For combato (1000, 0.06), the maxmum rug tme ca reach secods. O the other had, the heurstc algorthm ca obta a feasble soluto wth secod. Wth a local search phase, the GRASP does eed more tme to termate. Meawhle, solvg a lear relaxato for upper bouds seems to be a practcal way to gauge the qualty of solutos obtaed by other algorthms wth lmted tme TABLE I NUMERICAL RESULTS FOR DIFFERENT APPROACHES Ru. tme (s) Ru. tme (s) Ru. tme (s) Ru. tme (s) Gap (%) wth Gap (%) wth Gap (%) wth Para. comb. for MIP for UB for Heurstc for GRASP MIP Heurstc GRASP (, κ) Avg. Max. Avg. Max. Avg. Max. Avg. Max. Avg. Max. Avg. Max. Avg. Max. (400, 0.02) (400, 0.04) (400, 0.06) (600, 0.02) (600, 0.04) (600, 0.06) (800, 0.02) (800, 0.04) (800, 0.06) (1000, 0.02) (1000, 0.04) (1000, 0.06) Average

5 cosumed. As for the optmalty gaps, we foud that the average optmalty gaps wth MIP over all 300 staces s oly 0.03% ad the maxmum s oly 0.20%. That s, our approach to obta a upper boud of the jot optmzato problem s umercally tght. Meawhle, the optmalty gaps wth our heurstc s 0.18% o average ad 0.46% at the worst case. That s, our greedy heurstc algorthm ca geerate a cosderable hgh-qualty soluto much more effcetly compared to the approach based o MIP. As for the GRASP, the umercal results dcate that t does mprove the soluto qualty, especally, t ca sgfcatly mprove the soluto qualty at the worst case. As we ca see from Table 1, the optmalty gap at the worst case s reduced from 0.46% to 0.31% compared wth the aïve heurstc algorthm. To sum up, our heurstc algorthms are qute effcet to obta hgh-qualty feasble solutos wth reasoable tme cosumed. V. CONCLUSION I ths paper, we vestgate the capactated assortmet optmzato problem wth prcg. The customers purchase behavor s modeled usg the pared combatoral logt model. We frst formulate ths problem as a o-lear mxed teger program ad propose a twostep approach to obta the optmal soluto based o solvg a MIP ad a Lambert-W fucto. To further mprove the umercal performace, we develop a greedy heurstc ad a GRASP to obta hgh-qualty solutos qucly. The umercal expermets dcate that our heurstc ca obta hgh-qualty solutos whose devatos from the upper boud are oly 0.16% o average ad 0.48% at the worst case. We ow outle two possble future research drectos: (a) I our settgs, the prce sestvty s detcal for all products. However, there ca be scearos where customers have dfferet sestvtes over dfferet products. It would be of terest to vestgate the case wth dfferet prce sestvtes; (b) Whe the retaler ca determe the prces ad the assortmet dyamcally over fte sellg perods, our algorthms may ot rema effcet. Therefore, t s of great terest to study the correspodg problem a dyamc evromet. REFERENCES [4] J. Davs, G. Gallego, H. Topaloglu, Assortmet plag uder the multomal logt model wth totally umodular costrat structures, upublshed. [5] O. Besbes, D. Saure, Product assortmet ad prce competto uder multomal logt demad, Producto ad Operatos Maagemet, vol. 25, o. 1, pp , [6] K. E. Tra, Dscrete choce methods wth smulato, Cambrdge uversty press, [7] H. C. Wllams, O the formato of travel demad models ad ecoomc evaluato measures of user beeft, Evromet ad plag A, vol. 9, o. 3, pp , [8] A. G. Ko, Y. Xu, Optmal ad compettve assortmets wth edogeous prcg uder herarchcal cosumer choce models, Maagemet Scece, vol. 57, o. 9, pp , [9] W. Z. Rayfeld, P. Rusmevchetog, H. Topaloglu, Approxmato methods for prcg problems uder the ested logt model wth prce bouds, Iforms Joural o Computg, vol. 27, o. 2, pp , [10] R. Che, H. Jag, Capactated assortmet ad prce optmzato for customers wth dsjot cosderato sets, Operatos Research Letters, vol. 45, o. 2, pp , [11] S. Jagabathula, P. Rusmevchetog, A oparametrc jot assortmet ad prce choce model, Maagemet Scece, vol. 63, o. 9, pp , [12] F. S. Koppelma, C.-H. We, The pared combatoral logt model: Propertes, estmato ad applcato, Trasportato Research Part B: Methodologcal, vol. 34, o. 2, pp , [13] J. Prasher, S. Behor, Ivestgato of stochastc etwor loadg procedures, Trasportato Research Record: Joural of the Trasportato Research Board, o. 1645, pp , [14] A. Che, S. Ryu, X. Xu, K. Cho, Computato ad applcato of the pared combatoral logt stochastc user equlbrum problem, Computers & Operatos Research, vol. 43, pp , [15] A. Karoosootawog, D.-Y. L, Combed gravty model trp dstrbuto ad pared combatoral logt stochastc user equlbrum problem, Networs ad Spatal Ecoomcs, vol. 15, o. 4, pp , [16] H. L, S. Webster, Optmal prcg of correlated product optos uder the pared combatoral logt model, Operatos Research, vol. 65, o. 5, pp , [17] H. Zhag, P. Rusmevchetog, H. Topaloglu, Assortmet optmzato uder the pared combatoral logt model, upublshed. [18] J. Feldma, Space costraed assortmet optmzato uder the pared combatoral logt model, upublshed. [1] K. D. Che, W. H. Hausma, Techcal ote: Mathematcal propertes of the optmal product le selecto problem usg choce-based cojot aalyss, Maagemet Scece, vol. 46, o. 2, pp , [2] R. X. Wag, Capactated assortmet ad prce optmzato uder the multomal logt model, Operatos Research Letters, vol. 40, o. 6, pp , [3] G. Gallego, H. Topaloglu, Costraed assortmet optmzato for the ested logt model, Maagemet Scece, vol. 60, o. 10, pp , 2014.