Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School

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1 Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management n publc storage warehouses. We optmze the expected revenue of publc storage warehouses aganst the worst cases wth a max-mn revenue objectve, and the decson varables are manly the number of storage unts for each storage type. Wth the robust desgn, we can observe worst-case revenue mprovement. Introducton Publc storage warehousng s a mature ndustry n the U.S. and a fast-growng busness n Europe and Asa Pacfc. As the SSA (Self Storage Assocaton, the offcal assocaton of ths ndustry n the U.S.) reported, the number of such facltes has clmbed from only 6,60 facltes at year-end 984 to over 46,500 at year-end 200. In the year of 200, the total revenue of ths ndustry n the U.S. was $22 bllon (USD). Ths busness has become one of the fastest-growng sectors of the Unted States commercal real estate ndustry over the last 35 years (see []). The ndustry n Europe had boomed after a breathtakng growng from 2006 to Accordng to the 2008 ndustry annual report of the SSA UK (Self Storage Assocaton of the UK, the offcal assocaton of ths ndustry n the UK), the number of publc storage warehouse facltes n Europe had sharply ncreased from 2007 to 2008, partcularly n those emergng markets. For example, the ndustry had ncreased by 200% n Poland, 7% n Swtzerland, 64% n Denmark (see [2]). Publc storage warehousng usually operates as self-storage mode and thereby ts cost s low and stable. For publc storage warehouses wth an objectve to

2 maxmze the expected revenue and wth a stable cost level, revenue management rather than cost control s crtcal. A typcal publc storage warehouse contans storage spaces of dfferent storage types, each type wth a specfc number of storage unts. A customer rents one or multple storage unts of an approprate type for several months. However, the exstng storage types or the avalable number of storage unts for each type may not ft the needs of the market. The number of avalable storage unts of some types may be nsuffcent, whle other types are abundant. Ths results n ether lost customers and revenue, or neffcent utlzaton of capacty of one type, whch also may brng potental loss n another type. The man faclty desgn queston s to provde a faclty desgn mprovng the expected revenue that s wth a better ft between storage desgn (types and numbers) and market demand. Based on data of warehouses n Amerca, Europe and Asa, Gong et al.[3] propose a faclty desgn approach for publc storage warehouses wth three dfferent cases: an overflow customer rejecton model, and two models wth customer upgrade possbltes: one wth reservaton and another wthout reservaton. They analyze models for real warehouse cases manly by stochastc models and dynamc programmng (see [3], [4]). Whle these methods assume a probablty dstrbuton or a specfc stochastc process of demands, t s dffcult to accurately estmate the probablty dstrbuton of the total arrvng storage requests or assocated revenue n some warehouses. To provde a method to solve the practcal problem, we present robust optmzaton models to handle problems wth uncertan marketng data and parameters whch cannot be accurately estmated. Robust optmzaton s a developng lterature body recently, whch plays an mportant role n revenue management (see [5]). Robust optmzaton s a method for modelng optmzaton problems under uncertanty, to fnd optmal decsons for the worst-case realzaton of the uncertantes wthn a gven set. Typcally, the orgnal uncertan optmzaton problem s converted nto an equvalent determnstc form (called the robust counterpart) usng strong dualty arguments and then solved usng standard optmzaton algorthms. Soyster [6], Ben-Tal and Nemrovsk [7], and Bertsmas and Sm [8] descrbe the method for constructng robust counterparts for uncertanty sets of varous structures. Robust optmzaton has been appled n varous felds ncludng supply chan management, fnance, and logstcs. Goh and Sm [8] develop ROME, an algebrac modelng toolbox desgned to solve a class of robust optmzaton problems n the MATLAB envronment, to solve applcaton problems. AIMMS provdes robust optmzaton add-on, wth the ablty of handlng wth mult-perods optmzaton problems wthout fallng nto the trap of the curse of dmensonalty. Revenue management (see [0]) s a man concern of publc storage warehouses. Robust optmzaton plays an mportant role n revenue management

3 (see [4]). We optmze the expected revenue of publc storage warehouses aganst the worst cases wth a max-mn revenue objectve, and the decson varables are manly the number of storage unts for each storage type. Ths research s manly n the feld of faclty desgn (see []). Rosenblat and Lee[2] propose a robust approach to faclty desgn. Ths paper makes contrbuton to faclty plannng (see [] ) and proposes a faclty desgn method to handle rsks and uncertantes and wth an objectve to ncrease revenue. 2 Models The man research problem s to determne how many storage type unts should be constructed to ft demand. There are m dfferent storage type unts wth a specfc nteger storage sze area c, m. We assume that the random arrval processes of customers requestng a storage type unt, mare ndependent Posson processes wth arrval rates λ and that the occupaton tmes of a storage type unt are ndependent and dentcally dstrbuted random varables wth expected occupaton tme β. Customers requestng a storage type unt are called type customers. When a type customer arrves and storage type unts are not fully occuped, the customer wll be accepted at a prce of r per perod for a storage type unt. When all storage type unts are occuped, customers who ntally are nterested n a storage type unt, may accept wth probablty p to pay a prce of r + for a storage type + unt. A customer wllng to accept ths s called an upscaled customer. If a storage type + unt s avalable, an upscaled customer wll be served, otherwse the customer s lost. Let x, mbe the number of storage type unts bult for type customers at level and y, m the type + unts reserved for type customers upscaled to level +. A type customer who fnds upon arrval all the x storage type unts occuped may choose to be upscaled and use one of the y reserved unts, f one s avalable. If all y unts are occuped, the customer s lost. Customers of type m fndng upon arrval all xm unts busy are drectly lost. A customer of type + s not allowed to occupy one of the y unts reserved for upgraded type customers. Each rejected type customer s wllng to upscale wth probablty p.

4 To analyze ths model we frst look at the process followed by each type of customers separately. The long-run average revenue obtaned from type customers can be splt n the long-run average revenue obtaned from the x and y unts. The number of occuped storage type unts among the avalable x can agan be modeled by a queueng loss system. Due to our assumpton of exponentally dstrbuted occupaton tmes ths s an M / M / x / x loss queue. The long-run average revenue obtaned from the x unts s gven by long-run average revenue obtaned from the y unts s m rη+ ( x) β( Pr( x, y) ), where η ( x ) p (, ) λ B x ρ = m rρ ( B( x, ρ )). The = + = and r(, ) P x y the rejecton probablty that an upscaled type customer wll fnd all the reserved y unts occuped. Let Λ be the set of demand data for all storage types of D quarters n the examnng years, we calculate the worst performance as m m mn λ rρ ( B( x, ρ )) + rη ( x ) β P x, y ( ( )) d Δ, d=,..., D + r = = Gven the prce r for each storage type unt per unt of tme, the motvaton of robust desgn s to mnmze the loss n revenue due to varaton n demand, and to decde how many unts of each type to buld (and reserve) such that we can fnd the best one among the worst performances, and we present a robust model as follows, m m max mn λ,,..., ( (, )) ( ),... d Δ d= D rρ B x ρ + rη+ x β Pr x y R = = m st.. ( cx + c y) + c x C = x, y Z + m m + ( ( )) ( ) 3 Analyss We apply the followng random search robust optmzaton algorthm to calculate the desgn and solve model (R).

5 Algorthm: robust optmzaton for PS warehouses (I) For t = : T. (I.) Generate a sample constrant m cx C. = t x from (, ) U x x wth consderaton of the capacty (I.2)(A) For d = : D, compute the objectve value R ( x t ) from the pattern. (I.2)(B)Compute the worst performance of ths generated desgn t t patterns R mn ( Rd ( x ), d : D) (II)Compute R = max ( R t, t = : T) = =. (III) Return x correspondng to R as the robust desgn. d th d demand t x n D demand We take the S.P. Rotterdam warehouse as an example, to llustrate how to provde a robust desgn, such that a warehouse can acheve the best revenue performance among the worst revenues from demand data of D quarters. Table : Parameters of S.P. Rotterdam warehouse Items Class Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Types(m 2 ) Prce (Euro) Old desgn We use optmal results from Table 3 n [3], whch s by a senstvty analyss of S.P. Rotterdam warehouse desgn based on data of 8 quarters, to construct the search range ( x, y),( x, y ) for problem (R). For teratons t = : T, we generate τ t samples ( x, y ), τ =,...,t from the search range whle consderng the capacty constrant. For each of the samples ( x, y) τ we compute ts revenues for D dfferent demand patterns, and obtan the worst value τ R = mn { R (( ), ),,..., d x, y d d D} τ λ = for ths desgn sample. The optmal revenue correspondng to the robust desgn s now Rt ( ) = max { R τ, τ =,..., t} for teraton t. We can then return the ( x, y) value correspondng to R( t) as the robust desgn for teraton t. The theoretcal optmal robust revenue s R = lm Rt ( ). t

6 We present the searchng process { R(), t t : T} = n Fg.-. By settng the convergence crteron as Rt ( + ) Rt ( ) Rt ( ) 0.3%, we observe the search converges n the last 500 teratons. We therefore calculate the average value from the last 500 values { Rt ( ), t= 2500 : 3000}, and obtan the robust optmal revenue value as 4392, whch s a sgnfcant mprovement compared wth the current worst revenue value based on the current desgn. We use the desgn obtaned after 3000 teratons {50(6), 9(5), 9(6), 46(0), 2(5), 25(5), 9(4), 9(0)} as the robust desgn. Fgure : Robust optmzaton for S.P. Rotterdam warehouse 4 Concludng remarks Ths paper presents a robust desgn method for publc storage warehouses by applyng robust optmzaton and revenue management. In regard to avalable nformaton, t needs to further consder two stuatons: () Some warehouses mantan ncomplete market data. They cannot accurately estmate customer demand, ncludng average order arrval rates. But they can roughly provde a scope of customer demand by hstory data. Ths happens n small and mddle faclty provders and some large-scale companes wth relatvely unqualfed nformaton management. (2) Some warehouses can provde more market data. They can estmate customer demand nformaton, ncludng average order arrval rates. They cannot accurately specfy some parameters of ther market demand models; nstead

7 they can roughly provde a scope of parameters by hstorcal data. Ths happens n some world-class faclty provders (lke Shurgard) and some mddle-scale companes wth good nformaton management systems. Methods and technologes of robust optmzaton need mprovement n the further research. The lmtaton of random search robust optmzaton algorthm s n searchng speed and accuracy. It s nterestng to explore better algorthms for robust desgn of large scale publc storage warehouses. Acknowledgements Yemng Gong acknowledges the support from NSFC ( ). References [] Self Storage Assocaton. Self Storage Assocaton Fact Sheet, Alexandra, VA, USA (20). [2] Self Storage Assocaton UK, Annual Industry Report, Self Storage Assocaton of Unted Kngdom, Cheshre, UK (2008). [3] Gong, Y. de Koster, R., Gabor, A., Frenk, J.B.G., Increasng revenue of selfstorage warehouses by faclty desgn, EMLYON workng paper (20). [4]Gong Y. Stochastc modellng and analyss of warehouse operatons, Erasmus Unversty, Rotterdam, the Netherlands (2009). [5]Brbl, S.I, Frenk, J.B.G., Gromcho, J.A.S., Zhang, S. The role of robust optmzaton n sngle-leg arlne revenue management. Management Scence, 55,, (2008). [6]Soyster, A. L. Convex programmng wth set-nclusve constrants and applcatons to nexact lnear programmng. Operatons Research, 2, 5, 54 57(973). [7]Ben-Tal, A. and Nemrovsk. A. Robust convex optmzaton. Mathematcs of Operatons Research, 23, 4, (998). [8]Bertsmas, D. and Sm M.. The prce of robustness. Operatons Research, 52,, 35 53(2004).

8 [9]Goh J. and Sm M.. Robust Optmzaton Made Easy wth ROME. Operatons Research, 59(4), (20). [0]Tallur, K.T., Van Ryzn, G.J. The theory and practce of revenue management. Kluwer, Boston (2004). []Tompkns, J.A., Whte, J.A., Bozer, Y.A., Tanchoco, J.M.A. Facltes plannng. John Wley & Sons, USA (2003). [2]Rosenblat, M.J. and Lee H.L. A robust approach to faclty desgn. Internatonal Journal of Producton Research, 25(4), (987).