FINAL REPORT. To: The Tennessee Department of Transportation Research Development and Technology Program

Transcription

1 FINAL REPORT To: The Tennessee Department of Transportaton Research Development and Technology Program Project #: Truck Congeston Mtgaton through Freght Consoldaton n Volatle Mult-tem Supply Chans Prepared by: a Dnçer Konur and b Mhals M. Golas a Department of Engneerng Management and Systems Engneerng Mssour Unversty of Scence and Technology 206 Engneerng Management, 600 W. 14th Street, Rolla, MO Offce: , Fax: , Emal: konurd@mst.edu b Department of Cvl Engneerng, Unversty of Memphs 104 Engneerng Scence Bldg, 3815 Central Avenue, Memphs, TN Offce: , Fax: , Emal: mgkolas@memphs.edu March, 2014

2 Contents INTRODUCTION Explct Transportaton Modelng Shpment Consoldaton Stochastc Jont Replenshment Problem PROBLEM FORMULATION Sngle-Item Tme-based Order-up-to-level Inventory Consoldated Tme-based Order-up-to-level Inventory Consoldaton Decsons SOLUTION ANALYSIS Chromosome Representaton and Intalzaton Ftness Evaluaton Local Search Heurstc for Consoldaton Approxmaton Mutaton Termnaton NUMERICAL ANALYSIS CONCLUSIONS REFERENCES I

3 Lst of Fgures 1 Inventory Level n Tme-based Order-Up-To-Level Control for Sngle Item Lst of Tables 1 Problem Parameters Comparng Soluton Methods for ( P xk ) Comparng Approxmated and Smulated Results for A Gven Consoldaton Comparng Consoldaton Strateges for (P) Comparng Consoldaton wth Multple Truck Types to Sngle Truck Type II

4 INTRODUCTION Transportaton costs consttute the major part of the total costs n many retalng ndustres. Therefore, there has been tremendous amount of studes n the lterature, whch ntegrate nventory control and transportaton decsons as nventory control polces determne how much and how often to shp. There s an obvous trade-off between nventory holdng costs and order setup plus transportaton costs n many practcal scenaros: whle replenshng nventory more frequently wth smaller orders decreases nventory holdng costs, t results n more order setup payments and frequent shpments from the supplers; hence, ncreases order setup and transportaton costs. Consderng the non-lnear nature of nventory related costs, jontly controlled nventores of multple tems, and demand uncertantes, ntegrated nventory control and transportaton problems can be challengng. Ths study focuses on a retaler s ntegrated nventory control and transportaton problem for multple tems, each of whch has ts own stochastc demand. We explctly model nbound transportaton costs by takng nto account that a retaler can use dfferent freght trucks to shp an order. Furthermore, to utlze transportaton capacty better, the retaler can possbly consoldate shpments of dfferent tems. To aval consoldaton, t s assumed that a retaler adopts a tme-based order-up-to-level nventory control polcy, where the retaler replenshes each consoldated set of tems n equal tme ntervals (ths enables jont use of the transportaton capacty by the consoldated tems). The retaler s problem s to fnd the cost mnmzng consoldaton strategy,.e., of whch tems orders are replenshed together, the tme nterval between two consecutve orders of a set of consoldated tems, and the order-up-to-level for each tem wthn a consoldaton. Due to stochastc demand envronment, the retaler s objectve s to mnmze the expected costs. Whle the expected nventory holdng costs, order setup costs, and penalty costs assocated wth shortages are well defned, the dervaton of the expected nbound transportaton costs s cumbersome due to the fact that freght truck choces for each order of a set of consoldated tems are dynamc n nature. That s, the retaler can determne how many trucks of each truck type to be used for each order at order ntaton dependng on the order quanttes of the ndvdual tems n the consoldaton. Ths, n turn, makes the retaler s problem of expected cost mnmzaton a b-level optmzaton model wth nfntely many lower level problems (each one s correspondng to a combnaton of the demands of the tems wthn a gven consoldaton). In ths study, we frst formulate the retaler s problem for a gven consoldaton of tems. Here, a blevel mxed-nteger nonlnear optmzaton problem s modeled, where the retaler decdes on the common replenshment cycle length for the consoldated tems and the order-up-to-level for each tem wthn 1

5 the gven consoldaton. Then, a set parttonng problem s presented to fnd the best consoldaton strategy. As a soluton approach, we frst provde an approxmaton formulaton for a gven consoldaton and solve the approxmated formulaton wth a neghborhood search heurstc. Then, an evolutonary heurstc method s dscussed for the set parttonng problem of nterest. A set of numercal studes are conducted to justfy the approxmaton formulaton and use of heurstc methods. Furthermore, through a set of numercal studes, we demonstrate the cost savngs and envronmental benefts of the proposed tme-based order-up-to-level nventory control wth shpment consoldaton and explct freght trucks modelng n mult-tem stochastc nventory systems. Ths study contrbutes to the nventory control lterature and practce n the followng felds: explct transportaton modelng, shpment consoldaton, and stochastc jont replenshment problem. In the remander of ths secton, we revew the related lterature on each of these felds and explan the smlartes and dfferences along wth contrbutons of our study n the respectve feld. In Secton, the mathematcal models for sngle tem, a consoldaton, and set parttonng problem are formulated. Secton explans the detals of the methods used to solve the resultng problem. Secton documents the results of a set of numercal studes that llustrate the effcency of the heurstc methods and demonstrate the potental cost and envronmental benefts of the proposed consoldaton strategy. Secton summarzes the results and contrbutons of ths study and suggests future research drectons. Explct Transportaton Modelng It s well known that freght trucks s the most common transportaton mode. Majorty of the freght tonnage s shpped by trucks n the U.S. (FHWA, 2012). Two common practces of the truckng are lessthan-truckload (LTL) and truckload (TL) transportaton. Most of the ntegrated nventory control and transportaton studes assume LTL transportaton, where the shpment cost depends on the number (weght or volume) of tems shpped. On the other hand, n TL transportaton, the shpment cost depends on the number of trucks used. In ths study, we assume TL transportaton wth the avalablty of heterogeneous freght trucks for nbound shpment. Partcularly, TL transportaton wth sngle truck type has been studed wthn nventory control models. Aucamp (1982), Lee (1986), Toptal et al. (2003), Toptal and Çetnkaya (2006), Toptal (2009), Toptal and Bngol (2011), and Konur and Toptal (2012) are some of the studes that account for TL transportaton costs explctly n sngle-tem nventory control models. In partcular, smlar to these studes, TL transportaton costs are modeled consderng the per truck capactes and per truck costs n ths study. In mult-tem nventory settngs, there s lmted number of studes assumng TL transportaton. Ben-Khedher and Yano (1994) analyze a mult-tem determnstc jont replenshment 2

6 problem wth truckng costs as well as capacty constrants. They propose a heurstc method to solve the resultng NP-hard problem. In a smlar settng, Kesmuller (2009) analyzes a mult-tem stochastc nventory system wth perodc revew and they account for TL transportaton costs. Specfcally, they propose a perod order-up-to nventory polcy where the trucks used for shpment have to be fully loaded; nevertheless, t s noted that full truckloads polcy can be suboptmal for a retaler as t mght lead to ncreased holdng costs at such levels that decrease n shppng costs cannot counter balance t. A smlar observaton has been made by Toptal et al. (2003) n a sngle-tem model; they note that t mght be benefcal to have one of the trucks to be partally loaded. In the aforementoned studes, only a sngle truck type s consdered. We further generalze TL transportaton modelng by takng dfferent freght trucks nto consderaton. In case a retaler uses 2PL or 3PL for nbound transportaton, there mght be dfferent TL carrers avalable, each of whch has dstnct truck fleets. Even n the case of a sngle TL carrer, t mght be the case that the retaler can be forced to select among a set of dfferent freght trucks for hs/her nbound transportaton. In such a case, the retaler needs to dynamcally determne how many trucks of each truck type to use for the nbound shpment of each order. Ths study contrbutes to the mult-tem nventory control models by provdng generalzed formulaton for TL transportaton wth heterogeneous freght trucks. Specfcally, we consder dfferent per truck capactes and per truck costs for dstnct truck types avalable for nbound shpment. Furthermore, the aforementoned studes defne truck capacty n terms of the number of tems that can be carred. We extend truck capacty defnton by jontly regardng the weght and volume capactes for dfferent truck types. Shpment Consoldaton As mentoned prevously, transportaton costs consttute a sgnfcant part of total costs n many ndustres; therefore, utlzaton of transportaton capacty can substantally save costs. The practce of shpment consoldaton targets better utlzaton of the transportaton capacty by mergng shpments of small quanttes to acheve a shpment wth larger quantty that utlzes the transportaton capacty better. Ths, n turn, reduces costs due to economes of scale n the transportaton costs (Mutlu et al., 2010). Three common shpment consoldaton polces consdered as quantty-based, tme-based, and tmeand-quantty-based consoldaton (Cetnkaya et al., 2006). In the quantty-based shpment consoldaton, the customer demands are accumulated untl a specfed quantty s acheved; and, then a shpment s released. On the other hand, n the tme-based shpment consoldaton, the customer demands are accumulated for a specfed tme perod; and, then a shpment s released. In the tme-and-quantty- 3

7 based consoldaton, the customer demands are accumulated untl a specfed quantty s acheved or a specfed tme perod s ended; and, then a shpment s released. Çetnkaya (2005) provdes a detaled revew of coordnated nventory control models wth shpment consoldaton. In ths study, we assume a tme-based shpment consoldaton polcy, that s, an order s placed n equal tme ntervals. However, we note that we also formulate the decsons on whch tems to be consoldate. Stochastc Jont Replenshment Problem The jont replenshment problem (JRP) consders how to jontly replensh a set of dfferent products n a mult-tem nventory system. The man motvaton for jontly replenshng the dfferent products s the economes of scale of the order setup costs. Generally, order setup costs are defned by the transportaton costs of a shpment. The reader s referred to a revew of JRPs by Khouja and Goyal (2008) for dfferent settngs, models, and soluton approaches studed n the lterature for JRPs. In stochastc JRPs, each product has ts own stochastc demand. Balntfy (1964) proposes a can-order polcy for a stochastc JRP, where each tem has a must-order level s, a can-order level c, and an order-up-to-level S. In a can-order polcy, denoted by (s, c, S), an tem s ordered when ts nventory level reaches the must-order level, and any other tem, whose nventory level s below the can-order level, s then ordered wth t such that the order quanttes for the ordered tems buld ther nventory levels to the specfed order-up-to-levels. Whle Balntfy (1964) assumes contnuous nventory revew, Johansen and Melchors (2003) analyze the can-order polcy under perodc revew notng that replenshment opportuntes may only come once or twce a day and; therefore, a perodc revew model can be superor for some customers. Atkns and Iyogun (1988) analyze JRP strateges where the tems are ordered up to an order-upto-level R every tme perod of length T. These polces are referred to as (R, T ) polces and Atkns and Iyogun (1988) nvestgate two (R, T ) polces: a perodc polcy, where all tems are ordered wth each replenshment and a modfed perodc polcy, where a base set of tems s ordered wth each replenshment and the remanng tems are ordered at each specfed consecutve replenshment. In ths study, we adopt a (R, T ) type of polcy for a gven set of consoldated tems: the nventores of the tems n the consoldaton are replenshed every T tme unts up to ther ndvdual order-up-to-levels. Atkns and Iyogun (1988) concludes that the perodc (R, T ) type polces show more promse than the (s, c, S) type polces. However, Pantumsncha (1992) notes that dfferent polces can be superor to the others dependng on the specfc problem parameters. Vswanathan (1997) ntroduces a new class of polces known as the P (s, S) polcy. The P (s, S) polcy s a perodc revew polcy where the amount of tems on hand are revewed at ntervals of tme 4

8 T. If the amount of tems on hand s less than s then tems are ordered to brng the nventory up to S. They test ther algorthm aganst the same problems n Atkns and Iyogun (1988) and fnd that ther proposed polcy generally gves domnatng solutons and that the extra computatonal requrement s nomnal. Nelsen and Larsen (2005) use Markov decson theory and fnd an analytcal soluton to the Q(s, S) polcy, whch was lsted as a future research drecton by Vswanathan (1997). In the Q(s, S) polcy, the total number of tems are revewed contnuously but the tems themselves are only revewed once the total demand reaches Q. Nelsen and Larsen (2005) fnd the Q(s, S) model to be superor to the perodc revew P (s, S) models. Ozkaya et al. (2006) propose a new hybrd (Q, S, T ) polcy. The polcy s consdered to be both contnuous and perodc as orders are placed to the order-up-to level S whenever total demand level Q s reached or tme T has elapsed snce the last order. Usng the same problem settngs wth Atkns and Iyogun (1988) and Vswanathan (1997) as a benchmark, Ozkaya et al. (2006) fnd ther proposed method to be better 72% of the tme. All of the above models are unconstraned and Zhao et al. (2012) state that Inventory systems wth lmted and sharable-common resource exst wdely n the real logstcs feld, yet studes on such systems are lmted. Mnner and Slver (2005) develop a mult-product nventory replenshment problem where the nventory level at any tme s constraned by a budget or space lmtatons. They assume a Posson demand, zero lead tme, and no backorders and formulate the problem as a sem-markov decson process. Zhao et al. (2012) also study a constraned polcy, specfcally, the (r, Q) polcy wth a lmted sharable common resource. In the (r, Q) polcy, when an tem s nventory drops below r then Q unts of that tem are ordered. Betts and Johnston (2005) study a smlar model wth a constrant on the nvestment captal avalable. In ths study, the resource commonly shared s the transportaton capacty, whch s also a decson varable of the retaler at each replenshment. PROBLEM FORMULATION Consder a set of n tems ndexed by, I, where I = {1, 2,..., n}, such that each tem has a stochastc demand. Let f (D ) and F (D ) denote the probablty densty functon and cumulatve dstrbuton functon of tem s demand, D, over unt tme. We assume that the unt tme demand for any tem s normally dstrbuted wth mean λ and standard devaton σ. Thus, tem s demand over a perod of t tme unts s normally dstrbuted wth mean λ t and standard devaton σ t (see, e.g., Nahmas, 2009). We denote f (D (t) ) as the probablty densty functon of tem s demand over a perod of t tme unts, where D (t) s the random varable defnng tem s demand over t tme unts. Under the current settngs, the retaler s subject to nventory holdng, order setup, and shortage The problem formulaton and the soluton methods presented can be easly modfed for other demand dstrbutons. 5

9 costs. In partcular, let h denote the nventory holdng cost per unt per unt tme, a denote the order setup cost per each order, and p denote the penalty cost per unt shortage for tem. In addton to these costs, the retaler s subject to explct transportaton costs assocated wth each order. We assume that the retaler can use m dfferent truck types for nbound shpment. Let dfferent truck types be ndexed by j, j J, where J = {1, 2,..., m} such that a sngle truck of type j has weght-capacty of W j, volume-capacty of V j, and cost of R j. Furthermore, let each unt of tem have weght w and volume v. The retaler s assumed to adopt a tme-based order-up-to-level nventory control polcy. That s, for a sngle tem or a set of consoldated tems, the retaler wll place an order at dentcal tme ntervals such that each tem s order quantty s determned to ncrease the nventory level of that tem to a specfc pont. We assume that delvery lead tme s neglgble. If the retaler plans to manage tem ndvdually, hs/her decson varables would be order-up-to-level for tem, denoted by s, and the replenshment cycle length t. Fgure 1 llustrates the expected nventory level over tme for a sngle tem wth replenshment cycle length t, order-up-to-level s, and λ demand per unt tme. Fgure 1: Inventory Level n Tme-based Order-Up-To-Level Control for Sngle Item In the remander of ths secton, we frst formulate the case of a sngle-tem beng replenshed ndvdually. Then, a mathematcal formulaton s gven to determne the common replenshment cycle length for a set of consoldated tems along wth each tem s ndvdual order-up-to-level decsons n the consoldated set. Followng ths, the model to decde on whch tems to consoldate s presented. We note that the problem formulaton provded can be modfed to handle constant lead tmes. Specfcally, once the tme nterval for consecutve orders s determne, a retaler can ntate the order accordngly regardng the delvery lead tme 6

10 Sngle-Item Tme-based Order-up-to-level Inventory Consder that tem s ndvdually replenshed. As noted prevously, the retaler s subject to nventory holdng, order setup, shortage, and nbound transportaton costs. Due to stochastc demand, the retaler s objectve s to mnmze the total expected costs per unt tme assocated wth tem. Expected nventory holdng cost per unt tme amounts to h (s λ t 2 ). Order setup cost per unt tme s a determnstc varable dependng on t and t amounts to a t. Now, let n (s, t ) be the expected number of shortages wthn one replenshment cycle as functon of s and t. Then, expected shortage cost per unt tme amounts to p n (s,t ) t. Note that the number of shortages wthn a replenshment cycle depends on both the replenshment cycle length t and the order-up-to-level s ; hence, n (s, t ) s a functon of s and t. One can show that n (s, t ) = ( ) s D (t ) s f (D (t ) )dd (t ). Therefore, expected shortage per unt tme s p ( ) t s D (t ) s f (D (t ) )dd (t ). The only remanng cost term s the expected nbound transportaton costs. Recall that the retaler can use m dfferent truck types for nbound transportaton. At each order replenshment, the retaler needs to decde on how many of each truck type should be used. Let x j be the nteger number of type j trucks to be used for nbound transportaton of an order and x = [x 1, x 2,..., x m ]. The order quantty to be shpped wll be equal to the demand realzed durng the replenshment cycle,.e., D (t ). In ths case, the retaler wll determne the truck confguraton x that wll mnmze nbound transportaton costs to shp D (t ) unts. Therefore, the followng problem needs to be solved at each replenshment: IT C (D (t ) ) = mn x j J x jr j s.t. j J x jw j w D (t ) j J x jv j v D (t ) x j {0, 1, 2,...} j J. The objectve functon n the defnton of IT C (D (t ) ) gven n Eq. (1) s the total truckng cost. The frst and second constrants assure that the selected trucks cumulatvely have the suffcent weght and volume capacty to shp D (t ) unts, respectvely. The thrd set of constrants s the nteger defnton for the x j values. (Note that f D (t ) 0, x j = 0 j = 1, 2,..., m; hence, IT C (D (t ) ) = 0 for D (t ) 0.) Then, expected nbound transportaton cost per unt tme amounts to 1 t 0 IT C (D (t ) )f (D (t ) )dd (t ). The retaler s total expected costs per unt tme when tem s ndvdually replenshed, denoted by g (s, t ), amount to ( g (s, t ) = h s λ ) t + a + p 2 t t + 1 t 0 s IT C (D (t ) )f (D (t ) )dd (t ) ( ) D (t ) s f (D (t ) )dd (t ) where the frst, second, thrd, and forth terms of Eq. (2) are the expected nventory holdng, order 7 (1) (2)

11 setup, shortage, and nbound transportaton costs per unt tme. The retaler s optmzaton problem for ndvdually replenshed tem then reads as (P ) mn g (s, t ) (s,t ) s.t. t 0 s 0 IT C (D (t ) ) = mn x s.t. x j R j j J j J j J x j W j w D (t ) x j V j v D (t ) x j {0, 1, 2,...} j J. Consoldated Tme-based Order-up-to-level Inventory Now suppose that a set of tems are ordered together, that s, ther shpments are consoldated. The retaler s objectve s to determne the order-up-to-level for each tem n the consoldaton and the replenshment cycle length for the consoldaton so that the total expected costs per unt tme for the tems n the consoldaton are mnmzed. Any subset of the set of tems I s a possble consoldaton; thus, there are 2 n 1 subsets of tems that can be consoldated. Let each possble subset of tems be ndexed by k, k K where K = {1, 2,..., 2 n 1} and Ω k denote a subset. Furthermore, let T k denote the common replenshment cycle when Ω k s selected as a consoldaton,.e., t = T k Ω k. Smlar to sngle-tem case, a consoldated set of tems has nventory holdng, order setup, shortage, and nbound transportaton costs. Note that nventory holdng, order setup, and shortage costs of the tems n a consoldaton are ndvdual cost terms; therefore, total expected holdng, order setup, and shortage costs per unt tme for the consoldaton wll be equal to the sum of the expected holdng, order setup, and shortage cost per unt tme of each tem n the consoldaton. That s, the total expected holdng cost per unt tme of consoldaton Ω k k K s equal to the sum of the expected holdng costs per unt tme of the consoldated tems. The total expected holdng cost per unt tme of the consoldaton s, therefore, equal to Ω k h s T k 2 Ω k h λ. Smlarly, t follows that the total order setup cost per unt tme for Ω k amounts to 1 T k Ω k a, and the total shortage cost per unt tme for Ω k s equal to 1 ( ( ) ) T k Ω k p s D (T k) s f (D (T k) )dd (T k). Unlke the nventory holdng, order setup, and shortage costs for Ω k, the nbound transportaton costs wll not be equal to the sum of the ndvdual tems transportaton costs as dfferent tems can share truck capactes due to beng replenshed smultaneously. In partcular, at each replenshment, the retaler needs to decde on the number of trucks of each type to shp the realzed demands of the tems n 8

12 the consoldaton. Let D (T k) Ω k be the Ω k -vector of D (T k) values for Ω k. The followng problem then should be solved at each replenshment to determne the nbound transportaton cost of consoldaton Ω k : IT C Ωk (D (T k) Ω k ) = mn x j J x jr j s.t. j J x jw j Ω k w D (T k) j J x jv j Ω k v D (T k) x j {0, 1, 2,...} j J. Smlar to Eq. (1), the objectve functon n the defnton of IT C Ωk (D (T k) Ω k ) gven n Eq. (3) s the total truckng cost. The frst and second constrants guarantee that the selected trucks cumulatvely have the suffcent weght and volume capacty to shp D (t ) (3) Ω k, respectvely. The thrd set of constrants s the nteger defnton for the x j values. Now, let us assume that Ω k = {1, 2,..., l} such that l n. Then, expected nbound transportaton cost per unt tme amounts to 1 T k 0 IT C Ωk (D (T k) Ω k )f(d (T k) Ω k )dd (T k) IT C Ωk (D (T k) Ω k )f 1 (D (T k) 1 )f 2 (D (T k) 2 )... f l (D (T k) l )dd (T k) 1 dd (T k) 2... dd (T k) l. 1 T k 0 Ω k = The retaler s total expected costs per unt tme when tems n Ω k are consoldated, denoted by G k (S k, T k ), amount to G k (S k, T k ) = h s T k h λ + 1 a T k T k Ω k Ω k Ω k + 1 T k 0 Ω k p ( s ( ) ) D (T k) s f (D (T k) )dd (T k) IT C Ωk (D (T k) Ω k )f(d (T k) Ω k )dd (T k) Ω k (4) where S k s a Ω k -vector of s values for Ω k. The frst, second, thrd, and forth terms of Eq. (4) are the expected nventory holdng, order setup, shortage, and nbound transportaton costs per unt tme for the consoldaton Ω k. The retaler s optmzaton problem for consoldaton Ω k then reads as (P Ω k ) mn G k (S k, T k ) (S k,t k ) s.t. T k 0 s 0 IT C Ωk (D (T k) Ω k ) = mn x s.t. Let S k and T k denote an optmum soluton of (PΩ k ). Consoldaton Decsons x j R j j J x j W j w D (Tk) j J Ω k x j V j v D (Tk) j J Ω k x j {0, 1, 2,...} j J. Ultmately, the retaler s goal s to determne whch tems wll be consoldated and what wll be the common replenshment cycle length for each consoldaton and order-up-to-level for each set of tems n 9

13 the consoldatons. Therefore, the retaler needs to select whch subsets of tems wll be consoldaton such that each tem wll be replenshed wthn a sngle consoldaton. A gven consoldaton Ω k can be defned by c k values such that Let c k = y k = { 1 f tem s n consoldaton Ωk, 0 otherwse. { 1 f consoldaton Ωk s selected, 0 otherwse. Assumng that the retaler wll adopt the optmum common replenshment cycle length and order-upto-levels for any consoldaton Ω k,.e., S k and T k, the retaler s consoldaton problem reads as (P) mn (y) s.t. C(y) = k K y k G k (S k, T k ) c k y k = 1 k K I y k {0, 1} k K. where y s the bnary (2 n 1)-vector of y k values. The objectve functon of (P) mnmzes the total expected costs per unt tme. The frst set of constrants ensures that each tem s ncluded wthn one of the selected consoldatons. The second set of constrants are the bnary defntons for the decson varables. We note that (P) s a set parttonng problem, whch s known to be NP-hard (see, e.g., Garey and Johnson, 1979). Furthermore, defntons of S k and T k requre to solve b-level mxed-nteger non-lnear optmzaton problems. Therefore, n the next secton, we focus on an evolutonary heurstc method to solve (P) and dscuss an approxmaton formulaton that reduces (P Ω k ) to sngle-level mxednteger non-lnear optmzaton problems, for whch we dscuss another heurstc method. Soluton Analyss In ths secton, we propose a genetc algorthm based meta-heurstc approach for solvng problem (P), denoted by GA-P. GA-P has the followng four man steps: () chromosome representaton and ntalzaton, () ftness evaluaton, () mutaton, and (v) termnaton. In what follows, we dscuss the detals of each step. Chromosome Representaton and Intalzaton Note that the retaler can select at most n consoldatons (when each tem s ndvdually replenshed), that s, k K y k n. Therefore, a soluton to (P) can be presented by an nteger n-vector chrom = [c 1, c 2,..., c n ], where c denotes the consoldaton number that tem belongs to. Note that one should have 1 c n I. The mportant pont about defnng a soluton for (P) as a chrom vector s that 10

14 the correspondng consoldaton decsons are feasble for (P) as each tem s guaranteed to be wthn one consoldaton. For nstance, for a problem nstance wth n = 5 tems, let chrom = [3, 1, 2, 3, 2]; then, tems 1 and 4 form one consoldaton, tems 3 and 5 form one consoldaton, and tem 2 forms one consoldaton. That s, {1, 4}, {3, 5}, {2} are the three consoldatons selected. Furthermore, chrom representaton enables mutaton operatons to be smply executed. As an ntalzaton, we randomly generate nm number of chrom vectors by randomly generatng c values such that 1 c n I. Ftness Evaluaton Now suppose that a set of chromosomes are gven. For each chromosome, one can determne the number of consoldatons and the tems n each consoldaton as explaned above. The ftness value for a chromosome s the total expected costs of the consoldatons n the chromosome. Therefore, one needs to fnd the total expected costs per unt tme for each consoldaton of a gven chromosome and get ther summaton to fnd the ftness value of the chromosome. To do so, problem (P Ω k ) should be solved for each consoldaton assocated wth the chromosome. Note that (P Ω k ) s a b-level mxednteger non-lnear optmzaton problem due to the calculaton of expected nbound transportaton costs present n the objectve functon,.e., Eq. (3). Even the smplest b-level optmzaton problems, when optmzaton problems at both levels are lnear, are shown to be NP-hard (see, e.g., Hansen et al., 1992). Furthermore, one needs to solve (P Ω k ) at least once and at most n tmes for each chromosome to be evaluated. Therefore, an effcent method to solve (P Ω k ) s requred. In what follows, we frst dscuss an approxmated reformulaton for (P Ω k ), whch gves a sngle-level mxed-nteger nonlnear optmzaton problem; then, we dscuss a local search algorthm to solve the resultng sngle-level mxed-nteger nonlnear optmzaton problem. Approxmated Reformulaton for A Consoldaton In determnng S k and T k for a gven consoldaton Ω k, the retaler should consder how much nbound transportaton costs on average wll be pad. However, nbound transportaton decsons,.e., x are dynamc n the sense that the retaler wll fnd hs/her optmal truck choces wth every replenshment. Nevertheless, snce S k and T k heavly affect the replenshment quanttes, problem (P Ω k ), therefore, explctly ncludes the expected nbound transportaton costs n fndng S k and T k. Ths, n turn, results n the b-level optmzaton problem gven by (P Ω k ). Specfcally, the lower level of (P Ω k ) s requred n order to fnd exact expected nbound transportaton costs per unt tme. As aforementoned, b-level optmzaton problems are complex, we, therefore, approxmate (P Ω k ) wth a sngle-level optmzaton problem as follows. 11

15 Note that expected order quantty for each tem n Ω k wll be equal to the expected demand durng one replenshment cycle,.e., λ T k Ω k. Then, we approxmate Eq. (3) by defnng expected number of trucks of type j used for consoldaton Ω k, denoted by x jk. That s, we defne Eq. (3) by assumng that, on average, the retaler decdes to use x jk number of type j trucks n each replenshment of the tems n Ω k. Let x k be the m-vector of x jk values. Usng ths approxmaton, average shpment cost per replenshment of Ω k amounts to IT C Ωk ( x k ) = j J x kjr j. Then, the retaler s approxmated total expected costs per unt tme when tems n Ω k are consoldated, denoted by G k (S k, T k, x k ), are equal to G k (S k, T k, x k ) = h s T k h λ + 1 a + 1 p n(s, T k ) + 1 x kj R j. (5) 2 T k T k T k Ω k Ω k Ω k Ω k j J The only dfference between Eq. (5) and Eq. (4) s that Eq. (5) uses IT C Ωk ( x k ) whle Eq. (4) requres the soluton of Eq. (3) for any combnatons of demand realzatons of the tems n Ω k. Usng Eq. (5), the retaler s optmzaton problem for consoldaton wth approxmated total expected costs per unt tme reads as ( P Ω k) mn G k (S k, T k, x k ) (S k,t k, x k ) s.t. T k 0 s 0 x jk W j Ω k w λ T k j J Ω k x jk V j v λ T k j J Ω k x j {0, 1, 2,...} j J. ( P Ω k ) s a sngle-level mxed-nteger nonlnear optmzaton problem. We note that ( P Ω k) s NPhard as a specal case of ( P Ω k) when w = 0 I (or W j ) s an nteger knapsack problem for gven S k and T k. Therefore, we next develop an heurstc method to solve ( P Ω k). Local Search Heurstc for Consoldaton Approxmaton We propose a local search heurstc for solvng ( P Ω k), denoted by LSH-k. Partcularly, LSH-k works as follows. Gven x k, we frst determne S k and T k by solvng ( P Ω k ) wth the gven x k. Gven x k, ( P Ω k) reduces to the followng optmzaton problem: ( P xk ) mn (S k,t k ) s.t. G k (S k, T k, x k x k ) { j J T k mn x } jkw j Ω w λ, j J x jkv j k Ω v λ k T k 0 s 0 Ω k 12

16 ( P xk ) s a nonlnear optmzaton problem. A common method to solve such nonlnear models s the Interor-Pont (IP) method. Snce ( P xk ) needs to be solved many tmes wthn LSH-k (whch s also needed to be executed many tmes wthn GAP-P), we focus on developng an effcent method to fnd solutons for ( P xk ) n less computatonal tmes. In partcular, gven S k, f we overestmate the number of expected shortages for any tem wthn one replenshment cycle and assume t s equal to the expected demand for that { tem wthn one replenshment cycle,.e., n (s, T k ) = λ T k ; then, one can easly show j J that T k = mn x } jkw j Ω w λ, j J x jkv j k Ω v λ mnmzes G k (S k, T k, x k x k, S k ) over the feasble T k values of k ( P xk ). Furthermore, gven T k, Gk (S k, T k, x k x k, T k ) s separable n and convex wth respect to each s Ω k ; thus, t follows from the frst order condton that s that mnmzes G k (S k, T k, x k x k, T k ) wll be the soluton of F (T k) (s ) = 1 h T k p, where F (T k) ( ) s the cumulatve dstrbuton functon of tem s demand over T k tme unts (.e., cumulatve dstrbuton of the normal random varable, D (T k), wth mean λ T k and standard devaton σ Tk ). Therefore, we accept the soluton of ( P xk ), denoted by S k and T k, as gven n the followng equatons: T k = mn { j J x jkw j Ω k w λ, j J x } jkv j, (6) Ω k v λ F (T k) ( s ) = 1 h T k. (7) p In Secton, we compare Eqs. (6) and (7) to IP and t can be seen from Table 2 that Eqs. (6) and (7) are computatonally very effcent compared to IP. Furthermore, the soluton qualtes are very close over the problem nstances solved. Therefore, we use Eqs. (6) and (7) to solve ( P xk ). Once ( P xk ) s solved, we calculate G k ( S k, T k, x k ) as the cost value of x k. After that, we check all neghbors of x k. To do so, we ncrease and decrease (f possble) the number of trucks of each type by 1. That s, we ncrease x jk by 1 and decrease x jk by 1 (f x jk 1) for each j. Ths generates all neghbors of x k. If there s a neghbor wth lower cost value, we take the neghbor wth the lowest cost as the new soluton and repeat the neghbor search wth ths soluton. Ths process s repeated untl no neghbor wth lower cost value s determned. At termnaton, we are guaranteed wth a local mnmum. To avod gettng a hgh-cost local mnmum, we start the LSH-k wth multple x k. Intally, { we randomly generate m x k vectors such that 0 x jk u k where u k = } Ω w λ t max max k Ωk j J W j, v λ t max V j and t max = max Ωk { 2a h λ } (note that 2a h λ s the replenshment cycle length of tem assumng that σ = 0,.e., the economc order quantty model); thus, u k s the maxmum number of trucks needed to shp total order quantty of the tems n the consoldaton assumng that each tem s order quantty s gven by the economc order quantty and a sngle truck type s used. The detals of LSH-k for a gven startng soluton are explaned below. 13

17 Local Search Heurstc for ( P xk ) (LSH-k) Step 0: Let x k be gven for a consoldaton Ω k. Step 1: Calculate S k and T k usng Eqs. (6) and (7) and determne G k (S k, T k, x k x k ) Step 2: Step 3: For j = 1 : m Let x k[ j] = x [+j] jk = x k. If x [ j] jk > 0, let x [ j] jk = x [ j] jk 1; and, let x [+j] jk = x [+j] jk + 1 Step 4: Calculate G k (S k, T k, x k[ j] x k[ j] ) and G k (S k, T k, x k[+j] x k[+j] ) usng Eqs. (6) and (7) Step 5: End Step 6: If mn j J { G k (S k, T k, x k[ j] x k[ j] ), G k (S k, T k, x k[+j] x k[+j] )} < G k (S k, T k, x k x k ) Step 7: Set x k = arg mn j J { G k (S k, T k, x k[ j] x k[ j] ), G k (S k, T k, x k[+j] x k[+j] )}, go to Step 2 Step 7: Else, termnate and return x k Mutaton Now suppose that we have a populaton of evaluated chromosomes, that s, the total approxmated expected cost per unt tme for each chromosome s known. Let chrom dl be the d th d {1, 2,..., pop l } chromosome n the l th populaton, where pop l s the number of chromosomes n the l th populaton. Furthermore, let C(chrom dl ) the total approxmated expected cost per unt tme of chrom dl. Wthout loss of generalty, let C(chrom 1l ) < C(chrom 2l ) <... < C(chrom popll ). To generate the (l + 1) st populaton, we execute the followng three mutaton operatons: () Local Mutaton: A local mutaton s appled to the chromosomes that are randomly selected from the frst 45% of the pop l chromosomes wthn the l th populaton,.e., the best 45% of the populaton. Local search mutaton randomly pcks an tem from a selected chromosome and randomly ncreases or decreases c of the chromosome by 1. For a gven populaton of evaluated chromosomes, we generate 0.45pop l new chromosomes at the end of local mutaton operatons. () Cross-Over: Cross-over mutaton s appled to the chromosomes n the best 50% of the populaton. We randomly create pars of two chromosomes from the best 50% of the populaton and perform a random sngle-pont cross-over. Each par of chromosomes crossed-over generates two new chromosomes, one from each chromosome wthn the par. For a gven populaton of evaluated chromosomes, we generate 0.5pop l new chromosomes at the end of cross-over operatons. () Random Mutaton: Random mutaton s appled to create a number of chromosomes so that the new populaton has the same populaton sze wth the current populaton. Frst, the number of chromosomes needed after local mutaton and cross-over operatons s determned. Then, chromosomes are randomly selected from the best 50% of the populaton and random mutaton s appled. A random mutaton on a selected chromosome randomly generates a c value such that 1 c n for a randomly 14

18 selected tem. At the end of mutaton operatons, the newly generated populaton has the same number of chromosomes wth the prevous populaton. Termnaton If there s no mprovement n C(chrom 1l ) for L consecutve populatons or O populatons are evaluated, the GAP-P termnates. NUMERICAL ANALYSIS In ths secton, we focus on two sets of numercal analyss. In the frst set of numercal analyss, the subroutne defned by Eqs. (6) and (7) s compared to IP and the approxmated reformulaton of a consoldaton s tested wth a smulaton study. In the second set of numercal analyss, the cost and envronmental benefts of consoldatng tems and usng multple truck types for shpment are llustrated. In both of the numercal analyses, the demand per unt tme for any tem s assumed to be normally dstrbuted wth mean λ and standard devaton σ. The problem nstances are randomly generated usng unform dstrbutons wth the gven ranges n Table 1. Smlar numercal values are assumed for these parameters n the lterature on ntegrated nventory control and transportaton (see, e.g., Toptal et al., 2003, Toptal and Çetnkaya, 2006, Toptal, 2009, Konur and Toptal, 2012). In all of the Table 1: Problem Parameters λ U[1750, 2250] w U[1, 4] σ U[150, 250] v U[0.5, 2] h U[1, 5] W j U[200, 600] a U[50, 250] V j U[100, 300] p U[2, 10] R j U[150, 450] followng analyss, we consder 15 dfferent problem classes, each of whch corresponds to a combnaton of n = {5, 10, 15, 20, 25} and m = {5, 10, 15}. For each problem class, 10 problem nstances are generated. The values shown n the tables of ths secton for a gven problem class are the average values over all 10 problem nstances solved wthn that problem class. We frst compare Eqs. (6) and (7) to IP. Here, we assume that all of the tems are consoldated n one sngle group and the approxmated truck choces for the consoldaton s gven as we are comparng two alternatve soluton methods for problem ( P xk ). That s, x k s gven for Ω k such that Ω k = I. Gven the number of truck types, x k s randomly generated such that x jk [0, 5]. For each problem class, Table 2 shows average values, over the 10 randomly generated problem nstances, for T k and correspondng 15

19 G k (S k, T k, x k x k ) values along wth the cpu tmes (n seconds) for Eqs. (6) and (7) and IP. Furthermore, the cost dfference column gves the average dfference n G k (S k, T k, x k x k ) values between Eqs. (6) and (7) and IP. Table 2: Comparng Soluton Methods for ( P xk ) Eqs. (6) and (7) Interor-Pont (IP) Cost n m Tk Gk cpu Tk Gk cpu Dfference , , % , , % , , % , , % , , % , , % , , % , , % , , % , , % , , % , , % , , % , , % , , % Average , , % As t can be seen from Table 2, the average computatonal tme wth Eqs. (6) and (7) s sgnfcantly lower than the average computatonal tme wth IP. Moreover, whle the IP method results n lower approxmated costs,.e., Gk (S k, T k, x k x k ) values, Eqs. (6) and (7) were able to fnd good qualty solutons; the ncrease n costs s less than 4% on average. Fnally, T k values returned by each alternatve method are very close on average. Therefore, we can conclude that Eqs. (6) and (7) are effcent for solvng ( P xk ) and we use them n GAP-P. Next, we evaluate the approxmated reformulaton of a gven consoldaton. Recall that truck choce decsons are dynamc as the retaler can select the number of trucks of each type to shp each order. However, calculaton of expected transportaton costs resulted n b-level optmzaton problem (P Ω k ), whch has been approxmated by problem ( P Ω k ). Partcularly, n ( P Ω k), x k defnes approxmated number of trucks of each truck type to be used by the retaler for a gven consoldaton. To see how close G k (S k, T k ) and G k (S k, T k, x k ), we smulate the truck choce decsons as well as order quantty decsons for a gven consoldaton. Partcularly, gven a problem nstance, we assume that all of the tems are consoldated n one sngle group. Then, S k, T k, and x k values are determned usng LSH-k. After that, wth the determned S k and T k values, we smulate 1,000 replenshment cycles for the problem nstance 16

20 (to do so, for each tem I, 1,000 demand realzatons,.e., D (T k) values, are generated usng normal dstrbuton wth mean λ T k and standard devaton σ Tk ). At each replenshment of the smulaton, the best truck choces for the order are determned by solvng Eq. (3) wth CPLEX (as the number of decson varables are 15 maxmum, t was not very tme consumng to solve Eq. (3) at each of the 1,000 replenshment). As a result of smulaton, we fnd the mean value of the cost per cycle and then determne the mean value of the cost per unt tme, denoted by G k (S k, T k ). Furthermore, we fnd the mean number of trucks of each type used, denoted by x jk. Note that, n the approxmated formulaton, x jk s assumed to be gven by x jk values. Table 3: Comparng Approxmated and Smulated Results for A Gven Consoldaton Approxmaton Smulaton n m T k Gk (S k, T k, x k ) j J x jk G k (S k, T k ) j J x jk , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Average , , In Table 3, the average values over the 10 randomly generated problem nstances for T k, G k (S k, T k, x k ), j J x jk, G k (S k, T k ), and j J x jk are documented for each problem class. It can be observed that from Table 3 that G k (S k, T k, x k ) over estmates G k (S k, T k ) for most of the problem classes (and ths was the case n most of the problem nstances solved). Ths result was expected snce the approxmaton reformulaton does not defne the mnmum transportaton costs n each replenshment. Specfcally, Gk (S k, T k, x k ) over estmated G k (S k, T k ) by approxmately 17% on average. Nevertheless, x jk can over or under estmate x jk values and the same observaton holds for the total number of trucks used for nbound shpment; however, the dfference between j J x jk and j J x jk s wthn ±15% and 6% on average. Based on these observatons, we beleve that approxmaton reformulaton of a consoldaton s suffcently well reflectng the actual costs; hence, can be navely used to evaluate the 17

21 cost performance of a gven consoldaton and fnd good S k and T k values for a gven consoldaton Ω k. The followng numercal analyses document the cost and envronmental benefts of consoldaton. Specfcally, we compare three consoldaton polces: () consoldaton polcy, the consoldatons adopted as the soluton of problem (P) va GAP-P, () no-consoldaton polcy, when all of the tems are ndvdually replenshed, and () sngle-consoldaton polcy, when all of the tems are consoldated n a sngle group. For each polcy, we determne approxmated expected costs (denoted by C) and truck densty (denoted by φ). Truck densty s defned as the average number of trucks used per unt tme. Partcularly, for a gven consoldaton of tems, Ω k, truck densty, φ k, s defned as follows: j J φ k = x jk. T k Then, truck densty of a consoldaton polcy, s equal to the sum of the truck denstes of the consoldatons suggested by the polcy. Table 4 gves the average values over 10 problem nstances solved wthn each problem class for C and φ for each consoldaton polcy. Furthermore, we gve the average values of the percent ncreases n C and φ (denoted by C and φ, respectvely) due to adoptng no-consoldaton and sngle-consoldaton polces over the consoldaton polcy. Table 4: Comparng Consoldaton Strateges for (P) Consoldaton No Consoldaton Sngle Consoldaton n m C φ C φ C φ C φ C φ , , % 34.0% 25, % 9.1% 10 20, , % 23.6% 25, % 6.3% 15 19, , % 19.5% 25, % 4.3% , , % 8.4% 69, % 3.4% 10 42, , % 38.2% 63, % 14.0% 15 39, , % 25.5% 56, % 14.1% , , % 23.4% 164, % 23.4% 10 60, , % 22.9% 104, % 4.3% 15 56, , % 12.5% 93, % 10.3% , , % 20.4% 205, % 11.3% 10 78, , % 40.9% 156, % 29.8% 15 74, , % 33.3% 134, % 23.5% , , % 26.0% 287, % 7.1% , , % 49.0% 235, % 28.1% , , % 24.6% 184, % 9.1% Average 62, , % 26.8% 122, % 13.2% As t can be seen from Table 4, consoldaton polces heavly affect the costs and truck densty. Specfcally, a retaler can save n costs by effcently determnng whch tems wll be consoldated. We note that both sngle-consoldaton and no-consoldaton polces are suboptmal for problem (P); 18

22 therefore, as expected, consoldaton results n lower costs than no-consoldaton and sngle-consoldaton polces. Compared to no-consoldaton polcy, consoldaton can save costs over 50% on average; and, compared to sngle-consoldaton polcy, consoldaton can save costs over 75% on average over the problem nstances solved. Furthermore, effcent consoldaton can reduce truck densty. As expected, truck densty s the hghest on average for no-consoldaton polcy as utlzaton of truck capactes s mnmum n no-consoldaton polcy. Compared to no-consoldaton polcy, consoldaton can decrease truck densty over 25% on average; and, compared to sngle-consoldaton polcy, consoldaton can decrease truck densty over 10% on average. These observatons suggest that effcent consoldaton n mult-tem nventory systems can save costs and result n envronmental benefts sgnfcantly. Fnally, we compare the case when the retaler uses sngle truck type nstead of multple truck types for nbound transportaton. Specfcally, we assume that the retaler wll select the truck type whch mnmzes the total of the approxmated expected costs of the consoldatons selected,.e., sum of the costs defned n Eq. (5) over the consoldatons. To fnd the sngle truck type to be used, we fnd the consoldaton polcy assumng sngle truck type va GAP-P for each truck type, and select the one whch gves lower approxmated expected costs. Table 5 gves the average values over 10 problem nstances solved wthn each problem class for C and φ for nbound transportaton wth consderaton of multple truck types and sngle truck type. Furthermore, we gve the average values of the percent ncreases n C and φ (denoted by C and φ, respectvely) due to adoptng restrctng sngle truck type for nbound shpment. As expected and can be observed n Table 5, restrctng sngle truck type for nbound shpment ncreases costs. On average, sngle truck type nbound shpment ncreases costs by 2.3% compared to allowng use of dfferent truck types for nbound shpment. Furthermore, sngle truck type restrcton ncreases the truck densty by 4.9% on average over the problem nstances solved. Therefore, one can conclude that consderaton of dfferent truck types smultaneously for nbound shpment can have cost savngs as well as envronmental benefts. CONCLUSIONS Ths report studes a mult-tem nventory system wth shpment consoldaton and explct TL transportaton n a stochastc demand envronment. A tme-based order-up-to-level nventory polcy s proposed for a set of consoldated tems. Furthermore, a retaler s consoldaton decsons are formulated as a set parttonng problem. Due to the complexty of the problem, heurstc methods are developed. Frst, for a gven consoldaton, an approxmated reformulaton of the tme-based order-up-to-level 19

23 Table 5: Comparng Consoldaton wth Multple Truck Types to Sngle Truck Type Multple-Truck Sngle-Truck n m C φ C φ C φ , , % 4.8% 10 20, , % 0.8% 15 19, , % 9.7% , , % 3.5% 10 42, , % 3.1% 15 39, , % 1.7% , , % 6.3% 10 60, , % 5.4% 15 56, , % 4.7% , , % 15.3% 10 78, , % 3.2% 15 74, , % 0.3% , , % 2.0% , , % 6.7% , , % 6.2% Average 62, , % 4.9% nventory polcy wth heterogeneous freght trucks s provded. A local search heurstc s proposed for the approxmated reformulaton. Ths search heurstc s utlzed n a genetc algorthm to fnd good-qualty consoldaton strateges for the retaler s consoldaton problem. Ths study contrbutes to the lterature on mult-tem nventory systems by explctly accountng for transportaton costs when heterogeneous freght trucks can be used for nbound shpment, proposng a practcal nventory control polcy for a set of consoldated tems wth dstnct characterstcs, and developng a soluton method for determnng consoldaton strateges. Wth a set of numercal studes, the accuracy of the approxmated reformulaton of a consoldaton s presented. Furthermore, a set of numercal studes s conducted to llustrate the economcal as well as envronmental benefts of shpment consoldaton wth heterogeneous freght trucks. Specfcally, t s observed that shpment consoldaton not only saves costs but also reduces truck densty. Reduced truck densty mples less transportaton emssons and less truck congeston. A future research drecton would be to analyze dfferent nventory control polces for a gven set of consoldated tems. For nstance, quantty-based order-up-to-level polcy can be studed and compared to the tme-based order-up-to-level polcy examned n ths report. Furthermore, jont replenshment problem wth explct transportaton costs consderng the avalablty of dfferent truck types s a remanng problem to be nvestgated. 20