An Integrated Approach for Maintenance and Delivery Scheduling in Military Supply Chains

Transcription

1 An Integrated Approach for Mantenance and Delvery Schedulng n Mltary Supply Chans Dmtry Tsadkovch 1*, Eugene Levner 2, Hanan Tell 2 and Frank Werner Bar Ilan Unversty, Department of Management, Ramat Gan Israel Ashkelon Academc College, Department of Logstcs and Economcs Ashkelon 2900 Israel 3 Otto-von-Guercke-Unverstät, Fakultät für Mathematk PSF 4120 Magdeburg 39016, Germany dmtrytsadkovch@gmal.com; levner@acad.ash-college.ac.l; tellha@mal.bu.ac.l; frank.werner@ovgu.de 4TAbstract Ths paper focuses on an ntegrated demand-responsve schedulng of operatons wthn a two-echelon mltary supply network consstng of mantenance and transportaton operatons durng peacetme. The paper emphaszes on the contnuous operaton of mltary's tranng facltes. Integraton of operatons s carred out by embeddng an addtonal ntermedary (command) module nto the mantenance-transportaton model. We analyze two performance measures: tme of response (defnng the system's ablty to react speedly to mltary requrements) and mltary effectveness (whch defnes the ablty of the mltary supply network to delver the rght product at the rght tme). Usng nteger programmng technques combned wth a heurstc algorthm, we derve a new method for coordnatng mantenance and transportaton operatons n mltary supply networks. The standard smulaton tool ARENA 11.0 s used to mplement the ntegrated schedulng/plannng method. The results of a computatonal experment are presented. Keywords: Integrated schedulng; Heurstcs; Smulaton; Routng; Mltary supply network. MSC classfcaton: 90 B 35, 90 C 11, 90 C 27 November 6, 2015 * Correspondng author Dmtry Tsadkovch,Bar-Ilan Unversty,Department of Management,Ramat-Gan 52900,Israel Offce: , Fax: , Emal: dmtrytsadkovch@gmal.com 1

2 1. Introducton The prmary goal of large-scale commercal and mltary supply networks s to procure, produce, repar and dstrbute requested cvl or mltary commodtes, tems and other goods and servces n a tmely and cost-effectve way. Ths paper focuses on coordnatng mantenance and transportaton operatons n mltary supply networks consstng of supplers, mantenance unts, warehouses, dstrbuton centers, retalers and mltary bases ( Natonal Materals Advsory Board 2008). Snce an effectve logstc support of mltary operatons entals the rapd supply of the rght product to the rght locaton at the rght tme, lack of coordnaton among dfferent components that form the mltary supply network can mpar the effectveness and responsveness of mltary forces n the theater of operatons. There are numerous examples from mltary hstory that llustrate ths fact. For example, durng the Gulf War ( ) lack of a coordnated supply network prevented logstcans n US Marnes' 1st Battalon from transferrng crtcal tank parts to the army n tme (Krkland 2000). The key ssue n terms of mproved logstcs s havng the spare part, where t s needed. For example, durng the operaton Desert Storm (1991) $ 2.7 bllon worth of spare parts went unused accordng to a 1992 report from the US General Accountng Offce. It s estmated that, f the US Army had an effectve cargo-trackng, t would have saved about $ 2 bllon. In ths paper, we focus on ntegrated, demand-responsve schedulng of operatons wthn mltary supply networks n support of a contnuous operaton of tranng facltes. In our study we follow the gudelnes set out n the mltary standard MIL-STD 1378D (1990). Ths standard provdes detaled procedures for coordnatng all tranng efforts and tasks n mltary tranng programs. It also dentfes the general requrements of mantenance and transportaton modules wthn the varous tranng programs. In accordance wth the recommendatons of the standard, we select and analyze a two-echelon mltary supply network that ncludes mantenance (repar) and transportaton operatons. Mantenance nvolves restorng operatonal capabltes by mantanng and/or replacng damaged parts. A lack of ntegraton between these two operatons can cause a "domno effect". For example, a one-hour delay n supplyng a necessary spare part may result n a one-day delay n transportng the repared parts to the end-customers (combat unts or combat support unts); f the mltary tranng s based on a week-long cycle, the transportaton delay wll hold up the next tranng cycle by a week. Two performance measures: tme of response and mltary effectveness are ntroduced and then mathematcally descrbed n order to analyze the effectveness of the mltary supply network that we defne. To obtan an optmal soluton of the proposed ntegrated plannng-schedulng problem, we employ a new heurstc technque based on a two-stage search algorthm combned wth nteger programmng technques. The mplementaton of ths heurstc method s based on the dscrete event smulaton tool Arena

3 The paper s organzed as follows. Secton 2 overvews the related work. Secton 3 presents the descrpton of demand-responsve mltary supply networks. Sectons 4 and 5 ntroduce new ntegrated schedulng models. A heurstc soluton technque s gven n Secton 6. The results of the computatonal experments are presented n Secton 7. Secton 8 concludes the paper. 2. Lterature revew Ths secton focuses on three man categores: (1) mathematcal models n mltary supply networks; (2) the role of performance measures n supply chan management; and (3) analyss of models for ntegrated producton and transportaton operatons Mathematcal models n mltary supply networks Kress (2002) nvestgates the management of warfare logstcs n plannng, mplementng and controllng mltary operatons. Yoho et al. (2013) dscuss how defense logstcs has been studed n the past and how t s beng studed today. Proposng a research agenda for future work, they suggest that the most mportant drecton for mltary logstcs n the future s to mprove coordnaton of the relatonshps between dfferent actors n the supply system and mantenance and the overhaul of both complex and agng weapon systems. Akgün and Tansel (2007) present the deployment plannng problem (DPP) as a problem n preparng the physcal movement of mltary unts, postoned at geographcally dspersed locatons, from ther mltary bases to ther deployment destnatons, subect to varous schedulng and routng constrants and dfferent types of transportaton assets. To solve the DPP, a mxed nteger program model s formulated. Focusng on the desgn of an effcent mltary supply network for recoverng faled combat vehcles n unpredctable mltary envronments, Barahona et al. (2007) use a new ntegrated approach, called chan-centrc logstcs (NCL). Ths approach s based on controllng physcal nventory and dstrbuton processes across the whole supply network. Two mathematcal optmzaton models for the nventory and dstrbuton modules are formulated and solved by usng nteger programmng algorthms and smulaton. A shortfall of the NCL model s that the mantenance process s not consdered. McGee et al. (2005) use smulaton to descrbe the operaton of a mult-level nventory system for an ar force n order to analyze how effectvely transportaton can reduce costs and delay tmes and mprove overall readness n the logstcs network. The authors only focus on fndng an optmal transportaton utlzaton of the mult-echelon nventory system; the optmzaton and coordnaton of the mantenance operaton wthn the transportaton module are not taken nto account. Wlhte et al. (2014) consder a problem related to the structure of the mltary repar supply network n the USA Army. The authors explore when t s n the best nterest of the army to develop nternal capabltes (repar manufactures), rather than to use external ones. Overholts et al. (2009) focus on developng a mssle mantenance and securty forces schedulng model for the US Ar Force. Snce mantanng the 3

4 nter-contnental ballstc mssle weapon system requres that mantenance and securty teams travel to some 150 launch stes, a two-stage model that maxmally covers the locatons was developed. In order to nvestgate the mltary replenshment system of the Chnese Army, Chang et al. (2007) develop a smulaton model based on a system dynamc approach. Recently, Fan et al. (2010) use ths approach to study the bullwhp effect n a mltary mantenance supply system wthn the Chnese Army s repar and logstc system. Joo (2009) develops a dynamc approach for schedulng preventve mantenance at a depot wth lmted avalablty of spare modules and other constrants. A backward allocaton algorthm s proposed and appled to schedule the preventve mantenance of an engne module nstalled n advanced et traners The role of performance measures n supply chan management The dffculty n developng approprate performance measures depends on the complexty of the supply network structure and the number of echelons and the facltes n each echelon. Beamon (1999) provdes an overvew and evaluaton of the performance measures used n supply chan (SC) models. The author ndcates that the SC performance measurement system must contan at least one of three types of performance measures dentfed as necessary components n any SC performance measurement system: resources, outputs, and flexblty. Measurng the performance of the mltary chans s also mportant for recognzng troubled sectors, defnng success and assessng operatonal capabltes. Burns et al. (2010) argue that the prmary obectve of the mltary supply network (SC) s to attan a specfc state of readness at the lowest possble overall cost. They clam that the metrc for mltary SC success s readness for war and not the presumed proftablty of the SC. The combnaton of streamlned procurement, supply, dstrbuton, and lean practces s fundamental to create an effcent mltary supply chan management system that wll reduce the mnmal customer wat tme (see Lephart 2001). The effectveness of mltary supply networks can also be analyzed usng ndcators such as communcaton volume, logstcs volume, and watng tmes (see Gbson 2004). Once hndrances are dentfed, logstcans can mplement the rght procedures to unblock the flow of materal and ncrease the supply chan velocty. In order to develop an effcent mltary supply network for spare parts, the RAND Corporaton (see Johnson and Levte 2003) ntroduced for the US Army a new methodology, called defne-measure-mprove (DMI) that conssts of two maor ndcators, namely, customer wat-tme (CWT) and requston wat-tme (RWT). The CWT measures the supply network performance from the unt perspectve: the tme t takes to satsfy a request for a part necessary for makng a repar. In contrast, the RWT assesses how well the supply network for spare parts serves tself,.e., how much tme t takes to meet a unt s demand. Among other works n ths feld, we would lke to menton the paper by Wang (2006). In the works of Lephart (2001) and Gbson (2004), the authors omt a comprehensve defnton of mltary SC flexblty as well as performance measures to evaluate t, where SC flexblty s defned as the ablty of the SC to respond to the changng requrements of purchased components n terms of volume. Sokr 4

5 (2014) addresses these gaps and seeks to develop performance measures to assess the flexblty of a mltary SC. In hs study, the author dentfes two maor groups of flexblty measures: volume and delvery Analyss of models for ntegrated producton and transportaton operatons As a way to mprove performance measures and to render the whole supply network more effectve, many researchers n the feld consder the producton-nventory-dstrbuton process as an ntegrated system, where nventory s an ntegral part of a producton and dstrbuton system. Dhaenens-Flpo and Fnke (2001) ntroduce a mxed nteger programmng model based on mult-products, facltes and perods. Ther nterest s to smultaneously obtan the optmal quantty of manufactured products on a specfc producton lne and the optmal quantty of products transported from the stock to the end-customers. Low and Chang (2013) nvestgate a two-echelon supply network n whch the orders frst have been processed by a dstrbuton center and then delvered to the retalers wth respect to the prescrbed tme wndows. A nonlnear mathematcal model s formulated and solved by employng a genetc algorthm. The work by Stenrücke (2011) focuses on determnng an optmal schedulng for ntegrated producton-transportaton operatons wthn an alumnum supply network to mnmze total cost. A monolthc mathematcal model s formulated to tackle the problem. The obtaned results show that employng the monolthc model n determnng optmal solutons for practcal nstances s not a vable procedure. Therefore, the relax-and-fx heurstcs (RF heurstcs) are developed to obtan best possble solutons n reasonable computatonal tme. There are several other studes that focus on specal types of ntegrated producton and dstrbuton schedulng problems, where manufacturers produce ther product only when the customer places the order, that s, a stock of fnshed commodtes s forbdden (a so-called make-to-order approach) (see, for example, Chen and Varaktaraks 2005). The schedulng of the nternal ntegrated producton-transportaton operatons n manufactures s presented n the work of Levner et al. (2009). The authors develop a so-called herarchcally based heurstc approach (consstng of batchng and local search heurstcs, where they suggest to use adaptatons of the heurstcs suggested n Sotskov et al. (1996), Danneberg et al. (1996) or advanced strateges suggested n Brucker et al. (1996, 1997)) and obtan ntegrated schedules of both raw materals and the robot's transportaton operatons as well as transportaton and dstrbuton of batches of sem-fnshed products. In our study, we develop a new logstc and mathematcal approach to ncrease the effectveness of a mltary supply network. Ths s acheved by ntroducng a new optmzaton module, the "ntermedary, or commandng, module", between the mantenance and transportaton parts n order to manage to combne and coordnate the correspondng processes. The measurement of the effectveness of the ntegrated mltary supply networks s based on two performance measures: tme of response and mltary effectveness. To the extent these measures are mproved, the entre mltary supply networks wll be more effectve. In the present work, we use a 5

6 verson of the tradtonal ob shop schedulng problem to descrbe the mantenance operatons and employ the vehcle routng problem to present the transportaton module (followng the models by Dantzg and Ramser 1959 and Desrochers et al. 1988). 3. Demand-responsve mltary supply networks 3.1. Integrated approach to a mltary supply network In accordance wth the gudelnes of the mltary standard MIL-STD (1990), we analyze two key modules n mltary tranng programs, mantenance and transportaton. In terms of mantenance, once a confrmed order for repar parts s receved, the mantenance unt assgns prortes to avalable batches of repar parts and attempts to schedule and carry delver them as soon as possble (so that the batches wth a hgher prorty wll be processed frst). At the end of the repar process, the fxed batches are delvered to ther bases by employng the mltary vehcle fleet. The transportaton detachment also prortzes delveres and executes all the prortzed shpments as soon as possble. There are two basc approaches n modern management theory and practce. Both are well-known and wdely used: (1) the "make-to-order" polcy (.e., once a confrmed order for products s receved, products are produced, or, as n our study, repared), and (2) the "demand-responsve transport" polcy (offerng customers the avalable transportaton capacty n terms of vehcles and routes to satsfy ther unpredctable-n-advance demands). Nevertheless, t s worth notng that the mplementaton of the sequental modular approach n solvng ntegrated producton-transportaton problems may lead to obtan a local rather than a global optmum for the entre supply network. In order to properly analyze and create an effectve mltary supply network, we suggest shftng from the sequental modular approach (Fg. 1a) to the herarchcal (ntegrated) approach (Fg. 1b). The ntegrated approach expresses the ntegraton of two basc concepts mentoned above: "maketo-order" and "demand-responsve transport". To mprove the coordnaton between dfferent components wthn the ntegrated supply network and to prevent the domno effect from arsng n the frst place, we ntroduce an addtonal module nto the chan. Termed the "ntermedary or commandng module (IM), ts man role s to reduce the dscrepances n tme and common resources that are shared between dfferent components. Ths module mmcs the functons performed by the mltary commandng staff n modelng and managng ntegrated mantenance and transportaton operatons durng mltary exercses. Therefore, the coordnaton between these operatons should enhance the qualty and effectveness of army force tranng durng mltary exercses, reducng operatonal costs and mprovng the utlzaton of resources. 6

7 Demand - responsve mantenance Demand - responsve transportaton Integrated demand - responsve mantenance and transportaton Intermedary Module Mantenance Module Transportaton Module Mantenance Module Transportaton Module Fg. 1a. The modular structure of Fg. 1b. The structure of ntegrated sequental schedulng n a supply network. schedulng n a supply network. We defne and analyze two performance measures: tme of response and mltary We defne and analyze two performance measures: tme of response and mltary effectveness. The frst measure (tme of response) descrbes the ablty of the system to respond rapdly to the mltary requrements. If the latter requrement s senstve to a partcular choce of the performance measure dependng on the applcaton, then more than one performance measure can be assocated wth the tme of response. In partcular, customer watng tme (Barahona et al. 2007); requston wat-tme (Johnson and Levte 2003); and average customer wat-tme (Gbson 2004) measure the elapsed tme from the request to the delvery of the part to the end-customer. In ths paper, we study another performance measure: the total watng tme between mantenance and transportaton operatons. Ths measure s approprate for our model because the mantenance and transportaton modules functonng ndependently provde the mnmum tme for performng ther own operatons, wthout takng nto account the n-between tme. For example, n stuatons n whch the loadng zone s vulnerable to enemy bombng or mprovsed explosve devce (IED) attacks, the total watng tme between mantenance and transportaton operatons should be as short as possble. A reducton n ths tme perod can reduce the number of solders klled. In order to measure the ablty of the ntegrated mltary supply network to provde the rght product at the rght tme, we present and analyze the mltary effectveness performance measure. Ths measure, whch s the total amount of orders delvered to the mltary bases on tme, s smlar to the performance measures that can be found n Beamon (1999) (percent on-tme delveres). 4. Desgn of a new model for demand-responsve schedulng 4.1 Integrated mantenance-transportaton model and algorthm Transportaton and mantenance modules that are operated ndependently and separately can only be used to determne a local optmum. Ths, n turn, leads to a coordnaton between them n terms of the common resources; tme s usually not taken nto account. Consequently, the effectveness and demand-responsveness of the entre supply network may not be ensured. In order to gan the advantages mpled by coordnatng the transportaton and mantenance modules n terms of tme and shared common resources, we ntroduce, as noted above, an ntermedary module whose maor role s 7

8 to consoldate the work between the above modules. We then expand the transportaton-mantenance schedulng modules n two drectons. Frst, we extend and develop each of the ntal modules by ntroducng new varables and constrants n order to enhance ther demand responsveness. Second, along wth solvng the nteger programmng problems (IP) that have been obtaned wth an IP solver, we ncorporate a heurstc procedure n whch we combne several heurstc rules to help the decsonmaker to obtan a good ntal soluton (a so-called warm start ). Another mportant role of the heurstc procedure s to adust several (excessve) varables n order to decrease the problem's sze. The warm-start solutons generated by the heurstcs n the three modules are used by the IP solver (GAMS) whch works n two stages. Frst, t solves the IP problems formulated n the mantenance (MM) and transportaton (TM) modules. The results are then delvered to the ntermedary module (IM), where the GAMS solver solves an optmzaton problem formulated n ths module. The output of the IM serves as the nput for MM and TM, where the optmzaton problems are (agan) solved by GAMS. The two stages are teratvely repeated untl an acceptable soluton at the ntermedary module s obtaned (see Fg. 2). The computatonal procedure s descrbed n more detal n Secton 6. A practcal mplementaton of the suggested combnaton of the exact IP solver and the heurstc procedure s acheved wthn the ARENA 11.0 smulaton system. Mantenance Module Output Input Output Transportaton Module Input Input "Intermedary" module Output Fg. 2. The herarchcal two-stage algorthm 4.2 Intermedary module The man role of the ntermedary module s to dscover parameters that were not taken nto consderaton when the modular approach was mplemented. We assocate these parameters wth two maor categores: tme and capacty. The frst category nvolves delay tme parameters that occur between the MM and TM modules and are used to nvestgate and optmze response tme. On the other hand, the capacty parameters express the effectve utlzaton of all avalable mltary resources. 5. Mathematcal models The man purpose of the new models s to provde an optmal soluton for the ntegrated schedulng of mantenance and transportaton operatons that have been obtaned by coordnatng the three modules: MM, TM and IM. 8

9 5.1 Mantenance module (MM) The mantenance process starts when all the faled unts are collected and then delvered from the mltary bases to the repar factory, where they are grouped nto several batches accordng to ther type. In our model the unts of the same type (that s, of a certan batch) are to be processed on any machne successvely, one after the other, wthout a break. Due to ths, the set of unts of any batch wll be consdered as a sngle ob. Each ob (batch) s characterzed by a specfc repar route,.e., the sequence of the technologcal stages through whch the ob should pass. At any stage, a ob can be processed by a number of alternatve machnes. The reparng of all obs should be completed as soon as possble, that s, the total completon tme s to be mnmzed. Ths model s well-known n schedulng theory as a classcal ob shop problem (see Pnedo 2002). We extend the ob shop model by takng nto account the followng real-world constrants: (a) any ob n the recovery plant can be processed by alternatve, non-dentcal resources and (b) each ob has ts own due date. We ntroduce the followng notaton: Table 1:Parameters of the MM Parameter m The number of machnes n The number of obs (batches) The machnes, = 1,2,..., m Descrpton The obs, = 1,2,..., n (, ) Operaton of processng ob on machne N Set of all operatons, N = {(, )} A K Set of routng constrants of the form (, ) ( ', ). Such a routng constrant defnes that operaton (, ) should be processed on machne before beng processed on machne ' The number of the technologcal stages through whch ob passes wthn ts mantenance process k ( ) The technologcal stages through whch ob passes, k( ) = 1,..., K L, k ( ) p D Cluster (a group) of alternatve machnes avalable for ob on stage k ( ). When a ob s processed by a number of alternatve machnes at a stage k ( ), then these machnes compose a cluster L, k ( ) such that only one machne of ths cluster s selected at each stage Repar tme of ob on machne Due date for ob to be repared. The mltary commander s staff assgns ths parameter accordng to the extent and urgency of the mantenance actvtes requred Cost of a completon tme unt 9

10 Decson varables: y - Startng tme of processng ob on machne. B - Bnary operaton-machne varable: B 1f machne s assgned to perform ob, k( ) n stage k ( ) (that s, L k, ( ), k ( ) = the ablty of the ob to be processed on alternatve machnes. F - Completon tme for ob : ndex of the last machne n the last stage F ), and 0, otherwse. Ths varable s used to descrbe = Max( y + p ) = yi ( ), + pi ( ),,, where I( ) s the K of processng ob. n F = 1 - Total completon tme. The schedulng problem s to select alternatve machnes and to determne the startng tmes of the operatons n order to mnmze the total completon tme subect to the defned due dates and routng constrants. We suggest the followng mathematcal model for ths problem: Problem MM: Mn n F = 1 (1) subect to the constrants: y ' y p, for all (, ) ( ', ) A (2) y y' p' or y ' y p, for all (, ' ) and (, ), = 1,2,..., m, = 1,2,..., n (3) L,k() B,k( ) = 1, for all L = { L, k ( ) }, = 1,2,..., n, k( ) = 1,2,..., K (4) F D, for all = 1,2,..., n (5) y 0, for all (, ) N (6) B, k ( ) {0,1}, for all = 1,2,..., m, = 1,2,..., n, k( ) = 1,2,..., K (7) The optmzaton crteron s the mnmzaton of total completon tme (1). Inequaltes (2) ensure that the operaton ( ', ) cannot start before operaton (, ) s completed. Inequaltes (3) ensure that each machne can perform only one ob at a tme. Condton (4) ensures that only one alternatve machne should be chosen at each stage (that s, n each cluster). Inequaltes (5) express that the completon tme for any ob s restrcted by the due date. Inequaltes (6) ensure that the decson 10

11 varables y are non-negatve. Constrants (7) guarantee that the operaton-machne assgnment varables B, k ( ) are bnary. We observe that the consdered mantenance schedulng problem s formulated as a mxed nteger lnear problem wth dsunctve constrants. There are a sgnfcant number of software programs for solvng such a type of problem. In ths study, we use the standard MILP solver of the commercal optmzaton package GAMS (see Rosenthal 2010) to solve ths problem. Results of computatonal experments are presented n Secton Transportaton module (TM) After the mantenance operaton s completed, the mltary vehcle fleet consstng of the homogeneous trucks wth the same loadng capacty must transfer the repared batches to the mltary bases. The sze of the vehcle fleet nvolved n the transportaton s assumed to be known; t s chosen by the mltary commander s staff dependng on logstcal and securty condtons. Also, n order to secure the transportaton, armored vehcles can be used as a mltary convoy. Beng accompanyng by the convoy, the delvery trucks are able to employ shorter routes and move faster, and, therefore, they may reach the mltary bases n less tme. Each base has ts own demand and should be served by exactly one vehcle wthn the specfed tme wndows. The problem to be solved s to mnmze the followng cost components: () the total travel cost assocated wth transportng repared batches from the repar plant to ther fnal destnatons (by the trucks and the accompanyng mltary convoy), and () the total cost of loadng/unloadng operatons for all delvered batches. The consdered problem s a verson of the vehcle routng problem (VRP). The operatons research lterature contans many versons of the VRP and a sgnfcant number of computatonal methods for ther soluton (see, for example, the excellent survey by Irnch et al. 2014). In spte of an abundant lterature on the VRP, to the best of our knowledge, there s no model comprehensvely treatng the mltary logstcs factors, whch we ntend to consder n the present paper. Our model s smlar to the VRP models by Dantzg and Ramser (1959), Laporte et al. (1985) and Desrochers et al. (1988), however, n contrast to the model gven n Dantzg and Ramser (1959), our model takes the tme wndows and truck capacty constrants nto account; further, n comparson wth the models n Laporte et al. (1985) and Desrochers et al. (1988), we ntroduce two types of vehcles (delvery trucks and armed convoy) nvolved n the mltary logstcs; and, fnally, none of the above works handle correspondng transportaton and loadng costs. We ntroduce the followng notaton: 11

12 Table 2: Parameters of the TM Parameter G Graph, G = ( V, A) V A Descrpton Set of vertces, V = { 0} N, where { 0} ndcates the depot (repar factory) and N = { 1,2,..., n} s the set of mltary bases Set of arcs, A = ({ 0} N) I ( N {0}), where I N N s the set of arcs connectng the mltary bases, { 0} N contans the arcs from the depot to the mltary bases, whereas N {0} contans the arcs from the mltary bases to the depot d Demand at mltary base, N K C t t u, r c b The truck fleet sze Capacty of a truck Travel tme from mltary base to. We assume that the transportaton tme s symmetrc, that s, t = t and t = 0 Tme shft, that s, the maxmal possble decrease n the transportaton tme t assocated wth employng a mltary escort to secure the transportaton over an arc (, ) Tme wndows at base (lower and upper tme bounds, respectvely) Travel cost assocated wth an employng truck for transportaton from to Travel cost assocated wth employng a mltary convoy for securng transportaton from to α M Loadng cost at mltary base Any very large number Decson varables: s - Arrval tme at mltary base, 0 s. 0 = y - Load of truck arrvng at mltary base, 0 y. 0 = x - Truck bnary routng varable: x = 1f arc (, ) s used on the tour by a truck, and 0, otherwse. z - Convoy bnary routng varable: z = 1f arc (, ) s used on the tour by a convoy, and 0, otherwse. The transportaton problem s to obtan optmal routes to mnmze total travel cost assocated wth employng trucks and armed escort n the delvery process (, ) [ c + ] x b z as well as the total A 12

13 loadng cost α y subect to the constrants on the bounded vehcle fleet, tme wndows and N truck capacty. We propose the followng mathematcal model for ths problem: Problem TM. Mn [ c (, ) A x + b z ] + N α y (8) subect to the constrants: N x 0 K (9) x0 = x0 N N N N x = 1, N (11) x = 1, N (12) (10) z x, (, ) A (13) s + t t z s M 1 x ), (, ) I ({0} N) (14) ( u s r, N (15) y + d y M 1 x ), (, ) I ({0} N) (16) ( 0 y C, N (17) s, y 0, N (18) x, z {0,1}, (, ) A (19) The optmzaton crteron s gven n (8). Inequaltes (9) ensure that at most K trucks depart from the depot. Equaton (10) ensures that all trucks departed from the depot also have to return. Equatons (11) and (12) express that each mltary base must be served exactly once. Inequaltes (13) ensure that the mltary convoy cannot be employed on arc (, ) f ths arc s not used by a truck on the tour. Inequaltes (14) ensure that a truck cannot arrve to mltary base before s + t, f t travels from to. Inequaltes (15) ensure that all tme wndows are respected. Condtons (16) and (17) ensure the feasblty of the loads. Condton (18) expresses that the decson varables s, y are non-negatve. Fnally, condton (19) defnes the bnary varables. Inequaltes (14) and (16) provde the subtour elmnaton n the same way as n the model by Desrochers et al. (1988). Smlarly to the descrpton of the MM module, we agan use the standard MILP solver of GAMS for ts soluton. The computatonal results are presented n Secton 7. 13

14 5.3 Intermedary module (IM) We defne the man role of the ntermedary module as a coordnaton between MM and TM operatons n terms of capacty (effectve usage of the mltary resources) and tme (reducng delay tme that occurs between the MM and TM). In the IM, we desgn and solve the followng optmzaton problem: determnng the best batch-vehcle allocatons such that the total watng tme between the mantenance and transportaton operatons wll be mnmzed. We ntroduce the followng notaton: Table 3: Parameters of the IM Parameter Descrpton k The trucks, k = 1,..., K Allowed delay tme that elapses between the moment when repar of batch s fnshed γ and the moment when ts transportaton starts Interval of allowed values for the delay tme between all mantenance and transportaton [ TD ', TD "] operatons; ths parameter s assgned by the mltary commander s staff dependng on the urgency of the part needed TQ ST k F The entre number of unts of batch that has to be delvered to the customers Startng tme of transportaton for batch by truck k (ths parameter s receved as the output of the transportaton module). Completon tme for batch (ths parameter s receved as the output of the mantenance module). Decson varables: U - Fnsh tme shft, that s, the ncrease n the fnsh tme of a repar operaton of batch made for reducng the total watng tmes between the repar and transportaton operatons. U - Startng tme shft, that s, the ncrease/decrease n the startng tme of a transportaton operaton k of truck k. Qk - Number of repared unts of batch that are loaded to truck k. Problem IM. ST TD k ' Mn + U n k U n K = 1 k= 1 [ ST k + U ( F + U + K = 1 k= 1 k ) γ, for U k TD '' ( F + U )] (20) = 1,..., n, k = 1,..., K (21) (22) n = 1 Q k C, for k = 1,..., K K Q k = k =1 TQ, for = 1,..., n (23) (24) 14

15 U, U R, for = 1,..., n; k = 1,, K (25) k Q 0, for = 1,..., n, k = 1,..., K (26) k Accordng to (20), the obectve s to mnmze the total watng tme between the mantenance and transportaton operatons. Inequaltes (21) ensure that the delay tme between the nstant when repar of batch s fnshed ( F ) and the nstant when ts transportaton to the customer (.e, the base) by truck k starts ( ST ) s lmted. Inequaltes (22) ensure that the total tme gap between all the k mantenance and transportaton operatons s lmted. Inequaltes (23) ensure that the total amount of batches delvered by truck k does not exceed the capactyc. Equatons (24) ensure that all the requred quantty TQ of unts n batch s delvered to the end-customers. Condton (25) expresses that the varables U, U k are real numbers. Fnally, condton (26) expresses that the varables Q k are non-negatve. The IM s formulated as the MILP and we solve t by the standard MILP solver of the GAMS. The mode of cooperaton and teratve data exchange between blocks was descrbed n Secton Descrpton of the algorthm As a soluton to the ntegrated transportaton-mantenance schedulng problem, we present a heurstc search procedure that combnes dfferent heurstc schedulng rules. At frst, the heurstc rules are mplemented usng ARENA As a result, good feasble solutons are found and some of the (nteger) varables become fxed. Ths decreases the problem sze. The man fxed varables are the startng tme y of processng ob on machne (n the module MM) and the truck bnary routng varable x (n the module TM). The varables become fxed f durng the prescrbed number of teratons ther values reman the same. The solutons that emerge are used as startng solutons for solvng the nteger programs (IP) n the MM, TM, and IM modules. The regular nteger programmng solver GAMS/CPLEX (see Rosenthal 2010) s used to acheve ths obectve. Each entty n the smulaton model s assocated wth ether a batch n the MM or a vehcle n the TM. Each mantenance batch n MM s processed followng a prescrbed path consstng of dfferent types of repar machnes wth the gven due dates beng taken nto account. The best processng polcy s determned by usng and comparng varous heurstcs rules, namely: EDD, LIFO, FIFO, and slackbased prortes. The descrpton of these basc heurstc rules can be found n Pnedo (2002), and s not dscussed here. At the end of the mantenance process, the fxed batches are delvered back to ther bases by usng a mltary fleet consstng of vehcles wth dfferent capactes. These capactes are generated randomly at each teraton by usng the followng rule: New_Capacty=Unform [0.5,1]*Intal_Capacty. In turn, n TM we use the coeffcent weghted tme dstance (CWTD), a 15

16 heurstc procedure proposed by Galc et al. (2006). The ste wth the lowest decson measure wll be served frst. We extend Galc's algorthm as follows: each parameter n the heurstc algorthm (namely, tme, dstance and amount) s assgned a specfc weght that s updated randomly at each teraton accordng to the unform dstrbuton. The role of the IM s to coordnate the operatons between MM and TM. The IM s presented n the smulaton model n terms of: (a) an exstng, addtonal pool of vehcles provded by the external thrd-part logstc companes; (b) determnng new due dates for the obs n MM and new startng tmes for the transportaton operatons of the vehcles wth respect to the tme of response and mltary effectveness performance measures and (c) updatng the vehcle capacty. The teratve soluton procedure s termnated when the mprovement n the value of the obectve functon n the IM s less than a prescrbed gven value. The followng flowchart n Fg. 3 vsually presents the algorthm: Preprocessng (preparaton of the data) Mantenance Module: Schedulng the repar of faled obs by usng due date heurstcs Transportaton Module: Delverng repared obs by usng transportaton heurstcs Re-runnng the smulaton Intermedary Module: Calculatng new data (due dates and weghts) for the mantenance and transportaton heurstcs Whether the mprovement of the obectve functon n IM s less than a prescrbed gven value? No Yes Generatng ntal ("warm start") solutons Solvng teratvely Mxed Integer Programmng problems for mantenance, transportaton and ntermedary modules and generatng the fnal solutons Fg. 3. The flowchart of the algorthm. 16

17 7. Computatonal results In the operatons research lterature, there are two dstnct approaches for modelng ntegrated schedulng and logstcs systems. The frst approach, monolthc, s based on groupng all the condtons of several sub-problems nto a sngle problem and formulatng the latter as a large mxednteger program to be solved to optmalty (Sawk 2009, Stenrücke 2011). The second approach, referred to as modular or herarchcal, parttons a large-scale problem nto a network of lnked subproblems and optmzes them sequentally so that an output of one problem serves as an nput for another one. Then the computatonal tme s consderably reduced although the obtaned soluton s sub-optmal (Sawk 1999, Park 2005). We carred out two types of computatonal experments to determne, frst, whether or not our herarchcal approach was effectve n comparson wth the exact monolthc model and, second, the effcency of the sequental modular model and our herarchcal model. 7.1 Comparson of monolthc and heurstc approaches In order to estmate the effcency of the exact monolthc approach, we constructed an nduced parametrc nteger programmng problem n whch all the constrants of Problems MM, TM and IM of Secton 5 are grouped together nto one monolthc problem. Snce the structure of ths problem drectly follows from the problems ndcated, we do not need to present here ts formulaton. To fnd an exact soluton of the monolthc problem, we used the solver GAMS/CPLEX. Ths solver s based on the branch-and-bound enumeratve search and desgned for solvng non-lnear mxed nteger problems. The comparson was carred out on the NEOS server. The hardware specfcatons of the server machnes (NEOS-2 and NEOS-4) are as follows: Dell PowerEdge R410 servers, CPU - 2x Intel Xeon 2.8GHz (12 cores total), HT Enabled, 64 GB RAM. The problem nstances were generated randomly as shown n Table 4. Table 4 Data used n the comparson analyss. Data type Range Number of obs Unf(5,24) Number of machnes Unf(2,4) Number of vehcles Unf(2,3) Number of bases Unf(2,3) Repar tme (n hrs) Unf(0.1,0.3) Transportaton tme (n hrs) Unf(0.2,1) Tme wndows (lower bound) Unf(2.5,4) Tme wndows (upper bound) Unf(4.5,6) Servce tme (n hrs) 0.1 A total of about 200 dfferent nstances were solved. For all experments, the calculaton tme was lmted to 1 CPU-hour per problem nstance. For each nstance, the solver provded an exact but computatonally expensve soluton. The largest problem nstance for whch we could fnd an exact 17

18 soluton usng the GAMS solver had 82 obs grouped nto 9 batches; 10 machnes; 6 vehcles; and 14 customers. The nduced monolthc Mxed Integer Programmng Problem had 350 varables, among them 280 dscrete ones and contaned 580 constrants. For ths nstance, the runnng tme was 52 mnutes. The soluton of smaller nstances requred from eght to 50 mnutes. To solve all these nstances, the GAMS/CPLEX va NEOS server requred a total of 85 hours of CPU tme. In addton, all these problems were also solved by the ARENA 11.0 that spent from 50 seconds to ten mnutes per nstance. All the derved solutons turned out to be optmal. The computatonal experment showed that for the problem nstances that were consdered, the heurstc system yelded the exact soluton and, operated much faster than the monolthc one. These computatonal results qualtatvely agree wth the smulatons reported earler by Sawk (1999), Sawk (2009). 7.2 Comparson of sequental and herarchcal approaches In ths sub-secton, we compare numercally the sequental and herarchcal approaches. In order to do ths, we performed experments n the smulaton model wth random data usng two scenaros: (1) the amount of obs ranged from 25 to 200 and (2) there were two to twenty vehcles (the correspondng data and computatonal detals can be provded by the authors upon a request). The output conssted of the total watng tme between the mantenance and transportaton operatons and the total number of obs delvered to end-customers on tme. The computatonal results for the two scenaros are as follows: (a) Scenaro 1. The number of obs rose from 25 to 200. Fg. 4 presents the tme of response as a functon of the number of obs. Fg. 5 presents the mltary effectveness measure as a functon of the number of obs. Watng tme per ob (n hrs) Modular Integrated Tme of response (best soluton) Number of obs Watng tme per ob (n hrs) Modular Integrated Tme of response (average soluton) Number of obs Fg. 4. The tme of response as a functon of the number of obs. 18

19 115 Mltary effectveness (best soluton) 100 Mltary effectveness (average soluton) Number of obs delvered n tme Number of obs delvered n tme Modular Integrated Number of obs 0 Modular Integrated Number of obs Fg. 5. The mltary effectveness measure as a functon of the number of obs. The graph n Fg. 4 shows that the ntegrated approach yelds a better tme of response than the sequental modular approach. Wth an ncrease n the number of obs, the total watng tme n the ntegrated approach grows more slowly. The reducton n the watng tme s acheved by a better assgnment of the obs to the avalable vehcles. The obtaned results are shown n Fg. 5, where we can see that wth the ncrease n the number of obs, the ntegrated approach gves a better obs-ontme/total-number-of-obs rato. (b) Scenaro 2. The number of vehcles ncreased from 2 to 20. Fgs. 6. and 7, respectvely, present the tme of response and mltary effectveness as functons of the number of vehcles. Tme of response (best soluton) 2.0 Tme of response (average soluton) Watng tme per ob (n hrs) Watng tme per ob (n hrs) Modular Integrated Number of vehcles Modular Integrated Number of vehcles Fg. 6. The tme of response as a functon of the number of vehcles. Number of obs delvered n tme Mltary effectveness (best soluton) Number of obs delvered n tme Mltary effectveness (average soluton) Modular Integrated Number of vehcles Modular Integrated Number of vehcles Fg. 7. The mltary effectveness as a functon of the number of vehcles. 19

20 The graphs n Fg. 6 show that the ntegrated approach acheves a better tme of response performance measure than the modular approach. From Fg. 7, we can see that the ntegrated approach acheves a hgher mltary effectveness than the modular approach. Wth an ncrease n the number of vehcles n the model, the number of obs delvered on tme to the end-customers n the ntegrated approach ncreases more rapdly. The reason of ths s that the IM uses the avalable vehcle fleet more effectvely so that the number of obs delvered n tme to the end customers s larger than n the standard modular approach not usng IM. In both scenaros the average runnng tme of ARENA 11.0 was between two and 53 mnutes. 8. 4TConclusons Proectng and sustanng a mltary force, partcularly durng tranng exercses, hnges on the successful establshment and management of demand-responsve mltary supply networks. The large number of dfferent types of tems n mltary warehouses, the tght constrants on operatonal delays and changeable tme demands are the man factors that nfluence the responsveness of mltary supply networks. For repar and mantenance operatons, t s unrealstc to rely exclusvely on the local optmzaton of mantenance and transportaton operatons. The responsveness of the mltary supply network s formally descrbed and analyzed wth the help of two performance measures: tme of response and mltary effectveness. A hgh demand responsveness s obtaned by ntegratng the mantenance and transportaton modules. Integratng these separate modules nvolves ncorporatng an ntermedary module. Ths module ncludes a number of parameters that are not taken nto account when the repar and transportaton processes are planned separately. The role of the ntermedary module s twofold. On the one hand, ths module nterprets and mmcs the role of the mltary commandng staff n effectvely modelng and managng ntegrated mantenance and transportaton operatons durng mltary exercses. An effectve modelng and managng make mltary supply networks more demand-responsve and effectual. On the other hand, the ntermedary module makes t possble to acheve a global mathematcal optmum rather than a local optmum for a mltary supply network. To fnd an optmal soluton of the ntegrated mathematcal model, we ntroduced a new two-stage heurstc search method that we used to solve at the frst level two schedulng problems arsng n the ndependent mantenance and transportaton modules. Then, at the second level, we use the outputs of the two modules, obtaned at the frst level, as the nput for the ntegrated problem of the second level. The optmal soluton obtaned at the second level s ether accepted or, f necessary, leads to a change of the nput data of the modules at the frst level. Ths algorthm s solved by a combnaton of heurstc rules and mxed nteger programmng technques. 20

21 In order to compare the effcency of the exact and heurstc approaches, an equvalent monolthc nteger programmng (IP) problem was formed. A set of randomly generated IP problems was solved sequentally by the branch-and-bound method mplemented by the standard software (GAMS/CPLEX) and by our heurstc method whch was mplemented wth the ARENA smulaton system. The exact method proved to be computatonally expensve, ts runnng tme beng prohbtvely larger even for small sze problems. The comparson across the sequental modular and ntegrated approaches was based on measurng the tme of response and mltary effectveness. The numercal results demonstrate that the ntegrated approach acheves better mltary performance measures than the modular one. Coordnatng the schedulng decsons between dfferent partcpants mproves the effectveness of both the ndvdual partcpants as well as the entre mltary supply network. These observatons confrm those that have been made snce the 1990s n many other studes on supply chan management (see, e.g., Dhaenens- Flpo and Fnke 2001 and Krepl et al. 2006, among others). Note that the mathematcal models descrbed n ths paper, although beng ntated by mltary logstcs applcatons, can also be used n dfferent non-mltary commercal supply networks. A promsng drecton for future research would be to extend the suggested two-stage modelng approach for explorng mult-echelon supply networks. References Akgün, I., and Tansel, B, Ç., Optmzaton of transportaton requrements n the deployment of mltary unts. Computers and Operatons Research, 34, Barahona, F., et al., Inventory allocaton and transportaton schedulng for logstcs of networkcentrc mltary operatons. IBM Journal of Research and Development, 51, Beamon, B, M., Measurng supply chan performance. Internatonal Journal of Operatons and Producton Management, 19, Brucker, P., Hurnk, J., and Werrner, F., Improvng local search heurstcs for some schedulng problems I. Dscrete Appled Mathematcs, 65, Brucker, P., Hurnk, J., and Werrner, F., Improvng local search heurstcs for some schedulng problems. Part II. Dscrete Appled Mathematcs, 72, Burns, L., Tseng, F., and Berkowtz, D., Global network analyss n a mltary supply chan: usng a systems based approach to develop a next-generaton end-to-end supply chan performance measurement and predcton system. In Proceedngs of the 2010 Cambrdge Internatonal Manufacturng Symposum, Chang, P,C., Fan, C,Y., and Huang, W,H., A system dynamcs smulaton approach for mltary supply chan management. In Proceedngs of the Second IEEE Internatonal Conference on Innovatve Computng, 422. Chen, Z-L., and Varaktaraks, G, L., Integrated schedulng of producton and dstrbuton operatons. Management Scence, 51,

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