Enabling Greater Access to Home Meal Delivery

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1 Loyola Unversty Chcago Loyola ecommons Informaton Systems and Operatons Management: Faculty Publcatons & Other Works Qunlan School of Busness 2013 Enablng Greater Access to Home Meal Delvery Macek Nowak Loyola Unversty Chcago, Leo Gala Rochester Insttute of Technology Mke Hewtt Rochester Insttute of Technology Recommended Ctaton Nowak, Macek; Gala, Leo; and Hewtt, Mke. Enablng Greater Access to Home Meal Delvery. European Journal of Operatonal Research,, :, Retreved from Loyola ecommons, Informaton Systems and Operatons Management: Faculty Publcatons & Other Works, Ths Artcle s brought to you for free and open access by the Qunlan School of Busness at Loyola ecommons. It has been accepted for ncluson n Informaton Systems and Operatons Management: Faculty Publcatons & Other Works by an authorzed admnstrator of Loyola ecommons. For more nformaton, please contact ecommons@luc.edu. Ths work s lcensed under a Creatve Commons Attrbuton-Noncommercal-No Dervatve Works 3.0 Lcense.

2 Enablng Greater Access to Home Meal Delvery Leo Gala, Mke Hewtt 1 Department of Industral and Systems Engneerng Rochester Insttute of Technology Rochester, NY 14534, U.S.A, Emal: mrhee@rt.edu Macek Nowak Informaton Systems and Operatons Management Graduate School of Busness Loyola Unversty Chcago Abstract Non-proft organzatons lke the Meals On Wheels (MOW) assocaton of Amerca prepare and delver meals, typcally daly, to approxmately one mllon homebound ctzens n the Unted States alone. However, many MOW agences are facng a steadly ncreasng number of clents requestng meal servce wthout an ncrease n resources (ether fnancal or human). One strategy for accommodatng these requests s to delver multple (frozen) meals at a tme and thus make fewer delveres. However, many of the stakeholders (funders, volunteers, meal recpents) value the relatonshps that are developed by havng a clent receve daly delveres from the same volunteer. Further, meal recpents may be concerned wth the qualty of food delvered through a frozen meal. In ths paper, we develop a method for ntroducng consoldaton nto home meal delvery whle mnmzng operatonal dsruptons and mantanng clent satsfacton. Wth an extensve computatonal study, the savngs assocated wth varous levels and types of dsruptons are detaled. Keywords: OR n health servces, transportaton, nteger programmng, heurstcs. 1. Introducton On a daly bass, non-proft organzatons lke the Meals On Wheels Assocaton of Amerca (MOW) delver approxmately one mllon meals throughout communtes 1 correspondng author; emal: mrhee@rt.edu, phone: , fax: Preprnt submtted to Elsever January 25, 2013

3 n the Unted States. Wthn each of these communtes, many ndvduals aged 60 and older rely on government funded programs lke MOW to meet ther detary needs for sustanng a healthy lfestyle. In addton to the agng populaton, MOW serves ndvduals who are ncapable of sustanng themselves due to medcal lmtatons. The assstance MOW provdes enables ther clents to reman comfortable n ther homes nstead of requrng them to relocate to subsdzed housng or nursng homes, ether at personal or government expense. Indvduals who wsh to receve free or reduced-cost meal assstance must qualfy, where the qualfcaton process also determnes the number of meals the ndvdual should receve n a week. To delver meals, MOW reles on a workforce comprsed of both professonals and volunteers from each communty, and routes between 800,000 and 1.2 mllon volunteers annually n the Unted States [17]. MOW s seeng s a steadly ncreasng demand for meals. From 1980 to 2002, the demand for meals n the Unted States ncreased by 290%. Whle contrbutons from prvate organzatons and cost reducton efforts have helped MOW to ncrease the number of meals t delvers, ther capabltes have not grown at the same rate as demand. As a result, MOW agences are often forced to put some or all of a clents delveres on a wat-lst, although ths s clearly undesrable. Smlarly, MOW antcpates a sgnfcant ncrease n the number of senors n Amerca that face the threat of hunger (8.3. mllon n 2010, 9.5 mllon projected n 2025), and has stated that ts vson s to end senor hunger by 2020 [18]. One strategy for reducng costs and ncreasng capacty that some MOW agences have undertaken (whle there s a natonal assocaton, agences are locally owned and operated) s to delver frozen meals. An offcal from New York Cty stated, for us t s really been about creatng a more effcent system and not havng anyone on a watng lst. That same offcal mentoned that by delverng frozen meals twce a week to 40% of ts recpents they were able to reduce the number of professonal drvers they pad from seventeen to three [14]. Smlarly, delverng frozen meals has enabled MOW agences to serve recpents n rural areas that, gven lmted drver resources, were too costly to reach on a daly bass [12]. By usng frozen meals, MOW agences fnd the benefts of consoldatng delveres that many other transportaton provders have already dscovered. However, whle frozen meals allow for consderable cost savngs, there are several operatonal challenges. Whle the prmary purpose of a MOW agency s to provde nutrtous meals to those who can not provde one for themselves, many agences and ther supporters (both volunteers and funders) also see themselves as provdng human nteractons and relatonshps to those who may otherwse have lttle human contact [15]. As an example, many MOW agences lke to nvoke the slogan more than just a meal. 2

4 Thus, whle there are some clear effcences that can be ganed by delverng frozen meals, some fundng agences that support MOW programs are retcent to support ther delvery as they fear dong so wll dmnsh the health montorng capabltes of Meals On Wheels. Further, many have clamed that the qualty of the frozen meals s smply nferor to the hot meals [8]. Lke major parcel delvery companes such as FedEx and UPS, MOW faces the problem of makng home delveres n large szed communtes, contanng demand at vared locatons, and wth delvery routes lmted by vehcle capacty. However, whle ndustry leaders n parcel delvery have the engneerng and fnancal resources to nvest n technology that handles these routng scenaros and to answer strategc what-f questons regardng how delveres are made, many communty based MOW agences do not. Creatng a good set of routes that are repeated on a daly bass can be dffcult enough wthout an advanced routng algorthm. Consoldatng delveres on those routes may requre the creaton of a unque route for each day of servce, puttng an even greater stran on the MOW agency s lmted operatonal resources. The contrbuton of ths paper s a methodology that MOW agences can use to effectvely ntroduce frozen meal delvery whle lmtng operatonal dsrupton. Whle ths work s presented n the context of ts nspraton, Meals On Wheels, the methodology and results presented are pertnent to any organzaton that wshes to quantfy the tradeoffs between consoldatng delveres and mantanng hgh levels of customer servce, such as a parcel delvery or LTL carrer. In partcular, the approach presented here s confgurable to the operatonal realtes faced and qualty metrcs valued by the MOW agences. The focus s on a settng where clents currently receve hot meals fve days a week and the agency has an establshed set of delvery routes that are executed daly. Based on those routes, ths approach can mantan consstency wth hstorcal operatons by ensurng that a clent remans on the same route as before, and that clents are seen n the same order. Introducng consoldaton nto daly delvery routes s remnscent of vendormanaged nventory or the Inventory Routng Problem [1, 5]. However, we consder constrants that have not been ncluded n the nventory routng problem. For example, ths approach can support fxed routes [4, 9] by servng the clents n the same order on every route. Such a restrcton can have many operatonal advantages, ncludng makng t easer to synchronze the meal preparaton and vehcle loadng process. Ths problem s nherently perodc and thus shares characterstcs wth the Perodc Vehcle Routng Problem (PVRP) [11]. However, lke the work on the PVRP wth Servce Choce [10], the number of tmes an ndvdual s vsted s not an nput to the model but a decson varable. Some of the earlest work on the home meal delvery problem was presented n [3], 3

5 whch used space-fllng curves as the bass of a system that could be mplemented on two rolodex cards. Whle the approach presented here s fundamentally dfferent, the emphass on operatonal ease s shared, as the focus s on mnmzng the operatonal dsruptons assocated wth ntroducng consoldaton nto delvery operatons. Ths approach can also enforce varous qualty of servce-type metrcs. For an agency that s concerned wth the qualty dsparty between frozen and hot meals, ths approach can ensure that each clent receves at least a mnmum number of hot meals. For an agency that wshes to mantan the relatonshps ther drvers have wth clents ths approach can enforce a maxmum number of days between vsts and that a clent s always vsted by the same drver. Qualty-of-servce metrcs were also studed n the context of nventory routng n [7]. To determne the potental effcences assocated wth delverng frozen meals, the problem of ntroducng consoldaton s formulated as a mxed nteger program (MIP). Dfferent varants of ths MIP are consdered, based on the constrants to be enforced, e.g. fxed routes, a maxmum number of days between delveres to a clent, etc. A characterstc of the problem s exploted to lmt the soluton space, allowng for shorter computatonal tmes. Transportaton costs ncurred by executng daly routes wth and wthout consoldaton are compared n an extensve computatonal analyss that detals the constrants that most mpact transportaton costs. It s found that there s vrtually no mpact on the savngs from consoldaton when operatonal dsruptons are prohbted. The savngs decrease as qualty of servce s more strctly enforced, but even when consoldatng just two delveres nto one for each customer, transportaton costs are reduced by 10%. For MOW agences that are operatng at such tght margns, those savngs can have a very sgnfcant mpact. Fnally, t s shown that through consoldaton, more space may be created on the daly routes to servce addtonal clents whle stll cuttng transportaton costs. The remander of ths paper s organzed as follows. In Secton 2, the problem of ntroducng consoldaton s presented along wth several constrants used to lmt operatonal dsrupton and clent dssatsfacton. In Secton 3 the heurstc used n ths study s developed. Secton 4 contans the results of an extensve computatonal study of the benefts assocated wth consoldaton under varous settngs. Fnally, Secton 5 provdes manageral nsghts based on the results. 2. Home Meal Consoldaton and Delvery Problem We next descrbe the problem of ntroducng consoldaton nto routes that are executed by a set of delvery vehcles, v V, every day, t, over a plannng horzon, T, and how ths problem s modeled as a mxed nteger program (MIP). One week 4

6 consstng of fve days s used for ths problem (thus T = 5). In ths work, t s assumed that the soluton for a week can be used repeatedly to provde servce over a longer perod, such that frozen meals left at the end of one week may be used n the followng week. Each day the set of delvery vehcles begn at a common depot, D, vstng a set of clents, I, delverng a sngle meal to each, and returnng to the depot. To preserve the qualty of the meals delvered, they are stored n a contaner whch can hold at most K meals. In the computatonal study n Secton 4, K s the same for both frozen and hot meals. The cost of travelng from clent to clent j s denoted by c j. By delverng frozen meals and, n partcular, multple meals on a sngle day, the delvery route may skp a clent on certan days. Thus, the decson varable y vt {0, 1} s defned to denote whether or not vehcle v vsts clent I on day t T, qc vt Z to represent the number of cold meals delvered by vehcle v, and qh vt {0, 1} the number of hot meals delvered by vehcle v. Because hot and frozen meals can not be n the same meal contaner (and each vehcle carres at most one contaner), delverng a hot meal to one clent on day t precludes delverng frozen meals to any clent that day by that vehcle. Thus, the varable r vt {0, 1} s defned to ndcate whether or not vehcle v delvers hot meals on day t. The decson varable x vt j {0, 1} s defned to denote whether vehcle v travels from locaton D I to locaton j D I on day t T. Fnally, defne z v {0, 1} to represent whether clent I s served by vehcle v V. The baselne home meal consoldaton and delvery problem (HMCD) problem s then to: subject to mnmze v V y vt t T I D j I D c j x vt j z v = 1 I, (1) v V z v I, v V, t T, (2) qc vt + qh vt = T I, (3) v V t T qc vt T y vt I, v V, t T, qh vt y vt I, v V, t T, (4) r vt q vt h r vt I, v V, t T, (5) q vt c T T r vt I, v V, t T, (6) 5

7 S j S j I D qc vt + qh vt K v V, t T, (7) I j I j I x vt j = y vt I, v V, t T, (8) x vt j = y vt I, v V, t T, (9) x vt j = j I D x vt j I, v V, t T, (10) x vt j S 1, S I D, D 2, v V, t T. (11) The objectve s to mnmze the total cost assocated wth travel. Constrants (1) and (2) ensure that each clent s seen by exactly one delvery person durng the week. To help foster relatonshps between clents and delvery persons, as was shown to be benefcal n [19], when a clent s moved between vehcles on one day, she s moved to that vehcle on all days, ensurng that each clent s seen by at most one delvery person each week. Constrants (3) ensure that each clent receves a meal for every day of the tme horzon, whereas constrants (4) ensure that a clent s not delvered a hot or cold meal on a day unless the delvery vehcle vsts them on that day. Note that an ndvdual can receve at most T frozen meals and at most one hot meal n one delvery. Constrants (5) and (6) together ensure that f one clent receves a hot meal on day t then no other clent receves a frozen meal on that day. Note constrants (5) also ensure that when hot meals are delvered on a day, every clent receves a hot meal that day. Constrants (7) ensure that vehcle capacty for meals s not exceeded. Constrants (8, 9, 10, and 11) are standard routng constrants. Whle the HMCD ensures that each clent receves fve meals a week, t requres lttle else. From the delvery agences perspectve, t may yeld delvery routes that vary sgnfcantly by day, whch could complcate admnstratve tasks. From the clent s perspectve, the use of frozen meals lmts the number of delveres and the tme for nteracton between clent and volunteer, harmng ths relatonshp. Also, some clents only receve frozen meals, whch they may enjoy less than hot meals. Thus, we next dscuss constrants that can be added to the model to mtgate negatve sde effects assocated wth delverng frozen meals. Then, n Secton 4 an extensve computatonal study s reported regardng how dfferent combnatons of these constrants mpact costs. 6

8 2.1 Mnmzng Operatonal Dsrupton The modfcatons to the model dscussed n ths secton are meant to mnmze the degree to whch consoldaton can change the daly operatons of the Meals On Wheels agency. Ths can be done by ensurng that the new set of routes for the delvery vehcles follow a pre-defned order throughout the week, by fxng the assgnment of clents to drvers, or by mantanng consstency wth respect to the order clents are vsted n the current daly routes. Fxed Route: Fxed routes, or a set of routes that vst locatons n the same order every day, often offer advantages over routes that vary by day. For home meal delvery, a fxed route can make t easer to synchronze the meal preparaton and vehcle loadng process. Smlarly, whle some routes are executed by volunteers, some are done by professonal drvers, n whch case the same drver executes the route every day. In ths case, even though some clents may be skpped on some days, ensurng that they are always vsted n the same order can make executng the delvery route easer. Thus, to ensure the delvery routes follow a fxed orderng of clents, we defne the varable o v j {0, 1} to ndcate whether clent s vsted before clent j by delvery vehcle v on days they are both vsted and add the followng constrants to the HMCD: o v j + o v j = 1, j I, v V, (12) o v j + o v jk + o v k 2, j, k I, v V, (13) x vt j o v j, j I, v V, t T. (14) Constrants (12) and (13) are standard orderng constrants, wth the frst ensurng that ether precedes j or j precedes, and the second ensurng for a trplet, j, k the orderng j k s not allowed. Fnally, constrants (14) ensure that daly travel corresponds to the orderng prescrbed for the week. These are referred to as the Fxed Route constrants. Clent Consstency: The model focuses on ntroducng consoldaton nto routes that are already executed on a daly bass by the same MOW provder, and, typcally, n the same order on every day. However, the HMCD, even wth the fxed route constrants above, may change the vehcle, and therefore the drver, that servces a clent. A drver may have developed a bond wth hs or her clents and dsruptng that bond n the nterest of operatonal effcency may be dscouraged. Therefore, t may be of nterest to keep the same clents assgned to the same vehcles as on the 7

9 currently establshed routes. Consstency wth clents s mantaned by addng the constrant: y vt = 0 t T, v V, I f s not currently vsted by v. (15) Ths s referred to as the Clent Consstency constrant. When these constrants are added to the HMCD, t separates nto a sngle-vehcle varant of the problem for each vehcle. Order Consstency: The HMCD may also change the order n whch clents are vsted on currently establshed routes, even wth the fxed route constrants above. As wth the developed clent relatonshp, a drver may have an establshed route that she s famlar wth and dsruptng that route may add some ntangble operatng costs. Therefore, t may be of nterest to keep the order n whch clents are vsted the same as on the currently establshed routes. Consstency wth order of clents s mantaned by addng the constrant: x vt j = 0 t T, v V,, j I f j vsted before n current route. (16) Ths s referred to as the Order Consstency constrant. 2.2 Mnmzng Clent Dssatsfacton Whle consoldaton has been shown to yeld tremendous savngs n many settngs, Meals On Wheels s prmarly focused on the wellness and betterment of ts clents. Thus, the clent experence wthn a meal delvery program that delvers frozen meals s of utmost mportance. The human nteractons and relatonshps offered by Meals On Wheels may be mantaned by lmtng the number of days that can elapse between vsts. Also, a mnmum may be placed on the number of hot meals that each clent receves as most prefer these to frozen meals. Delvery Frequency: The y t varables may be used to derve a constrant ensurng that a clent must be seen at least once wthn a specfc span of days, ensurng that there are never too many days between vsts. For example, to allow no more than two days between vsts (recall that T = 5), the followng constrants are added to 8

10 the HMCD: v V v V v V v V v V y v1 y v2 y v3 y v4 y v5 + v V + v V + v V + v V + v V y v2 y v3 y v4 y v5 y v1 + v V + v V + v V + v V + v V y v3 1 I, y v4 1 I, y v5 1 I, y v1 1 I, y v2 1 I. (17) These types of constrants are referred to as Delvery Frequency constrants. Note that because each y vt s bnary, the vald nequalty y v1 + y v2 + y v3 + y v4 + y v5 2 may be derved from these constrants wth a roundng argument. Hot Meal Delvery: To ensure that at least H hot meals are delvered to each ndvdual, the followng constrant s added to the HMCD: qh vt H I, v V. (18) t T Ths s referred to as the Hot Meal constrant. Some analyss of these constrants can be useful n restrctng the soluton space of the HMCD. By guaranteeng a mnmum number of hot meals and a maxmum tme between vsts, the schedulng optons are strctly lmted. Specfcally, a lemma regardng the schedule follows: Lemma 2.1. Gven a tme horzon of fve days ( T = 5), f at least two hot meals must be delvered to each ndvdual (H = 2) and at most two days are allowed between vsts, then there s an optmal soluton to HMCD where hot meals are delvered on days two and four (Tuesday and Thursday). Proof Consder a soluton ( q c, q h, x, ȳ, r) to the HMCD where at least two hot meals are delvered and at most two days pass between vsts. Suppose hot meals are delvered on days t, t. For vehcle v, set the value of all varables assocated wth day two equal to the values of the varables assocated wth t and day four varables 9

11 equal to the values of the t varables, whle smlarly settng the varables assocated wth t equal to the day two varables and t equal to day four (.e. set x v2 x vt j, xj vt ȳ vt, y vt = x v2 j, qc v2 = q c vt, qc vt = ȳ v2 j = = q c v2, qh v2 = qvt h, qvt h = qv2 h, rv2 = r vt, r vt = r v2, y v2 = and smlarly for days four and t ). Ths may be repeated for all vehcles v V. Changng the day on whch a route s executed does not have an effect on the cost of that route, so the cost of ths new soluton s the same as the orgnal soluton. Ths new soluton s stll feasble because the number of hot meals delvered to each clent durng the week remans the same. Also, by ensurng everyone s delvered to on days two and four, there can be at most two days between vsts (days fve and one f an ndvdual does not receve a delvery on those days). Thus, any optmal soluton to the HMCD can be translated to a soluton where hot meals are delvered on days two and four, and because the cost of that new soluton s the same, t s also optmal. For nstances of the HMCD where the premse of Lemma 2.1 apples, we can fx r v2 = r v4 = 1, r v1 = r v3 = r v5 = 0 for all v V, whch n turn reduces the problem to one where decsons only need to be made for days one, three, and fve. Smlarly, one can argue that f three hot meals must be delvered to each ndvdual, then there s an optmal soluton where they are delvered on days one, three, and fve. Whle ths lemma has algorthmc mplcatons n that nstances of the HMCD may be solved n much less tme by fxng r vt varables, t also has practcal mplcatons. Specfcally, ths lemma mples that an agency only needs to prepare hot meals on specfc days and thus resource requrements for meal preparaton may be re-allocated. 3. Introducng consoldaton nto daly delvery routes One of two algorthms s used to ntroduce consoldaton nto daly delvery routes, wth the choce of algorthm dependent on whether Clent Consstency s enforced. Each algorthm s based on solvng restrctons of the HMCD, and thus can be used (unchanged) when the model s extended to nclude other constrants such as the Fxed Route or Delvery Frequency constrants. In descrbng these algorthms, the constrants descrbed n Sectons 2.1 and 2.2 are collectvely referred to as Operatonal Consderatons. As alluded to above, when Lemma 2.1 apples, restrctons of the HMCD are further constraned by fxng the approprate r vt varables to 1 or 0. Wth Clent Consstency: As noted prevously, when Clent Consstency s enforced, such that the constrants y vt = 0 t T, v V, I f s not currently vsted by v 10

12 are added to the HMCD, then the problem separates nto a sngle-vehcle varant of the problem for each vehcle. In ths case, consoldaton s acheved by solvng V restrctons of the HMCD, one for each vehcle, whch we do to (near-) optmalty wth a commercal nteger programmng software package. Specfcally, Algorthm 1 s executed. Algorthm 1 Introducng consoldaton whle mantanng drver-clent assgnments Requre: Set of Operatonal Consderatons to enforce 1: Gven the pre-exstng daly routes used by the MOW provder, create ntal soluton where these routes are executed every day of the tme horzon 2: for all v V do 3: Create nstance of HMCD where the set of clents, I, corresponds to those vsted by v n daly routes 4: Add to HMCD constrants that model Operatonal Consderatons 5: Solve resultng nstance of HMCD 6: end for Wthout Clent Consstency: When Clent Consstency s not enforced, the HMCD does not separate by vehcle, leavng an optmzaton problem that, for realstcallyszed nstances, s dffcult to solve to near-optmalty wth off-the-shelf software. As a result, to ntroduce consoldaton nto daly delvery routes, a heurstc s used that repeatedly solves restrcted versons of the HMCD n whch certan values are fxed durng executon of the heurstc. These restrcted versons are created by frst parttonng the set of vehcles nto subsets, V1, V 2,..., V k V, and then, the clents seen by vehcles n a subset V, I( V ), are further parttoned nto subsets, Ī 1 ( V ),..., Īk( V ) I( V ). Gven these subsets, the followng varables of the HMCD are fxed to ther values n the current soluton: z v, y vt, x vt j, v V \ V z v, y vt, v V, I( V ) \ Ī( V ). That s, all but the delvery quantty varables are fxed for a subset of vehcles. Then, for the remander of the vehcles, a subset of the clent-assgnment varables are fxed (but none of the routng varables are fxed). Ths s done n the context of executng Algorthm 2 The k-means clusterng algorthm [16] s used to partton the vehcles and clents nto the subsets. The purpose of k-means clusterng s to partton n data ponts nto k clusters such that each data pont s n the cluster whose mean s closest. Whle 11

13 Algorthm 2 Introducng consoldaton whle allowng drver-clent assgnments to change Requre: Set of Operatonal Consderatons to enforce 1: Gven the pre-exstng daly routes used by the MOW provder, create ntal soluton where these routes are executed every day of the tme horzon 2: whle not stop do 3: Partton vehcles nto groups V 1, V 2,..., V k 4: for all vehcle groups V do 5: Partton clents I( V ) delvered to by some v V nto groups Ī 1 ( V ),..., Īk( V ) 6: for all clent groups I j ( V ) do 7: Formulate HMCD wth constrants that model Operatonal Consderatons 8: Fx varables z v, y vt, x vt j, v V \ V and then z v, y vt, I( V ) \ Ī( V ) 9: Solve resultng nstance to get new soluton 10: f mproved soluton s found then 11: Set current soluton to mproved soluton 12: end f 13: end for 14: end for 15: end whle v V, 12

14 determnng the optmal set of k clusters for a fxed k s NP-Hard, many computatonally effectve and effcent heurstcs have been developed. In partcular, we use the heurstc k-means++ [2] whch has been shown to have approxmaton guarantees. To use the k-means++ algorthm to partton a set of objects nto groups each object must be represented by a data pont. Once ths representaton of each object s determned, the k-means++ algorthm may be executed to cluster the data ponts nto groups. The representatons for vehcles and clents s dscussed next. Parttonng vehcles: To partton the vehcles nto groups a 2 T -dmensonal data pont s created for each vehcle contanng the average (x, y) coordnates for that vehcle for each day over the problem horzon. Assume the coordnates (x, y ) are gven for each clent and that vehcle v sees m t clents on day t. The average coordnates for each day are calculated as avgx t = seen on t x /m t and avgy t = seen on t y /m t. The data pont for vehcle v s then [avgx 1 avgy 1... avg x T avg y T ]. As an example, consder a two day plannng horzon (T = 2) and a vehcle that on day one vsts clents at locatons (20, 40), (10, 90), and (0, 10) and on day two vsts clents at locatons (10, 10), and (50, 50). Then, the data pont for that vehcle would be [ ]. Parttonng clents: To partton the clents nto groups, each clent s represented by a 2-dmensonal data pont that contans the (x, y) coordnates for that clent. Whle the k-means++ algorthm s the prmary mechansm for parttonng vehcles and clents nto groups, to ntroduce dversty nto the search, the heurstc perodcally parttons them randomly. 4. Computatonal Analyss For these experments, eght of the Chrstofdes nstances are modfed from [6], specfcally, nstances one through four and sx through nne. There are then two nstances each wth 50, 75, 100 and 150 customers. The nstances are modfed so that each customer has a demand of one unt per day (n our context, a meal) and each vehcle has a capacty of eght. The number of vehcles s adjusted so that the utlzaton of each vehcle s roughly 90%, wth 7, 10, 13 and 20 vehcles for the problems n ncreasng sze. As the algorthm ndcates, the ntal soluton s the pre-exstng routes daly routes used by the provder executed on each day of the problem horzon, such that each clent receves a hot meal on each day. Gven that pre-exstng routes do not exst for the modfed Chrstofdes nstances, a heurstc presented n [13] s used to create these daly routes wth costs that are wthn 5% of the best known soluton for each of the eght orgnal Chrstofdes nstances. Note 13

15 that most MOW provders do not utlze advanced routng technques such as ths heurstc, and thus ther current daly routes may not be as cost effcent. After the heurstc establshes the baselne daly routes provdng servce to every clent on each day, ether Algorthm 1 or 2 s appled, dependng on whch s approprate under the varous condtons descrbed n Secton 2 and comparsons may be made wth the baselne routes. When solvng nteger programs n the context of executng ether Algorthm 1 or 2, Gurob 4.6 s used wth the optmalty tolerance parameter set to 1%. When solvng nstances of HMCD n the course of executng Algorthm 1, soluton tme s lmted to ten mnutes. When solvng restrctons of HMCD n the course of executng Algorthm 2, soluton tme s lmted to one mnute. Fnally, the complete runtme for Algorthm 2 s lmted to one hour. Two questons are explored n ths computatonal analyss. The frst s, gven a set of clents, I, that are currently beng served, how much money can be saved by delverng frozen meals to the clents n I, as compared to delverng a hot meal every day to each of those clents? The second s, gven a set of clents, I, that are currently beng served and a set of clents, U, that are currently unserved, by delverng frozen meals how many clents n U can be served a meal whle stll savng money, as compared to delverng a hot meal every day to each clent n I? These questons are answered n the next two subsectons for the dfferent settngs dscussed n Sectons 2.1 and Savngs through consoldaton Table 1 presents several metrcs for nstances wth consoldaton that have no addtonal constrants and nstances wth the constrants used for mnmzng operatonal dsrupton. These metrcs nclude the averages of the savngs n travel cost compared to the baselne case where only hot meals are delvered, and the number of hot meals and vsts a clent receves n a fve day week. As mght be expected, usng frozen meals to allow for load consoldaton leads to consderable cost savngs. However, these results show that cost savngs may be found wthout sgnfcantly alterng current operatons. When mplementng the constrants to mnmze dsrupton from current operatons, the cost savngs are vrtually equvalent, as are the other metrcs of nterest. These fndngs ndcate that the cost savngs related to consoldaton are not a result of major changes n the routes used to delver servce. The heurstc used produces base daly routes that delver hot meals every day that are wthn 5% of the best known soluton, wth lttle room for mprovement from swappng clents between routes or reorderng them wthn a route. Consoldaton of delveres results n the elmnaton of clents from routes on certan days and the routes are robust 14

16 enough that removng stops does not open up savngs opportuntes assocated wth alterng the route n other ways. Thus, the order of clents served and the assgnment of clents to drvers may be mantaned whle travel costs are reduced by skppng multple stops. If mplemented n a real world settng wth lower qualty routes, ths algorthm may fnd more opportuntes to mprove the soluton through route adjustments. However, these results show that consoldaton can lead to savngs even wthout makng those adjustments. From a manageral perspectve, ths s mportant as t ndcates that operatonal costs may be reduced wth mnmal change management requred for the pool of drvers. Constrant Cost savngs Hot meals per clent Vsts per clent No operatonal constrants 21.21% Fxed route 21.14% Clent consstency 21.21% Order consstency 21.04% All constrants 21.03% Table 1: Evaluaton of constrants mnmzng operaton dsrupton Whle the mpact on drvers may be mnmzed, consoldaton leads to clents recevng more frozen meals and fewer vsts. Table 2 presents the results when constrants are placed on the amount that a clent may be nconvenenced through consoldaton, attemptng to lmt dssatsfacton wth the MOW servce. Tests were not run wth certan combnatons of mnmum hot meals and maxmum days between vsts as they were not feasble (e.g., four frozen meals must be delvered the day before there are three days between vsts). These constrants have a much greater mpact on the savngs assocated wth consoldaton than the operatonal constrants. As mght be expected, the savngs decrease as more hot meals are requred per week and less consoldaton s allowed. The mnmum number of hot meals allowed by the constrants are delvered to the clents for each constrant settng. The savngs also ncreased as more days were allowed between vsts. However, a 10% cost savngs stll results when a mnmum of three hot meals are requred. Allowng two frozen meals per customer s equvalent to removng one day of servce to each customer. Whle ths s not a drastc adjustment to the MOW operatons, wth such tght margns, even a savngs of 10% can be sgnfcant. To see the effect of fxng r vt varables when Lemma 2.1 holds, Algorthm 2 was run twce for each of the nstances, once wth the varables fxed and once wthout. Ths was done for the settng wth a maxmum of two days between servce, a mnmum 15

17 Maxmum days between servce Mnmum number of hot meals Cost savngs Hot meals per clent Vsts per clent % % % % % % % % % Table 2: Evaluaton of constrants mnmzng clent dssatsfacton of two hot meals per clent, and no addtonal operatonal constrants. For each executon and nstance, Algorthm 2 found the same soluton. However, wth the r vt varables fxed, Algorthm 2 was able to fnd that soluton an average of almost eght mnutes faster than when the varables were not fxed. Smlarly, fxng the r vt varables enabled Gurob to solve nstances of the HMCD n much less tme, as evdenced by the fact that Algorthm 2 was able to execute 48% more teratons n one hour when dong so. 4.2 Expandng the clent set For these experments, ntal daly routes are agan generated for each nstance. Then, assumng I clents are n an nstance, 0.1 I new clents are randomly dstrbuted across the regon spanned by the orgnal I clents. All of these new clents are placed on one route for a dummy vehcle, wheren the transportaton costs for ths route are multpled by a large number. Algorthm 2 s then executed on ths nstance, wth the dummy vehcle ncluded n each vehcle subset and each clent subset contanng all of the clents served by ths vehcle. Thus, when solvng restrctons of the HMCD n the course of executng Algorthm 2, clents may be moved from one real vehcle to another, as well as from the dummy vehcle to a real vehcle. By modelng a watng lst of clents wth ths dummy vehcle, t s possble that a clent only receves one or two meals over the tme horzon. Ths s n lne wth how agences operate, as some clents may be assgned delvery on fewer than fve days a week f that s what the current routes wll allow. Table 3 presents several metrcs to evaluate the addton of new clents when usng consoldaton. These metrcs are presented for the four problem szes, defned by the number of vehcles used to delver the meals. The frst values show the ncrease n 16

18 the total number of meals served when the new clents are added, as well as the average number of addtonal meals that each vehcle delvers. As the problem sze ncreases the total number of addtonal meals served ncreases, whle the effcency of the vehcles decreases. The heurstc s effectve wth the larger problems, but as problem sze ncreases t becomes dffcult to fnd avalable capacty for as many new clents on the routes. Intal meals served Meals served wth addtonal clents Addtonal meals per vehcle Number of vehcles Table 3: Addtonal meals served wth ncreased clent base Table 4 presents the savngs wth consoldaton over the baselne routes that delver all hot meals, comparng the nstances wth and wthout the addtonal clents. Despte the ncrease n routng costs from servng addtonal clents, consoldaton stll leads to consderable savngs, whle also allowng for the delvery of more meals. The savngs wth addtonal clents ncrease wth problem sze as a smaller percentage of the new clents may be nserted nto the routes, leadng to less of an ncrease n routng costs. Conversely, the savngs wthout addtonal clents decreases as the problem sze ncreases. Due to the larger problem sze and soluton space, the heurstc cannot explore as many of the avalable opportuntes for consoldaton. Savngs wthout addtonal clents Savngs wth addtonal clents Baselne cost per meal Cost per meal wthout addtonal clents Cost per meal wth addtonal clents Number of vehcles % 9.8% % 11.6% % 14.9% % 14.0% Table 4: Impact on consoldaton savngs wth an ncreased clent base An analyss of the costs per meal for the three problem scenaros ndcates not only the consderable savngs per meal from consoldaton, but the reducton n cost as the problem sze ncreases. Further, whle the cost per meal s ntally greater 17

19 wth the new clents for the smallest problem sze, t drops slghtly below the cost wthout the new clents for the problems wth 13 and 20 vehcles (or 100 and 150 customers). In combnaton wth the results n Table 3, ths ndcates that whle the heurstc can not fnd as much avalable capacty wth the larger problem szes, effcency s mantaned wth routng those addtonal clents that can be nserted. 5. Concluson In ths paper a model and soluton methodology are presented that wll enable home delvery companes to balance the effcences ganed from consoldaton wth customer satsfacton and ease of operaton metrcs. The results provde several nsghts that may be benefcal to a MOW agency manager, or any transportaton provder nterested n consoldatng loads whle mnmzng operatonal dsrupton. The order n whch clents are servced and the assgnment of drvers to clents do not have to be altered n order to reduce costs through consoldaton. Ths may have a greater mpact on other transportaton provders focused more on lmtng operatonal dsrupton and less on customer servce. However, whle MOW agences may be more clent-centrc, a methodology that does not mpact ther servce provders s lkely to be vewed favorably. Whle savngs from consoldaton decrease as customer servce requrements ncrease, a non-neglgble beneft may be found even wth a mnmal amount of consoldaton. As descrbed earler, many MOW agences are strugglng to reman operatonal wth lmted fundng and even a small cost savngs may be what allows an agency to stay n busness. Consoldaton can lead to a reducton n transportaton cost, whle also allowng for prevously unserved clents to receve meals. Clents may be unhappy wth frozen meals; however, t may be easer to convnce them of the beneft by ndcatng how many addtonal people the program can ad when some hot meals are replaced. The lemma mples that when certan customer servce lmts are n place, an agency only needs to prepare hot meals on specfc days. Ths allows for operatonal plannng to be smplfed and for resource requrements for meal preparaton to be re-allocated. As more people requre the servce provded by MOW and fundng becomes harder to fnd, the cost savngs found through the use of frozen meals wll be dffcult to 18

20 resst, despte some dssatsfacton on the part of clents. However, the methodology presented n ths paper may be used to make the transton easer by lmtng dsrupton to both clents and servce provders. [1] Andersson, H., Hoff, A., Chrstansen, M., Hasle, G., and Løkketangen, A. (2010). Industral aspects and lterature survey: Combned nventory management and routng. Computers & Operatons Research, 37(9): [2] Arthur, D. and Vasslvtsk, S. (2007). k-means++: The advantages of careful seedng. In Proceedngs of the eghteenth annual ACM-SIAM symposum on Dscrete algorthms, pages Socety for Industral and Appled Mathematcs. [3] Barthold III, J., Platzman, L., Collns, R., and Warden III, W. (1983). A mnmal technology routng system for Meals On Wheels. Interfaces, 13(3):1 8. [4] Beasley, J. (1984). Fxed routes. Journal of the Operatonal Research Socety, 35(1): [5] Campbell, A., Clarke, L., Kleywegt, A., and Savelsbergh, M. (1998). Fleet management and logstcs, chapter The nventory routng problem, pages Kluwer Academc Publshers. [6] Chrstofdes, N. and Elon, S. (1969). An algorthm for the vehcle-dspatchng problem. Operatons Research, 20(3): [7] Coelho, L., Cordeau, J., and Laporte, G. (2012). Consstency n multvehcle nventory-routng. Transportaton Research Part C: Emergng Technologes, 24: [8] DeSo, J. (2008). Crtcs say frozen Meals-On-Wheels are n poor taste. The Vllage Voce, 2/28/2008, s ay f ro.php. [9] Erera, A., Savelsbergh, M., and Uyar, E. (2009). Fxed routes wth backup vehcles for stochastc vehcle routng problems wth tme constrants. Networks, 54(4): [10] Francs, P., Smlowtz, K., and Tzur, M. (2006). The perod vehcle routng problem wth servce choce. Transportaton Scence, 40(4):

21 [11] Gaudoso, M. and Paletta, G. (1992). A heurstc for the perodc vehcle routng problem. Transportaton Scence, 26(2): [12] Gbson, H. (2004). Natonal Meals On Wheels rolls out nto rural areas. Today s Caregver, 11/1/2004, m eals o n w heels.htm. [13] Hewtt, M. and Nowak, M. (2012). Estmatng the cost of contnuty of care n home health care. Workng paper. [14] Kelley, T. (2008). Taste-testng the latest Meals On Wheels. New York Tmes Cty Room, 5/7/2008, [15] Kyle, S. J. (2012). Causes: Meals On Wheels provdes nourshment for body, soul. Coloradoan, 10/5/2012, Meals-Wheels-provdes-nourshment-body-soul. [16] Marsland, S. (2009). Machne learnng: an algorthmc perspectve. Chapman & Hall/CRC. [17] Meals on Wheels Assocaton of Amerca (2009). MOWAA Annual Report. [18] Meals on Wheels Assocaton of Amerca (2012). MOWAA s pledge: End senor hunger by [19] Smlowtz, K., Nowak, M., and Jang, T. (2013). Workforce management n perodc delvery operatons. Transportaton Scence, Forthcomng. 20

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