CENTER FOR MULTIMODAL SOLUTIONS FOR CONGESTION MITIGATION (CMS)

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1 Final Repot to the CENTER FOR MULTIMODAL SOLUTIONS FOR CONGESTION MITIGATION (CMS) CMS Poect Numbe: _8-4_ Title: Chaacteizing the Tadeoffs and Costs Associated with Tanspotation Congestion in Supply Chains fo peiod _6/5/8_ to _/3/9_ Joseph Geunes, Ph.D. and Dince Konu (gaduate student) Industial and Systems Engineeing, 33 Weil Hall, PO Box 6595, Gainesville, FL 36; x geunes@ise.ufl.edu Januay,

2 TABLE OF CONTENTS DISCLAIMER AND ACKNOWLEDGMENT OF SPONSORSHIP... 3 LIST OF TABLES... 3 LIST OF FIGURES... 3 ABSTRACT... 4 EXECUTIVE SUMMARY... 5 CHAPTER BACKGROUND... 6 CHAPTER RESEARCH APPROACH... 9 Stage-two Decisions: Maket-Supply Game... Impact of Vaiation in α... 5 Stage-one Decisions: Facility Locations... 7 CHAPTER 3 FINDINGS AND APPLICATIONS... Analysis : Effects of Taffic Congestion... Analysis : Efficiency of the Heuistic Method... 3 Analysis 3: Accounting fo Congestion in Decision Making... 5 CHAPTER 4 CONCLUSIONS, RECOMMENDATIONS, AND SUGGESTED RESEARCH... 7 REFERENCES... 8 APPENDIX A Poofs of Popositions... 3 APPENDIX B Heteogeneous Cost Case Pape CMS Final Repot

3 Disclaime The contents of this epot eflect the views of the authos, who ae esponsible fo the facts and the accuacy of the infomation pesented heein. This document is disseminated unde the sponsoship of the Depatment of Tanspotation Univesity Tanspotation Centes Pogam, in the inteest of infomation exchange. The U.S. Govenment assumes no liability fo the contents o use theeof Acknowledgment of Sponsoship This wok was sponsoed by a gant fom the Cente fo Multimodal Solutions fo Congestion Mitigation, a U.S. DOT Tie- gant-funded Univesity Tanspotation Cente. LIST OF TABLES Table page Data Intevals fo Poblem Classes Aveage Statistics ove Poblem Classes Data Ranges fo Poblem Classes Compaison of Total Enumeation and Weight-based Heuistic Method Compaison of Total Enumeation and Weight-based Heuistic Method fo Each m Data Categoies fo Poblem Classes and Statistics of Cases (i) and (ii) fo Poblem Classes and Solution fo Example... 6 LIST OF FIGURES Figue page Pattens of Each Column in Table... 3 CMS Final Repot 3

4 Abstact We conside distibution and location-planning models fo supply chains that explicitly account fo taffic congestion effects. The maoity of facility location and tanspotation planning models in the opeations eseach liteatue conside facility opeations and tanspotation costs as sepaable (e.g., linea) by oigin-destination pais. Ou goal is to undestand how congestion costs and effects, which ae not sepaable, influence supply chain location and distibution decisions. We study a competitive facility location and maket-supply game with multiple fims competing in multiple makets in a congested distibution netwok. As a esult of location and quantity decisions, fims ae subect to location-specific tanspotation costs, convex taffic congestion costs and fixed facility location costs. The unit pice in each maket is a linea deceasing function of the total amount shipped to the maket by all fims; that is, we conside an oligopolistic Counot game and analyze the two-stage Nash Equilibium. We discuss the esults of extensive numeical studies that illustate the effects of taffic congestion on a fim's equilibium location and quantity decisions and demonstate the efficiency of ou solution appoaches fo finding equilibium solutions. CMS Final Repot 4

5 Executive Summay We conside distibution and location-planning models fo supply chains that explicitly account fo taffic congestion effects. The maoity of facility location and tanspotation planning models in the opeations eseach liteatue conside facility opeations and tanspotation costs as sepaable (e.g., linea) by oigin-destination pais. Ou goal is to undestand how congestion costs and effects, which ae not sepaable, influence supply chain location and distibution decisions. We fist study a competitive facility location and maket-supply game with multiple identical fims competing in multiple makets in a congested distibution netwok. As a esult of location and quantity decisions, fims ae subect to location-specific tanspotation costs, convex taffic congestion costs and fixed facility location costs. Fist, we study the supply quantity decisions fo any fim when the location choices of the fims ae identical. An oligopolistic Counot game is analyzed to detemine a Pue Nash Equilibium (PNE) fo these quantity decisions, and we povide analytical esults on the effects of taffic congestion costs on the equilibium quantities flowing fom supply facilities to makets. We then focus on the location decisions of the fims. As fims ae identical, fims will choose identical facility locations, and we theefoe study the optimal location decisions fo any individual fim. We discuss the esults of extensive numeical studies that illustate the effects of taffic congestion on a fim's location and quantity decisions and demonstate the efficiency of ou solution appoach. We then study a set of heteogeneous competitive fims consideing the location of uncapacitated facilities at a set of candidate locations in ode to seve a set of makets. Each fim incus fim-specific (linea) tanspotation costs, as well as convex congestion and fixed location costs as a esult of location and distibution volume decisions. The unit pice in each maket is a linea deceasing function of the total amount shipped to the maket by all fims; that is, we conside an oligopolistic Counot game and analyze the two-stage Nash Equilibium. This poblem is efeed to as the location-supply game, o competitive location game, and we fist study the fims' maket-supply decisions fo given facility locations, i.e., the game's second stage. We fomulate the poblem of finding the equilibium supply quantities as a vaiational inequality poblem and povide a solution algoithm. Then we focus on the location decisions, i.e., the game's fist stage. We povide ules to obtain a dominant location matix, and use these ules in a heuistic solution appoach to seach fo an equilibium location matix. Numeical esults on the efficiency of the heuistic method ae documented. CMS Final Repot 5

6 A Facility Location Poblem with Supply Competition in a Congested Netwok Dinçe Konu and Joseph Geunes. Backgound Reseach on taffic netwok equilibium poblems, toll picing (congestion picing), and methods to mitigate taffic congestion have typically focused on the welfae of individual oad uses. Howeve, ecent studies identify the negative impacts taffic congestion has on supply chain opeations. In paticula, the pefomance of logistics opeations is affected by taffic congestion, and these impacts ae moe dastic in Just-in-Time (JIT) poduction systems. Despite the fact that taffic congestion affects supply chain opeations, most of the studies combining taffic congestion and supply chains ae based on empiical data and lack theoetical esults. Anothe poblematic point is that taffic congestion effects ae exogenous in past liteatue, and these effects ae analyzed indiectly by assuming that inceased congestion eithe implies inceased tavel times o deceased tavel time eliability. Moe impotantly, taffic congestion effects ae only studied in the context of a distibution netwok of a single fim. In this study, we focus on the effects of taffic congestion on supply chain opeations by modeling taffic congestion costs endogenously. We study two pimay supply chain decisions: facility location decisions and supply quantity decisions. McKinnon et al. (8) note that companies may estuctue thei distibution systems due to inceased congestion. Moeove, Rao et al. (99) mention that changes in facility locations ae often a long-tem eaction to inceased taffic congestion. Fo example, Lee (4) points out that when 7- Japan (SEJ), a convenience-stoe company, located stoes in key locations, SEJ was subect to moe damatic effects of taffic congestion. Sankaan et al. (5) also note that the effects of taffic congestion depend on the facility location choices of a company. Theefoe, it is impotant to gain a bette undestanding of the effects of taffic congestion on facility location and distibution flow decisions. We study these factos in a competitive envionment, i.e., when multiple fims compete in common makets. McKinnon (999) pesents suvey esults on the negative effects of taffic congestion on the efficiency of logistics opeations. In a simila study, McKinnon et al. (8) note that, on aveage, taffic congestion accounts fo 3 pecent of the total delay times in shipments of the companies completing the suvey. This ate can be highe (up to 34 pecent) in some industies McKinnon et al. (8). Fo instance, Fenie et al. () point out that taffic congestion is one of the most impotant factos affecting cost and sevice in gocey etailing in the UK. Sankaan et al. (5) also document the esults of a suvey and mention the effects of taffic congestion on supply chain opeations. Weisbod et al. () povide a systematic eview of the studies at the intesection of taffic congestion and supply chains and discuss how taffic congestion affects costs and poductivity. Anothe steam of eseach studies taffic congestion in JIT systems. Rao et al. (99) note that JIT systems equie small lot sizes, which esults in inceased taffic congestion. Moeove, thei suvey esults indicate that companies ae awae of the associated congestion impacts and Rao et al. (99) popose shot-tem and long-tem methods to mitigate the effects of congestion. Moinzadeh et al. (997) study the elationship between small lot sizes CMS Final Repot 6

7 and taffic congestion fo a company s distibution system, with multiple etailes using a common congested oad. Rao and Genoble (99) also study the effects of JIT eplenishment and the esulting taffic congestion on distibution costs. One othe field of eseach that combines taffic congestion and supply chains focuses on feight distibution. Figliozzi (9) studies the effects of taffic congestion on the costs associated with commecial vehicle tous, while Figliozzi (6) and Figliozzi et al. (7) analyze feight tous in congested uban aeas. Golob and Regan (; 3) also study the impacts of taffic congestion on feight opeations. The model we fomulate consides facility locations and supply quantity decisions in a competitive envionment on a congested distibution netwok. In paticula, we study a competitive location game with multiple fims competing in multiple makets. The competitive location poblem we study assumes the following settings. Competing fims ae non-coopeative and they must simultaneously detemine thei facility locations (fist stage decisions); then, the supply quantities flowing out of these facilities into each maket (second stage decisions) must be detemined. Note that a fim may locate moe than one facility. Makets and possible facility locations ae epesented as vetices in a netwok. Fims ae assumed to compete unde a homogeneous cost stuctue; that is, they have identical cost paametes. Fo this eason, we assume that fims make the same facility location decisions when maximizing expected pofits (when a unique Pue Nash Equilibium does not exist). We assume an oligopolistic Counot competition in the second stage, i.e., the total supply to a maket detemines the pice in that maket. The fist competitive location poblem was intoduced by Hotelling (99). In this study, each of two competing fims ties to maximize its maket shae by locating a single facility on a line. Hotelling s poblem is then extended by Teitz (968) to the case in which fims may locate moe than one facility. Studies exist in the liteatue that conside competitive location poblems unde diffeent assumptions. The numbe of competing fims, the natue of stategic decisions of the competing fims and the competition assumptions ae distinctive chaacteistics of diffeent studies in the liteatue. Competitive facility location games focus mainly on the location decisions of competing fims and equilibium conditions ae analyzed to detemine fims location decisions along with othe stategic decisions, such as picing, poduction levels and capacity planning. Eiselt and Lapote (996), Eiselt et al. (993) and Plastia () povide eviews of competitive facility location poblems unde diffeent assumptions studied in the liteatue. Most of the competitive location poblems in the liteatue assume Counot competition. Spatial competition of two fims with Counot competition is studied by Labbé and Hakimi 99). This study is extended to multiple fims by Saka et al. (997). Both of these studies assume that fims locate a single facility. Pal and Saka () conside spatial competition in a Counot duopoly whee the competing fims may locate moe than one facility. The distinguishing assumption of these studies is that competing fims ente each maket by supplying a positive quantity to each maket. Rhim et al. (3) and Sáiz and Hendix (8) elax this assumption and conside the case of fee enty. The settings of the competitive location poblems studied in Rhim et al. (3) and Sáiz and Hendix (8) ae simila to the settings of ou poblem. In both of these studies, the competition basis is that of Counot and fims detemine the location of thei single facility and the quantities to be supplied fom this facility to each maket, if they CMS Final Repot 7

8 choose to ente any maket. While a homogeneous cost stuctue is assumed by Rhim et al. (3) and Sáiz and Hendix (8) study a heteogeneous cost stuctue. Ou study extends the poblems studied by Rhim et al. (3) and Sáiz and Hendix (8) by allowing fims to locate moe than one facility. Moeove, fims ae subect to nonlinea taffic congestion costs. Konu and Geunes (9) study a geneal competitive facility location game whee fims ae subect to nonlinea congestion costs and allowed to locate moe than one facility. The poblem we study is a special case of thei poblem in which we assume that supply fims ae identical in tems of the costs they incu. Ou goal is to analyze the effects of taffic congestion on the fims equilibium facility location and supply quantity decisions. Consideing the special case involving identical supply fims enables us to explicitly analyze and chaacteize how congestion costs affect the stuctue of equilibium decisions, and to use this analysis to povide insights into how equilibium solutions change in esponse to changes in congestion levels and costs. We use a two-stage solution appoach simila to those in Labbé and Hakimi (99), Ledee and Thisse (99), Pal and Saka (), Rhim et al. (3), Sáiz and Hendix (8) and Saka et al. (997). Fist, we study the second stage decisions fo any fim when the location choices of the fims ae identical. The Pue Nash Equilibium (PNE) concept is used in the analysis of a Counot oligopoly to detemine the equilibium supply quantities. We povide analytical esults on the effects of taffic congestion costs on the equilibium quantities flowing fom supply points to makets in this stage. Then, we focus on the location decisions of the fims. We note that fims choose identical facility locations in the case of a unique PNE location decision. Howeve, fo othe cases, since the equilibium concept does not chaacteize what fims will actually do, we use the maximization of expected pofits as an obective, assuming that any location decision is equally likely fo each fim. We show that a mixed stategy Nash Equilibium (MSNE) implies that it is equally likely fo any fim to choose any given location decision. Thus, when fims ae homogeneous, they will end up with identical facility locations, and theefoe, we study the optimal location decision set fo the individual fim. A heuistic solution method to detemine a good location decision fo a fim is also povided. We pefom extensive numeical studies that illustate the effects of taffic congestion on a fim s location and quantity decisions. The est of this pape is oganized as follows. In Section, we discuss the detailed poblem setting and solution appoach. In Section 3, we study the popeties of equilibium supply quantities and popose a solution algoithm, given that fims make identical facility location decisions. Moeove, we analyze the effects of inceased taffic congestion on equilibium supply quantities. Section 4 discusses the ationale behind the assumption that fims choose identical facility locations, and a total enumeation scheme and heuistic method ae povided to chaacteize facility location decisions. In Section 5, we povide the esults of extensive numeical studies that chaacteize: (i) the effects of congestion on facility location and supply quantity decisions, (ii) the efficiency of the heuistic method and, (iii) the impacts of accounting fo congestion in the decision making pocess. Concluding emaks, a summay of the contibutions of ou study, and futue eseach diections ae noted in Section 6. CMS Final Repot 8

9 . Reseach Appoach We study a set of k competitive fims consideing the location of facilities at m possible locations in ode to seve custome makets at n locations. Each fim incus tanspotation, taffic congestion and facility location costs as a esult of thei location and distibution volume decisions. Moe explicitly, fims ae subect to linea tanspotation costs in the quantity shipped fom a facility to a maket and the taffic congestion cost incued is convex in the total quantity shipped fom a facility to a maket. A fixed facility location cost exists fo each location i. Moeove, we assume that any open facility is effectively uncapacitated and, hence, a fim will not open moe than one facility at a given location. The notation we use thoughout the text is summaized below. We will define additional notation as needed. : index fo fims, R = {,,, k} i: index fo locations, i I = {,,, m} : index fo makets, J = {,,, n} q : quantity shipped fom the facility of fim at location i to maket q : total quantity shipped to maket by fim, q q q : total quantity shipped fom location i to maket, q : total quantity shipped to maket, i : k m n matix of q values ( ) : q = R i I x : m-vecto epesenting location decisions of fim X : m k matix epesenting location decisions p q pice function fo maket q = i I q = g ( q ): taffic congestion cost function fo the link fom location i to maket c : tanspotation cost pe unit of flow fom location i to maket f i : fixed cost of opening a facility at location i f (x ): total facility location cost fo fim We assume the unit pice in each maket is a linea and deceasing function of the total quantity of flow into the maket. Explicitly, the unit pice in maket, p, is defined by the function p ( q ) = a b q, () whee the paametes a and b > epesent the level of maximum demand and the pice sensitivity in maket. We assume that the tanspotation cost is linea in the quantity of flow on link (i, ) and c epesents a pe unit tanspotation cost. It should be noted that c can be assumed to include pe-unit poduction costs as well. That is, a paamete v i > specific to location i can be included within c to account fo pe-unit poduction cost at location i. The function g, which is a function of the total quantity of flow on link (i, ), detemines the taffic congestion cost coefficient on link (i, ). In paticula, we assume that g ( q ) = α q () whee α > denotes the taffic congestion cost facto. Hence, the congestion cost incued by a fim using link (i, ) inceases with the total quantity of flow on the link as well as with the R q CMS Final Repot 9

10 quantity sent by the fim on that link. In paticula, the congestion cost fo fim is α q q = when q =. On the othe hand, when q >, the congestion cost of fim equals α q q > and is convex and inceasing in q when the quantities sent by othe fims on the link ae fixed. Thus, the fim s congestion cost is a nondeceasing convex function of the quantity sent by the fim on the link. This choice of functional fom eflects the natue of taffic congestion, as congestion costs incease in volume at an inceasing ate. This is compatible with the note in Weisbod et al. (), which emphasizes that companies with highe shipping levels ae subect to a highe level of congestion elated costs. The pofit function of fim eads as J i I R i I J i I J i I R Π (, X) = p q q c q q g q f ( x ), (3) whee the fist tem is the evenue gained by seving makets, the second tem is the total tanspotation cost, the thid tem is the total taffic congestion cost, and the last tem is the total facility location costs. Konu and Geunes (9) define the fims pofit function in a simila way, although they conside the case in which both c and α may be diffeent fo individual fims. The supply fims fist decide whee to locate facilities and then detemine how much to supply makets fom each of thei facilities. Clealy, unde competition, a fim s esulting pofit afte making location and supply decisions depends on the location and supply decisions of all othe fims. This implies a two-stage decision and associated solution appoach. Stage-one decisions coespond to fim location decisions, while Stage-two decisions coespond to maket supply decisions fo each fim. Ou solution appoach fist solves the Stage-two decisions fo a fixed set of location decisions, assuming each fim chooses the same location decision vecto. We will employ the Nash Equilibium concept of Nash (95) to detemine the fims supply quantity decisions and povide a method to find Pue Nash Equilibium (PNE) quantities sent fom any location to any maket by each fim... Stage-two Decisions: Maket-Supply Game In this section, we study the second-stage game, which detemines the fims supply quantity decisions fo a given location decision. This esticted game to detemine the equilibium quantity decisions is efeed as the Maket-Supply Game. Note that, unlike the pevious studies by Rhim et al. (3) and Sáiz and Hendix (8), the fims not only compete based on pice, but also as a esult of the congestion cost functions on supply links. This Maket-Supply Game is a non-coopeative game in which the supply fims ae the playes. Fims simultaneously detemine how much to send fom facilities to makets. To detemine the fims flows, we use the PNE concept, i.e., no fim will be bette off by alteing its supply quantity decisions unde the given location decisions. Now let us assume that the location decision fo each fim, i.e., the vecto x fo each =,,, k, is pe-detemined. That is, X is fixed. Since f (x ) is fixed fo the given X = X, it can be omitted fom Equation (3) fo the analysis of the Maket-Supply Game. Using the notation intoduced in the pevious section and Equations (), (), and (3), the pofit function of fim fo the given X = X eads as CMS Final Repot

11 ( ) ( a bq ) q cq αqq J i I i I Π X= X =. (4) The function in Equation (4) is stictly concave in each q, as b > and α >. Note that q = fo all J, i I, whee I denotes the locations whee fim has facilities fo the given X = X. The quantity decision fo any fim will depend on the quantity decisions of the othe fims; thus, we can apply the Nash Equilibium concept in ou solution appoach. A Nash equilibium solution fo the Maket-Supply Game will satisfy the fist ode conditions, Π X= X / =, fo a set of q values such that q >. Explicitly, fo any Nash ( ) q equilibium solution the following equation must hold wheneve q > : a b [ q + q ] c α [ q + q ] =. (5) Note that Equation (5) depends on the total quantity supplied to maket, the total quantity supplied by fim to maket, the total quantity supplied fom location i to maket and the quantity supplied fom location i to maket by fim. On the othe hand, Equation (5) does not depend on the othe maket paametes o vaiables and, hence, the quantity decision of any fim fo maket can be made independently fom the decisions elated to the othe makets. That is, each maket can be analyzed sepaately. Let Π ( X= X ) denote the pofit function of fim at maket, whee denotes the vecto of quantity decisions of the fims at maket fo the given location decision X = X. Then, it follows fom Equation (4) that Π ( X = X ) = p ( q ) q c q α q q. (6) i I i I In the est of this section, we focus on the Maket-Supply Game fo maket, since the Stage-two decisions fo each maket can be analyzed sepaately. The esults that hold fo maket will also hold fo the Maket-Supply Game acoss all makets. It follows fom Equation (6) that Equation (5) gives the fist ode equilibium condition fo quantities such that q >. Let q denote the equilibium quantities. Ou goals ae then (i) to detemine the locations and the fims such that q > and (ii) to simultaneously solve Equation (5) fo each q > fo the given X = X. At this point, we assume that X consists of identical columns, whee the th column coesponds to the location decision of fim. That is, x = x R, whee x denotes any column of X. The ationale behind consideing such location decisions will be explained in the following section when we study Stage-one decisions. This choice of X enables us to detemine the quantity decisions using an iteative scheme and to analyze the effects of taffic congestion cost factos on the equilibium quantity decisions. Howeve, the equilibium quantity decisions fo any given X can also be solved using a vaiational inequality appoach. See Konu and Geunes (9) fo an application of the vaiational inequality fomulation to detemine equilibium quantity decisions fo a competitive location-quantity game. Now, suppose that the location decision matix X consists of identical columns and, thus, the numbe of facilities at any candidate location is eithe k o fo some positive k. Note that we do not need to conside locations whee no fim has located a facility. Theefoe, we only study CMS Final Repot

12 quantity decisions at supply locations with k facilities. In the next poposition, we show that the quantity supplied fom location i to maket is the same fo each fim. Poposition. Suppose that X consists of identical columns, i.e., x = x R. Then q = / k R, whee denotes the total quantity flow on link ( i, ) at equilibium. Poof: All poofs can be found in the Appendix. Poposition implies that if we know the total equilibium quantity supplied fom location i to maket, denoted by, we also know the quantity that each fim with a facility at location i supplies to maket. Fo the given X, with x = x R (i) q = / k, (ii) q =, (iii) q = / k and (iv) q =, whee i I, it follows fom Poposition that i I I denotes the set of locations with k facilities associated with X. Recall that Equation (5) gives the fist ode conditions fo any q > ; that is, it gives the fist ode condition when >. Substituting (i)-(iv) into Equation (5), we get δ b b i I i I k α k δ γ γα + + = =, (7) i I δ = a c and γ = ( k+ ) / k. Note that we may have at most k m such fist ode whee conditions defined fo maket. Nevetheless, the fist ode conditions associated with a location use the same equation fo each fim, given by Equation (7). Theefoe, we focus on simultaneously solving at most m fist ode conditions, one fo each location, defined by Equation (7). The next poposition povides conditions that must be satisfied by the total equilibium quantity on a link ( i, ). Poposition. The equilibium quantities must satisfy the following conditions: a > if and only if δ > γ b, Poposition implies that the ( ) i I ( b) = if and only if δ γ b. i I δ values ae impotant in detemining the active locations at maket. A location is efeed to as active at a maket wheneve thee is a positive supply fom this location to the maket. Similaly, a fim is efeed as active at a maket wheneve thee is a positive supply by this fim to the maket. The next poposition is a diect esult of Poposition and states the activeness elations between two locations. Poposition 3. Suppose that δ δ fo locations i i i i, I. Then in an equilibium solution ( a) If i >, then i >, (b) If i =, then i =. Poposition 3 highlights the impotance of soting locations accoding to thei δ values which, CMS Final Repot

13 fo a given maket, is equivalent to soting supply locations based on the c values. In paticula, if we know that l locations ae active at maket, these locations should be the l locations with geatest δ values (o, smallest c values). Note that if thee exist locations with identical δ values, it follows fom Poposition 3 that eithe all of these locations ae active o none of them is active at maket. Moeove, fo both of these cases, the equilibium supply quantities at maket emain unchanged egadless of the soting ode of tied values of δ, as the same fist ode conditions given in Equation (7) will be solved. Now let us sot locations accoding to thei δ values, and without loss of geneality, let us assume that δ δ ( i + ). Theefoe; if l locations ae active at maket, these locations ae,,,l with l I, whee I denotes the cadinality of the set I. Then fo any fim at any location i, i l (since q > as > ) the following fist ode condition must be satisfied: δ γb ( + + L+ ) γα = i l. l In matix notation, the fist ode conditions can be epesented as δ α + b b L b δ b α + b L M γ M M M O b M δ b b α b l L + l l =. It follows fom the above epesentation that we can find the values easily by inveting the l l matix fo a given set of active locations. Note that inveting this matix basically involves solving the fist ode conditions fo locations,,,l togethe. Howeve, ou aim is to detemine the set of active locations and then find the equilibium quantities. In the next poposition, we povide an algoithm that detemines the set of active locations at a maket as well as the total equilibium flows fom these locations. The algoithm is based on Popositions and 3. Poposition 4. Suppose that X consists of identical columns, i.e., x = x R. Then Algoithm, stated below, detemines the numbe of the active locations and the coesponding equilibium flow quantities. Algoithm. Given x = x R, the numbe of fims, b, δ and α values fo maket : Step. Set = geatest i I. i I. Sot the emaining locations such that location has the δ value. If δ >, set l = and go to Step ; othewise = CMS Final Repot 3

14 Step. Step. Fo location l, find in matix fom δ ( ) l l by solving the following set of equations epesented α + b b L b ( l) ( ) δ b α b l + L M γ M M M O b M ( ) δ b b α b l + l L l l =. ( l) If l > and l < I, set l= l + and go to Step. If stop, locations,,,l ae active and = locations,,, l ae active at maket. = fo i l. ( l) ( l) l > and i I. Else if, = ( l ) ( l) l (8) l = I,, stop; fo i l and We next analyze Algoithm, which will be helpful in chaacteizing the effects of the taffic congestion cost facto on the equilibium flow quantities. Suppose that thee ae l active th ( ) locations at maket. Conside the w iteation of Step in Algoithm. Let w be the tentative quantities calculated at the th w iteation using Equation (8). (Note that th the w iteation, we have tentative equilibium quantities fo locations to w, which ae the solutions to ( w) ( w) ( w) ( w) δ γα = γb ( + + L + ) i w. (9) w It follows fom Equation (9) that ( w) ( w) ( w) δ γα = δ γα = = δ γα. w l w ( l) = L () th Equations (9) and () imply that, fo location s, s w, in the w iteation w ( δs δ ) δl + b ( w) i= α s =. () w α s γ αs+ b i= α Equation () gives the equilibium quantity fo location s when w=l. In the next poposition, we show that the quantity supplied fom an active location deceases as the numbe of active locations inceases at each iteation of Algoithm, wheeas the total quantity supplied to the maket inceases..) In Poposition 5. (a) w+ l. > + fo location s, s w and w+ l. (b) ( w) ( w ) s s w w+ ( w) ( w+ ) < i= i=, The following subsection chaacteizes the effects of the taffic congestion cost facto on the total equilibium quantity flow fom a location to maket, as well as on the total equilibium quantity supplied to maket. Poposition 5 will be used in the analysis of these effects. Ou CMS Final Repot 4

15 goal is to undestand how vaiation in α influences the equilibium solution... Impact of Vaiation in α In this section, we analyze the changes in the equilibium supply quantities when the congestion cost facto on one of the links connecting a supply location to a maket inceases. Note that we assume the facility location decisions of the fims ae fixed and identical, i.e., fims have the same facility locations. Suppose that locations ae soted such that location has the geatest δ value. Hence, when thee ae l locations active in a maket, these locations will be the fist l locations unde Poposition 3. We fist note that when thee ae l locations active initially, an incease in the taffic congestion cost facto fo one of the active locations will not esult in any of the initially active locations becoming inactive. The next poposition povides a fomal poof of this. Poposition 6. Conside α s and α s such that α < α, and suppose that locations to l ae s s active unde the α s value, s k. Then locations to l ae also active unde the α s value. Poposition 6 implies that when the taffic congestion cost facto fo one of the initially active locations inceases, it is possible that the total numbe of active locations may incease. Moeove, the initially active locations will continue to be active. Next, we study the cases (i) when the numbe of active locations emains the same and (ii) when the numbe of active locations inceases. (i) When the numbe of active locations emains the same, we know that all of the initially active locations will emain active. That is, the set of active locations emains the same. This case also captues the situation when all of the locations ae initially active. In this case, the quantity supplied fom the location fo which the taffic congestion cost facto inceased, will decease. On the othe hand, the quantity supplied fom the othe locations will incease. Moeove, the total quantity supplied to the maket deceases. We fomalize this discussion in the next poposition. Poposition 7. Suppose that α s and α s ae such that active unde α s and locations emain the same. Then (a) l l > i= i=, whee s α < α, and that locations to l ae s s α, s l; that is, the numbe of active locations and the set of active and > fo i= s and, to maket unde the α s and α s values, espectively. < i s. Moeove, (b) denote the equilibium quantities supplied fom location i Statement (a) of Poposition 7 implies that each fim will educe the quantity that it supplies to maket on link ( i, ) if the set of active locations does not change when the taffic congestion cost facto inceases on the link. On the othe hand, each fim will incease the quantity it supplies to maket on the othe links in this case. Moeove, it follows fom Statement (b) of CMS Final Repot 5

16 Poposition 7 that the total quantity sent to maket by any fim will decease. This discussion highlights the fact that fims will educe thei supply to maket and, hence, incease the pice in maket, while deceasing thei tanspotation costs by supplying less, to balance the incease in thei taffic congestion costs. Next we study the case when the numbe of active locations inceases. (ii) When the numbe of active locations inceases, the total quantity supplied fom the location fo which the taffic congestion cost facto inceases will decease. On the othe hand, the total quantity supplied fom the othe locations that wee initially active may incease o decease. Howeve, if the total quantity supplied fom one of the initially active locations (fo which the taffic congestion cost facto emains the same) inceases (deceases), the total quantity supplied fom the othe initially active locations (with unchanged taffic congestion cost factos) also inceases (deceases). The next poposition fomalizes this discussion. Poposition 8. Suppose that α s and α s ae such that l ae active unde fo i = s. Moeove, (b) if l l+ i > = i= α < α, and suppose that locations to s s α s, s l, and locations to l + ae active unde, whee < fo a location i, i s, then and location i to maket unde α s and α s, espectively. α s. Then (a) < i l, i > s and denote the equilibium quantities supplied fom Poposition 8 implies that each fim will educe the quantity it supplies to maket on link ( s, ) if the numbe of active locations inceases when the taffic congestion cost facto inceases on link ( s, ). On the othe hand, each fim may incease o decease the quantity it supplies to maket on the othe links in this case. Howeve, the eaction of the fims will be the same fo the quantity decisions on the othe links, i.e., if fims incease (decease) the flow on link ( i, ), i s, they will incease (decease) the flow on any link ( i, ), i s. Moeove, when fims incease (decease) the flow on link ( i, ), i s, the total quantity supplied to maket and the total quantity supplied to maket by any fim deceases (inceases). When the total quantity supplied to maket by a fim deceases, this implies that all of the fims decease supply to maket, inceasing the pice in maket to balance the incease in the taffic congestion costs. Nevetheless, when the total quantity supplied to maket by a fim inceases and the numbe of supply points inceases, this illustates how fims may choose to divet flow to maket using links that ae not as close to maket but ae less congested. Ou discussion of Popositions 7 and 8 implies that inceased congestion hampes efficient planning of supply chain activities, because it pushes fims to supply a maket using eithe moe congested links o links that ae not close to the maket. In Section 5, we give the esults of extensive numeical studies to chaacteize the effects of inceased taffic congestion on the facility location decisions of the fims as well as the supply quantity decisions. CMS Final Repot 6

17 .3. Stage-One Decisions: Facility Locations In this section, we study the fims supply facility location decisions. We fist discuss the ationale behind ou pio assumption that all fims make identical location decisions. Then, we seek the best location decision of a single fim, unde the assumption that it will also be the best location decision of the othe fims..3.. Identical location decisions Suppose that we ae able to detemine the optimal supply quantity decisions fo any given location decision matix X, which implies that we can detemine the total pofit, including the facility location costs, fo any given X (see Konu and Geunes, 9). In the next poposition, we show that if thee exists a unique PNE location decision, then each fim chooses the same facility locations in equilibium. Poposition 9. Suppose that thee exists a unique PNE location matix, R, whee x denotes the column vecto decision fo each fim in X. X. Then, x = x Poposition 9 also follows fom the fact that location decisions of the fims fom a multi-playe symmetic (stongly symmetic; Bant et al., 9) game with a finite numbe of stategies (Nash, 95). Fo symmetic games, it is well known that a symmetic equilibium exists, eithe unde pue stategies o mixed stategies (Nash, 95). Theefoe, when thee exists a unique PNE location matix, it will be a symmetic PNE, i.e., each fim makes the same location decisions. Futhemoe, Poposition 9 implies that when thee exists a unique PNE location matix, the seach fo an equilibium location matix can be esticted to location decisions such that each fim chooses the same facility locations. We can thus use the method descibed in the pevious section to chaacteize the pofit of each such location matix and, hence, choose the best among all solutions with identical columns to detemine the unique PNE. On the othe hand, it is possible that multiple PNE location decisions exist, o that a PNE location decision does not exist. While uniqueness of PNE location decisions implies existence of a symmetic PNE (which is the unique PNE location matix itself as implied by Poposition 9), in the case of multiple PNE location decisions, it is possible that none of the equilibium points unde pue stategies is symmetic. Cheng et al. (4) show that at least one PNE exists fo multi-playe symmetic games with two stategies. That is, if thee exists a single location, the game coesponding to the location decisions of the fims has a PNE solution. We note that the single location case can be solved by consideing solutions with each fim eithe locating o not locating a facility at the single location. It easily follows fom the discussion in the pevious section that fo any such configuation, the quantity decisions of the fims with a facility will be identical. Moeove, Rhim et al. (3) pove the existence of a PNE in a competitive facility location game in which fims ae allowed to locate at most one facility, by noting that the game can be modeled as a congestion game unde the assumption that each maket will be supplied fom a single location. It is a well-known esult that congestion games have PNE points (Rosenthal, 973). Nevetheless, the game we study cannot be modeled as a congestion game due to the fact that fims may locate moe than one facility. It is noted by Cheng et al. (4) that even fo symmetic games with two stategies, the existence o CMS Final Repot 7

18 uniqueness of a symmetic PNE (i.e., when each playe chooses the same stategy) is not guaanteed. Ami et al. (8) show that fo supemodula, doubly symmetic games, thee exists a Paeto dominant symmetic PNE. Howeve, the location decisions fo ou poblem do not constitute a doubly symmetic game. When a symmetic PNE solution does not exist, this implies that eithe multiple PNE solutions exist o no PNE location exists. Fo both of these cases, as peviously noted, the coesponding mixed stategy Nash equilibium (MSNE) will be symmetic. Next, we study MSNE fo such cases unde the following assumptions: Assumption. Given the location decisions of othe fims, a fim will neve locate an additional facility if locating this facility educes pofit. Assumption. Given the location decisions of othe fims, if locating an additional facility does not change the fim s total pofit, the fim will add this facility. Assumption 3. Given the location decisions of othe fims, thee do not exist multiple distinct location decisions containing an identical numbe of facilities that esult in the same pofit level fo any fim. Note that Assumptions -3 imply that, given the location decisions of othe fims, a fim will have a unique choice of location vecto. In the next poposition, we show that, unde Assumptions -3, a MSNE exists such that the pobability of a fim choosing any paticula location vecto x is eithe o equal to some value ρ such that ρ >. Poposition. Suppose that Assumptions -3 hold and that no fim will choose a location decision that is weakly o stictly dominated. Then, thee exists a mixed stategy Nash equilibium with ρ ( x ) = ρ o ρ ( x ) = fo any location vecto x, fo all R, whee ρ ( x ) denotes the pobability that fim will choose location vecto x and ρ >. It follows fom the poof of Poposition that when thee does not exist a unique symmetic PNE location decision, fims will assign the same pobabilities to location vectos that ae not dominated in a mixed stategy and dominated location vectos will be assigned pobability. Moeove, due to the symmety of the mixed stategy equilibium, fims will assign the same pobability to each paticula location vecto. The poblem with using the equilibium concept as a decision mechanism fo location decisions is that it fails to explain and chaacteize fims actual decisions when multiple PNE solutions exist o when no PNE location decision exists. We aleady know fom Poposition 9 that when the PNE is unique, all fims will choose the same locations and, hence, we can seach ove one fim s decisions to find an equilibium solution, as the pofits of the fims will be the same when the location decisions ae the same. Nevetheless, when multiple PNE solutions exist o when no PNE solution exists, we cannot chaacteize the fims actions using the PNE concept. Thus, if we assume that fims detemine facility locations puely based on thei expected pofits (assuming that any location vecto is equally likely fo any fim), then since fims ae homogeneous, they will make the same decisions. We can theefoe detemine fims location CMS Final Repot 8

19 decisions by choosing the best among all location matices with identical columns. Moeove, as noted in Poposition, in the case of multiple o no PNE location solutions, when fims detemine the pobability of choosing a location vecto, they will assign the same pobabilities, and the pobabilities associated with location vectos that ae not weakly o stictly dominated ae the same fo each fim. Detemining the best among all location matices with identical columns will be equivalent to assuming that any location vecto is equally likely as well, because a location matix that consists of identical weakly o stictly dominated location vectos will not esult in highe pofits fo any fim. Theefoe, fom this point on, we focus on detemining the best location decision of a single fim, assuming that the othe fims will choose the same locations. We note that the coesponding solution is a PNE when thee exists a unique PNE location decision and it is the best symmetic PNE when thee exist multiple symmetic PNE points. Fo both of these cases, the esulting solution will be a Subgame Pefect Nash equilibium (Selten, 975). Now suppose that eithe x o x is the best location decision fo fim. To detemine which of these is bette fo fim, we need to compae the pofits of fim given X = X and X = X, whee each column of X equals x and each column of X equals x. Note that we can find the total pofit fo fim associated with X and X by detemining the pofit fom supplying makets using the method descibed in the pevious section, and then subtacting the facility location costs associated with x and x. A total enumeation scheme would detemine the pofit fo each X such that X has identical columns, and pick the one with maximum pofits. In case of altenative optimal solutions, Assumptions - can be used as a selection tool. The esulting matix X will give the best location decision fo fim as well as fo all othe fims. Howeve, total enumeation equies evaluating exponentially many location decisions fo a fim. In paticula, a fim must detemine the pofit fo m location decisions, and choose the one with the maximum pofit. As total enumeation is computationally budensome, we next povide a heuistic method intended to be epesentative of how individual fims may appoach simultaneous location decisions in pactice. Ou heuistic method fist chooses the numbe of facilities to be located based on a anking of locations deived fom the poblem paametes and then, chooses the best locations fo these facilities. The compaison of the heuistic method with total enumeation that we late povide in Section 5 will chaacteize conditions unde which the method of analyzing location decisions in two steps leads to optimal o nea-optimal pefomance..3.. Heuistic method fo identifying location matix Because we conside a simultaneous game in which a playe may not possess all elevant infomation associated with the othe playes, it is impossible to povide a geneal chaacteization of how an individual fim will appoach the decision poblem (and to, theefoe, chaacteize the solution that will esult). In an attempt to emulate a easonable appoach that might be taken by an individual fim unde such conditions, we have constucted a anking-based heuistic appoach in which potential locations ae anked in a pefeence ode based on poblem data. The heuistic method we povide is thus based on assigning weights to locations. In paticula, the weight of location i is detemined by the expession CMS Final Repot 9

20 ω = c + α + f. () i ( ) i n J The weight of location i, ω i, is the sum of aveage tanspotation cost coefficients and the squae of taffic congestion cost factos fom location i to all of the makets, plus the facility location cost at location i. As a esult, the weight facto fo a location will be lowe fo locations with low facility location costs and low aveage tanspotation and congestion costs. Theefoe, a location with lowe weight is moe favoable. The heuistic method has two phases. In the fist phase, a fim decides on the numbe of facilities to locate as follows. Suppose that a fim is planning to locate l facilities, l m. We assume that the locations of these l facilities will be the l locations with the lowest weights, and we compute the pofit associated with such a location decision. We epeat this pocess fo each l m, and assume that the fim chooses the numbe of facilities that povides the maximum pofit. In the second phase of the heuistic method, a fim detemines the best locations fo the numbe of facilities detemined in the fist phase. Below, we povide a step-by-step desciption of the algoithm. Algoithm. -Phase Heuistic method: Phase I: Detemining the numbe of facilities to be located Step. Calculate the location weights using Equation (). Sot locations in noninceasing ode of weight. Set l =, l = π = and go to Step. Step. Step. Constuct x l by locating facilities at locations to l and detemine the pofit of any fim, π l, using Algoithm with X = X l, whee each column of X l is x l. Go to Step. l If π π, π = π l and l = l. If l m set l= l + and Go to Step. If l = m, go to Step 3. Step 3. If π, π = π l and and stop. = l l and, go to Step 4. Else, set π = and l =, Phase II: Finding the best location decision with l facilities Step 4: Find the best location decision with l facilities by enumeating the location decisions containing l ones (locations). Retun the best solution. Algoithm assumes that a fim detemines facility locations in two phases; fist, the numbe of facilities to be located is decided and then the locations fo these facilities ae detemined. We note that Algoithm povides the best location decision of a fim when the fim believes that all othe fims will utilize the same weight anking based appoach in deciding the numbe of facilities to be located. CMS Final Repot

21 In the next section, we pesent the esults of a numeical study to analyze the effects of taffic congestion costs on the quantity and facility location decisions of the fims. Moeove, we povide numeical esults on the efficiency of the heuistic method descibed in Algoithm. 3. Findings and Applications Ou numeical studies focus on thee kinds of analysis. We fist conside the effects of taffic congestion cost factos on the fims best decisions. Following this, we chaacteize the efficiency of the heuistic method povided in the pevious section. We then compae the fims best decisions (i) when fims conside taffic congestion costs in decision making and (ii) when fims disegad taffic congestion costs in decision making. 3.. Analysis : Effects of Taffic Congestion Ou fist analysis documents the effects of taffic congestion cost on the best decisions of the fims. We geneate data fo ou computational tests in the following way. We conside fou poblem classes, whee each poblem class diffes in tanspotation costs, c, and facility location costs, f i. Fo each of the classes, we use all combinations of k {3,5}, n {3,5,7} and m {3,5, 7,}, esulting in 4 combinations of the values of k, n, and m. Fo each of these combinations, we geneate poblem instances and each poblem instance is solved fo 6 diffeent intevals of taffic congestion cost facto, α, stating fom and inceasing to 8 in incements of.5. This way we can analyze the effects of inceasing congestion cost on the facility location and supply quantity decisions of the fims. Fo evey poblem, we let a U [5, 5] and b U [, ], whee U[ l, u] denotes the unifom distibution on [ l, u]. Table gives the distibution inteval of c and f i values in each poblem class. Table. Data Intevals fo Poblem Classes -4 In each poblem class, we solve 4 poblem instances, and each poblem instance is solved 6 times, once fo each inteval of α values. Fo each poblem instance, we detemine the best location decision (using total enumeation) and the coesponding equilibium quantity decisions fo a single fim. We document the following aveage statistics ove 96 poblem instances (4 in each in each Poblem Class) fo each inteval of α values in Table : a given fim s numbe of facilities (# of fac.), total quantity supplied to makets (Supply uant.), total tanspotation costs (Tans. Cost), total taffic congestion costs (Cong. Cost), total facility location costs (Loc. Cost) and total pofit. A gaph of each statistic in Table is given in Figue. The following conclusions can be dawn by analysis of the statistics shown in Table. CMS Final Repot