# Cost Models for Vehicle Routing Problems Stanford Boulevard, Suite 260 R. H. Smith School of Business

Save this PDF as:

Size: px
Start display at page:

Download "Cost Models for Vehicle Routing Problems. 8850 Stanford Boulevard, Suite 260 R. H. Smith School of Business"

## Transcription

1 /02 \$17.00 (c) 2002 IEEE 1 Cost Models for Vehicle Routing Problems John Sniezek Lawerence Bodin RouteSmart Technologies Decision and Information Technologies 8850 Stanford Boulevard, Suite 260 R. H. Smith School of Business Columbia, MD University of Maryland College Park, MD Abstract Most algorithms for solving vehicle routing problems use the criterion of minimizing total travel time in forming their routes. As illustrated in the paper, even in the case of a homogeneous fleet, solutions using other criteria may be preferred to the solution generated by minimizing total travel time. This situation becomes more pronounced when considering a vehicle routing problem where the fleet is non-homogeneous. In this paper, we explore the uses of more complex cost models in deriving solutions to vehicle routing problems, especially in the case of a non-homogeneous fleet. We develop a function called the Measure of Goodness that can be used as the criterion for solving these problems. We show how the Measure of Goodness criterion or special cases of this criterion can be used to compare solutions, generate an initial fleet mix or used as the criterion in a mixed integer program for carrying out between route swaps of the entities to be serviced. The vehicle routing problem emphasized in this paper is the Capacitated Arc Routing Problem with Vehicle/Site Dependencies (CARP-VSD). 1.0 A Dilemma Consider the point-to-point vehicle routing problem (VRP) whose locations to be serviced are displayed in Figure 1. In this problem, we use Euclidean Distances to measure the distance between the stops and the stops and the depot. We assume there are no service times at the stops, the maximum length of any route is 35 units and all vehicles are identical; i.e., we have a homogeneous fleet of vehicles. We want to develop a set of routes to service these stops. The criterion ordinarily used to form these routes is to minimize total travel time. The following two solutions (Solution 1 and Solution 2) can be generated to this example using well-known algorithms: Solution 1, which is displayed in Figure 2, requires three vehicles. The length of each route is 22.1 units so that the total length of the three routes is 66.3 units. This three- route solution can be generated using the Clark and Wright heuristic algorithm. Solution 2, which is displayed in Figure 3, requires two vehicles. The length of each route is units so that the total length of the two routes is units. This tworoute solution can be generated using a route first-cluster second or giant tour heuristic approach. Which solution is preferred? How can we find a solution to this dilemma? The three route solution is shorter (66.3 units rather than units) but requires an extra vehicle (3 vehicles versus 2 vehicles). Since the algorithms were formed using heuristic algorithms, neither solution is guaranteed to be optimal whatever optimal means. Traditional algorithms that do not require the number of vehicles to be specified as input can find the three route solution whereas other algorithms that do not specify the number of routes in advance can find either the two route solution or the three route solution. Some algorithms require the number of routes to be specified a-priori. If

2 /02 \$17.00 (c) 2002 IEEE 2 an algorithm needs to know the number of vehicles to use, then the question arises as to how to estimate the number of vehicles to use in the solution. In the above example, just minimizing total travel time may not be adequate to decide upon the preferred solution. One way to determine the preferred solution is to assign costs to the various metrics that can be computed from a solution. By assigning costs to the metrics of the problem, the user can decide if the capital cost associated with an additional vehicle offsets the operating cost of 3.36 additional units of route length in the solution. The previous example describes one possible use of a cost model - to assist the user in determining the preferred solution to a VRP in the case of a homogeneous fleet. To find the preferred solution, the user determines a Measure of Goodness of a solution by forming a weighted linear combination of many of the factors that can be computed from a solution to a VRP. The user can then compare the Measure of Goodness for the various solutions generated in order to rank the solutions or select the solution that is preferred. -1,10 1,10-10,1-10,-1 Depot 0,0 10,1 10,-1 Figure 1. Locations to be serviced. -1,10 1, ,1 2-10,-1 Depot 0,0 2 10,1 10,-1 Figure 2. Three route solution.

3 /02 \$17.00 (c) 2002 IEEE 3-1,10 1, ,1 10,1 Depot 2 2 0,0-10,-1 10,-1 Figure 3. Two route solution. 2.0 Comparing Solutions for a Non- Homogeneous Fleet Comparing solutions to a VRP for a non-homogeneous fleet can become a more complex situation than the same situation for a homogeneous fleet (as described in the previous section). Solutions 3 and 4 illustrate the ambiguities that can arise in solving a non-homogeneous fleet VRP. Solution 3: The fleet consists of 5 large vehicles and 38 hours of travel time. Solution 4: The fleet is comprised of 2 large vehicles, 2 medium size vehicles and 2 small vehicles and 44 hours of travel time. The question as to what solution is best becomes more difficult to answer. Nonhomogeneous fleet vehicle routing problems need a cost model that can compare solutions similar to the above two solutions in order to rank the solutions or determine the preferred solution. In some situations, the 5 large vehicle solution may be preferred to the 6 vehicle solution; in other situations, the 6 vehicle solution may be preferred to the 5 vehicle solution. The decision as to the preferred solution can be based on the capital and operating costs of both solutions as well as other factors such as the amount of overtime and the compactness of the routes. All of these factors can be combined into a Measure of Goodness of a solution. It is possible for a feasible solution with a more vehicles and more deadheading to be preferred to a feasible solution with less vehicles and less deadheading if its capital cost is significantly smaller. In this paper, we want to discuss the use of cost models to generate and evaluate solutions to vehicle routing problems. Our analysis will focus on the use of cost models to generate and evaluate solution to the Capacitated Arc Routing Problem with Vehicle-Site Dependencies (CARP-VSD). The analysis presented in this paper is based on the PhD dissertation of Sniezek (Sniezek, 2001). Although this paper concentrates on analyzing the CARP-VSD, we believe that the ideas presented in this paper can be adapted for analyzing other classes of vehicle routing problems. 3.0 Capacitated Arc Routing Problems The CARP VSD is a variant of the Capacitated Arc Routing problem (CARP). In most CARPs, the following holds: The total travel time in any solution is the sum of the service times on the streets and the total deadhead time in the solution. The total service time on the streets is assumed to be a constant and is generally much larger than total deadhead time in the solution which is a variable.

4 /02 \$17.00 (c) 2002 IEEE 4 Since, in most CARPs, the total deadhead time (a variable) is generally small when compared to the total service time (a constant), basing an algorithm strictly on total deadhead time may give misleading results. A vehicle class is defined to be a set of vehicles with identical characteristics. In a CARP with a non-homogeneous fleet, the fleet of vehicles used in a solution can come from more than one vehicle class. A homogeneous fleet CARP has one vehicle class. The CARP-VSD is a CARP where a non-homogeneous vehicle fleet and at least one street (arc or edge) in the underlying network has a vehicle/site dependency. A street has a vehicle/site dependency if vehicles from certain vehicle classes cannot service or traverse the street. The vehicle/site dependency on a street determines the vehicle classes that can service the street, the vehicle classes that can travel along the street but not service the street, and the vehicle classes that can neither travel along the street nor service the street. The Composite Approach presented in Sniezek (2001) is a procedure for finding a feasible solution to the CARP-VSD. To determine this feasible solution, the Composite Approach breaks the network of streets to be serviced into partitions, assigns a vehicle from a vehicle class to each partition and exchanges streets between partitions. The Composite Approach is integrated with well-known travel path generation procedures to derive an overall algorithm for solving the CARP-VSD. The uses of the cost model described in this paper is key to the successful use of the Composite Approach. Residential solid waste collection serves as the motivating example for solving the CARP-VSD. 4.0 Uses of Cost Models for Solving the CARP-VSD The Composite Approach uses cost models for the following purposes. a. To compare alternative solutions to the CARP-VSD. b. To determine an initial fleet mix for the CARP-VSD. c. To be the objective function of a mixed integer program that breaks a network into feasible partitions for the CARP-VSD. Critical to the use of these cost models is the Measure of Goodness of a solution to the CARP-VSD mentioned in section 1 of this paper. In the next section, we want to quantify the notion of the Measure of Goodness for the CARP-VSD. 5. Measure of Goodness for the CARP-VSD The principal cost factors to any solution to the CARP-VSD are now defined. The Measure of Goodness for any solution to the CARP-VSD is a linear combination of these principal cost factors along with a measure of the compactness of the partitions. 5.1 Principal Cost Factors to a Solution to the CARP-VSD a. Capital Cost of the Vehicles. Each partition (and route) requires a vehicle and the vehicles in a solution can vary in size. Associated with each vehicle is an amortized daily capital cost that can be determined from the purchase cost of the vehicle, maintenance cost, insurance cost, etc, along with the expected lifetime of the vehicle. This daily capital cost is vehicle class dependent since larger vehicles generally have a larger daily capital cost than smaller vehicles. Let \$CAP k represent the daily capital cost of a vehicle from vehicle class k. b. Salary Cost of the Crew Operating the Vehicle. Associated with each partition is a crew that operates the vehicle. A fixed daily salary including benefits and overhead can be established for a crew. Since the number of persons in a crew can be vehicle class dependent, the salary cost can be vehicle class dependent. In sanitation routing, some vehicle classes require a one person crew while other vehicle classes require a two or three person crew. (As a note, in some countries, sanitation crews have as many as 10 persons in the crew.) Let \$SAL k represent

6 /02 \$17.00 (c) 2002 IEEE 6 VLC p = \$HOUR k * MAX [(TIME p - TARGET k ), 0] (2) VRC p = \$MILE k * DIST p + \$TIP k *Z p (3) 5.4 Total Cost of a Solution The total cost of a solution is the sum of the costs of each individual partition; i.e. Total Solution Cost = (FC p + VLC p + VRC p ) (4) p If we assume that there is only one vehicle class, no overtime and one trip per route, then, for any partition p, \$CAP k, \$SAL k, \$MILE k and \$TIP k do not depend upon vehicle class k, \$HOUR k = 0 and Z p = 1. Thus, the only variable in the cost of any partition p is DIST p and the total variable cost of a solution is the total travel time over all of the routes, DIST p. Minimizing DIST p is p p the typical objective of most VRPs. 5.5 Partition Compactness In many arc routing applications, the partitions must be compact and the interlacing of the routes must be avoided. Let C p be the measure of partition compactness for partition p. Assume arc a is assigned to partition p. There are two nodes that define the endpoints of arc a. D ap is defined as the distance from the closer of the two nodes making up the endpoints of arc a to the seed point of partition p. Given this measure of D ap, the measure of compactness for partition p is C p = (D ap ) 2 where the sum is over all arcs a a assigned to partition p (5) Let GEOSPREAD be the measure of compactness for a solution. Then, GEOSPREAD is computed as follows: GEOSPREAD = p C p (6) 5.6 Measure of Goodness for a Solution MG, the Measure of Goodness for a solution can then be computed as follows: MG = A* Total Solution Cost + B*GEOSPREAD (7) In (7), A and B are parameters that allow for a tradeoff between the total cost of a solution and the route compactness of a solution to be considered. 5.7 Discussion The evaluation of a solution in Sections allows the user to estimate the total solution cost, measure the compactness of each route in a solution and use MG to measure the goodness of the solution. The parameters A and B in (7) and the parameters used to compute (4) give the user great flexibility in trading off between the cost of a solution and the route compactness of a solution. The above functions are developed specifically for the CARP-VSD; however, similar functions can be developed for other vehicle routing applications. 6. Comparison of Alternative Solutions to a CARP-VSD The Total Solution Cost (4) or the Measure of Goodness of a solution (7) allows us to compare alternative solutions to a CARP-VSD. Thus, any solution can be measured in terms of distance, time, vehicle configuration, fixed vehicle costs and route compactness. Using (4) to quantify a solution can help answer questions such as the following; Which solution is preferred: A four vehicle solution composed of large vehicles or a five vehicle solution composed of small vehicles? In the above cost model, most of the quantities can be determined from a solution and only a few quantities need to be estimated. Thus, this cost model can serve as a good base for comparing the quality of solution derived by different algorithms to a given vehicle routing problem. 7. Use of the Cost Model to Determine an Initial Fleet Mix for the CARP-VSD The algorithm developed in Sniezek (2001) for determining an initial solution to the CARP-VSD needs an initial fleet mix. We have used the cost model concept described in Section 5 to formulate a mixed integer programming problem in order to find an initial fleet mix. Input to the mixed integer program are the costs identified with each vehicle class and the value of the Vehicle Class Workload Matrix V. The generation of the V matrix is described in detail in Chapter

7 /02 \$17.00 (c) 2002 IEEE 7 3 of Sniezek (2001) and briefly described below in Section Estimating the Daily Cost of Each Vehicle Class As described in Section 5, the total cost associated with a partition includes fixed costs (Capital Costs + Crew Salary Costs), variable labor costs (Overtime Cost), and variable routing costs (Mileage Costs + Disposal Costs). Each of these costs is a function of the vehicle class servicing a partition. An example that illustrates the computation of these costs is given in Tables 1-3. Table 1 shows the computation of fixed costs by vehicle class, Table 2 shows how to estimate the variable routing costs by vehicle class, and Table 3 gives an estimated total cost comparison by vehicle class. The following was assumed in generating these tables: Crew size by vehicle class is specified and each member of the crew costs \$150/day, including benefits. Daily capital cost of a vehicle from any vehicle class is amortized over 1000 workdays (approximately 4 years). The daily capital cost times 1000 gives the total capital cost of the vehicle, by vehicle class. The tipping fee is \$100. The typical number of trips to the disposal facility is assumed known by vehicle class. The mileage on each route is estimated and includes the mileage between the routes and the disposal facility and the mileage to and from the depot. Each mile is assumed to cost \$2. The overtime associated with each route is zero. This is not a requirement of the solution. However, for estimating costs, the initial assumption is that no overtime will be required. Vehicle Daily Crew Daily Daily Class Capital Cost Size Crew Cost Fixed Cost Table 1 Daily fixed cost comparison among vehicle classes. Vehicle Disposal Disposal Route Mileage Daily Class Trips Cost Mileage Cost Routing Cost Table 2 Estimated daily routing cost comparison among vehicle classes. Vehicle Fixed Variable Variable Labor Total Class Cost Routing Costs (OT) Costs Vehicle Cost Table 3 Estimated daily total cost comparison among vehicle classes.

8 /02 \$17.00 (c) 2002 IEEE Determining the V Matrix A detailed discussion of determining the V = (V(i,j)) matrix is given in Chapter 3 of Sniezek (2001). The entry, V(i,j), in the V matrix represents the estimated number of vehicles from vehicle class j required to service all arcs whose largest vehicle class that can service these arcs is vehicle class i. A sample V matrix for a four vehicle class problem is given in Figure 4. This V matrix is based on an actual 1800 arc and edge street network from Philadelphia, PA. This network contains the actual site dependencies that exist in Philadelphia. Thus, the element V(3,1) = says that we estimate that vehicles from vehicle class 1 are needed to service all of the streets whose largest vehicle class that can service these streets is vehicle class 3. Many of our computational tests are based on this network V = Figure 4: V matrix for example 7.3 The Mixed Integer Program to Determine the Initial Fleet Mix The following mixed integer program uses the estimated daily cost of a vehicle class and the V Matrix to determine an initial fleet mix. The solution to this mixed integer program allocates enough vehicles to satisfy the workload requirements of each row of the V matrix while minimizing the estimated cost of the fleet mix. The formulation of this mixed integer program is the following Variables in the Mixed Integer Program The variables in the mixed integer program are as follows: X j = The total number of vehicles used from vehicle class j. X ji = The number of vehicles used from vehicle class j for allocating (satisfying) the work in row i of the V matrix. S i = The excess work (slack) available after satisfying the requirements for row i of the V matrix. This slack is present because the required number of vehicles needed to satisfy a row of the V matrix may be non-integer while the number of vehicles allocated must be integer. This slack can be used toward meeting the requirements of subsequent rows of the V matrix Data Elements Input to the Mixed Integer Program The data elements input to the mixed integer program are as follows: C j = Estimated total daily cost of a vehicle from vehicle class j as determined in Table 3. The matrix V with data elements V ij (see section 7.2 for a description of the V matrix) QMAX j = The maximum number of vehicles available from vehicle class j. QMIN j = The minimum number of vehicles from vehicle class j that must be used in the solution. This data element will typically be zero but is included in case there is a contractual obligation to use a certain number of vehicles from a specific vehicle class or some other condition specific to the organization. VCLASS = The total number of unique vehicle classes that can be used in the fleet mix. The V Matrix has VCLASS rows and VCLASS columns Formulation of the Mixed Integer Program The mixed integer program (IFM) given in (8) to (15) determines an initial fleet mix for the CARP-VSD. The formulation of the IFM is as follows:

9 /02 \$17.00 (c) 2002 IEEE 9 (IFM) Min Z = i j C j X ji (8) subject to the following conditions: i ( j= 1 VCLASS i= j (V i1 /V ij )X ji ) + S i-1 S i = V i1 i (9) X ji = X j j (10) X j QMAX j j (11) X j QMIN j j (12) S 0 = 0 (13) All X jj and X j variables are integer and nonnegative (14) All S variables are nonnegative (15) The constraints of the mixed integer program represent the following. (8) minimizes the estimated total cost of the fleet mix chosen. (9) ensures that enough vehicles from the vehicle classes are used to satisfy the workload for each row of the V matrix. V i1 /V ij is a straightforward proration formula on workload. If V i1 = 12 and V ij = 3, then it is assumed that a vehicle from vehicle class j can cover the workload of 4 vehicles from vehicle class 1. As such, only one vehicle from vehicle class j is needed to cover the work of 4 vehicles from vehicle class 1. (10) tallies the number of vehicles used from each vehicle class. (11) and (12) are the upper and lower bounds on the total number of vehicles from each vehicle class that can be used in the fleet mix. (13) states that there is no slack available toward satisfying row 1 of the V matrix. Recall that the slack variables are introduced to account for an integer number of vehicles being used to satisfy a non-integer demand in a row of the V matrix. Since we start with row 1, there can be no slack yet realized from a previous row of the V matrix and as such, S 0 is set to zero. (14) ensures that the number of vehicles used in the fleet mix are both nonnegative and integer. (15) ensures the slack variables are nonnegative Example Using the cost estimates and V matrix defined above and assuming that all vehicles have to be purchased, the IFM becomes the following mixed integer program (IFM2): (IFM2) MIN Z = 920X X X X X X X X X X 44 SUBJECT TO X 11 + S 0 - S 1 = X X 22 + S 1 - S 2 = X X X 33 + S 2 - S 3 = X X X X 44 + S 3 - S 4 = X 1 - X 11 - X 12 - X 13 - X 14 = 0 X 2 X 22 X 23 X 24 = 0 X 3 X 33 X 34 = 0 X 4 X 44 = 0 S 0 = 0 All variables nonnegative All X ij and X variables are integer. The solution to IFM2 is X 1 = 2, X 2 = 6, X 3 = 3, and X 4 = 7 and the value of the objective function of IFM2 is Thus, the initial fleet for this example is the following: two vehicles from vehicle class 1, six vehicles from vehicle class 2, three vehicles from vehicle class 3, and seven vehicles from vehicle class 4 and the estimated total cost of the solution is \$16,290/day Company Owned Fleet Example If we assume that the company already owns 25 Class 1 vehicles and only Class 1 vehicles can be used (a homogeneous fleet), then there is no capital cost associated with these 25 vehicles and the daily cost of the Class 1 vehicles would drop from \$920 to \$820. This fleet would have an estimated cost of \$20,500/day (\$820*25). The 25 Class 1 vehicle fleet is the minimum number of Class 1 vehicles needed to cover total demand.

10 /02 \$17.00 (c) 2002 IEEE 10 Assuming the daily cost of a vehicle from vehicle class 1 is \$820, then the fleet mix determined by the IFM is two vehicles from vehicle class 1, six vehicles from vehicle class 2, three vehicles from vehicle class 3, and seven vehicles from vehicle class 4. The total cost of the IFM is \$16,090. This fleet mix is \$4,410/day less expensive than the company owned fleet of 25 Class 1 vehicles. In other words, the company saves \$4,410/day by purchasing vehicles and incurring additional capital costs rather than using the vehicles that they already own. The salvage value of the company owned vehicles is not considered in the IFM Using the IFM as a Cost Estimation Tool We then analyzed the IFM to determine if the IFM could be used as a tool to estimate the cost of a CARP-VSD. We evaluated 42 different fleet mixes by the objective function of the IFM. Then, each of the 42 fleet mixes was processed by a partition generation heuristic called the Vehicle Decomposition Algorithm (VDA). The VDA is described in Sniezek (2001). The VDA is initiated with the fleet mix found from the IFM. The VDA partitions the area to be serviced into partitions as dictated by the fleet mix found from in the IFM. To complete a VDA solution, the travel paths are generated, deadhead travel distance and time as well as actual trips to the disposal facility are determined and overtime is computed. The determination of the VDA solution requires considerably more data preparation and software development that the determination of the IFM solution. We used the estimated vehicle costs computed in Table 3; that is to say, we assumed that the company had to purchase the entire fleet. The results from our 42 scenarios are listed in Table 4. These problems are described in Chapter 4, Section 3 of Sniezek (2001). The results were very encouraging. We observed the following. In 8 of these 42 trials, the estimated cost found by the MIP was within 1% of the cost found from the VDA solution. In 20 of these 42 trials, the estimated cost found by the MIP was within 2% of the cost found from the VDA solution. In 28 of these 42 trials, the estimated cost found by the MIP was within 3% of the cost found from the VDA solution. In total, 41 of the 42 trials, the estimated cost found by the MIP was within 6% of the cost found from the VDA solution. The worst estimated cost found by the MIP was 6.33% from cost found the VDA solution. The average deviation of the total estimate cost found by the MIP from the cost found by the VDA solution over all 42 scenarios was 2.31%. In 38 of the 42 scenarios, the cost found by the VDA solution was lower than the estimated cost found by solving the IFM. It appears that the estimated cost of a solution found from the IFM with the assumed fleet mix appears to be close to the actual cost of the VDA solution. Thus, we concluded that the IFM could be a very useful tool for determining a fleet mix and for estimating the cost of the fleet mix without having to generate the partitions and travel paths, thereby avoiding considerable data preparation. Therefore, the MIP may be very useful in a sketch planning environment where we are only trying to estimate the cost of a fleet mix. In this case, we want to get an idea of what the best fleet mix might be without having to generate a feasible solution for each case. As Sniezek (2001) shows, generating a complete solution to the CARP- VSD for large problems can be complex and lengthy computationally.

11 /02 \$17.00 (c) 2002 IEEE Use of the Cost Model in the Objective of an Mathematical Program for Partitioning a Street Network The Composite Approach proposed by Sniezek (2001) for generating a solution to the CARP-VSD involves solving a mixed integer program (MPP) as a between-route route improvement procedure. A description of the MPP can be found in Sniezek (2001). Input to the MPP is a subset of the routes that contain no more than 250 arcs and edges. The MPP has the capability of breaking the arcs and edges in the subset of the routes into partitions and assigning a vehicle from the proposed fleet mix to each of the partitions. The objective of the MPP is the Measure of Goodness (see equation (7)). In the objective function, we can vary the weights A and B in (7) to try to find a better solution that emphasizes either total cost or compactness or, in some cases, try to improve both total cost and compactness by assigning them equal weight in the objective function. Further, we add the following two constraints to the MPP. Total Solution Cost < Value of Total Solution Cost from Previous Solution (16) GEOSPREAD < Value of GEOSPREAD from Previous Solution (17) Constraints (16) and (17) insure that the solution found by the MPP is no worse than the previous solution. Using the Measure of Goodness as the objective function along with constraints (16) and (17) gives the user tremendous flexibility in trying to find an improved solution. The user has complete control on the values of A and B and the values of the right hand sides of (16) and (17). If desired, the user can choose to improve GEOSPREAD while allowing for a slight increase in cost. Conversely, the user can choose to decrease cost by allowing for a little less compactness. Several scenarios of using the composite procedure in this context are described in Sniezek (2001). 9.0 Conclusions In this paper, we have shown that the use of most complex cost models for the analysis of vehicle routing problems can lead to finding good and implementable solutions to vehicle routing problems. Further, these cost models can be used to find an initial fleet mix or as the objective function in a mathematical programming problem that performs between-route swaps. The motivating example in this analysis is the CARP/VSD. We further have shown that the cost model function called the Measure of Goodness, can generate a reasonably accurate solution to the CARP-VSD without having to generate a complete of routes and schedules. This allows us to estimate the cost of alternative non-homogeneous fleets in a very rapid manner. The notion of a cost model, although not unique in transportation applications, is relatively unique when comparing vehicle routing solutions to the same problem. In the 1970s, transit systems generally used these cost models to estimate the costs of bus systems. In these unit cost models, the estimated cost of a proposed timetable for a transit system was simply the following: Cost = A * (Number of Buses) + B * (Number of Miles Traversed) These cost models for analyzing bus systems were extended in Bodin, Rosenfield and Kydes (1978) and Bodin, Rosenfield and Kydes (1981) to include factors such as the number of bus drivers needed, overtime, maintenance costs, etc. Further, these cost models allows for the use of overtime in generating routes and analyzing solutions. These models confirm a long-term conjecture of the authors that using overtime can help to generate less expensive solutions to vehicle routing problems because of the savings in capital cost that results. These cost models allow for the tradeoff between overtime and increased capital cost to be fully explored.

12 /02 \$17.00 (c) 2002 IEEE 12 Problem Fleet Mix Net Percent Name Used IFM Cost VDA Cost Difference Difference VSD1-ST \$16,025 \$15,858 (\$167) -1.04% VSD1-ST \$16,989 \$16,735 (\$254) -1.50% VSD1-ST \$18,200 \$17,048 (\$1,152) -6.33% VSD1-ST \$17,045 \$16,409 (\$636) -3.73% VSD1-ST \$16,290 \$16,022 (\$268) -1.65% VSD1-ST \$17,354 \$16,737 (\$617) -3.56% VSD1-ST \$32,050 \$31,205 (\$845) -2.64% VSD1-ST \$32,912 \$32,125 (\$787) -2.39% VSD1-ST \$31,793 \$31,985 \$ % VSD1-ST \$34,090 \$32,709 (\$1,381) -4.05% VSD1-ST \$32,401 \$31,758 (\$643) -1.98% VSD1-ST \$31,707 \$32,072 \$ % VSD1-ST \$48,199 \$47,640 (\$559) -1.16% VSD1-ST \$49,368 \$47,146 (\$2,222) -4.50% VSD1-ST \$48,598 \$47,741 (\$857) -1.76% VSD1-ST \$52,055 \$49,658 (\$2,397) -4.60% VSD1-ST \$48,531 \$48,143 (\$388) -0.80% VSD1-ST \$49,410 \$48,013 (\$1,397) -2.83% VSD2-ST \$18,236 \$18,006 (\$230) -1.26% VSD2-ST \$19,059 \$18,366 (\$693) -3.64% VSD2-ST \$19,572 \$19,174 (\$398) -2.03% VSD2-ST \$18,479 \$18,029 (\$450) -2.44% VSD2-ST \$35,492 \$35,690 \$ % VSD2-ST \$40,045 \$38,197 (\$1,848) -4.61% VSD2-ST \$39,736 \$38,644 (\$1,092) -2.75% VSD2-ST \$38,738 \$36,482 (\$2,256) -5.82% VSD2-ST \$52,684 \$53,141 \$ % VSD2-ST \$59,104 \$57,294 (\$1,810) -3.06% VSD2-ST \$60,398 \$59,163 (\$1,235) -2.04% VSD2-ST \$55,144 \$54,499 (\$645) -1.17% VSD3-ST \$18,885 \$17,809 (\$1,076) -5.70% VSD3-ST \$19,877 \$19,121 (\$756) -3.80% VSD3-ST \$18,542 \$18,405 (\$137) -0.74% VSD3-ST \$18,513 \$18,418 (\$95) -0.51% VSD3-ST \$35,934 \$35,365 (\$569) -1.58% VSD3-ST \$39,754 \$37,614 (\$2,140) -5.38% VSD3-ST \$37,084 \$36,929 (\$155) -0.42% VSD3-ST \$37,026 \$36,666 (\$360) -0.97% VSD3-ST \$53,963 \$53,337 (\$626) -1.16% VSD3-ST \$59,631 \$56,633 (\$2,998) -5.03% VSD3-ST \$55,626 \$54,509 (\$1,117) -2.01% VSD3-ST \$55,539 \$54,889 (\$650) -1.17% Total \$1,500,078 \$1,465,384 (\$34,694) -2.31% Table 4 - Estimated solution costs versus actual solution costs.

### VEHICLE ROUTING PROBLEM

VEHICLE ROUTING PROBLEM Readings: E&M 0 Topics: versus TSP Solution methods Decision support systems for Relationship between TSP and Vehicle routing problem () is similar to the Traveling salesman problem

### Residential waste management in South Africa: Optimisation of vehicle fleet size and composition

Residential waste management in South Africa: Optimisation of vehicle fleet size and composition by Wouter H Bothma 26001447 Submitted in partial fulfillment of the requirements for the degree of Bachelors

### INTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models

Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is

### Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 36 Location Problems In this lecture, we continue the discussion

### Charles Fleurent Director - Optimization algorithms

Software Tools for Transit Scheduling and Routing at GIRO Charles Fleurent Director - Optimization algorithms Objectives Provide an overview of software tools and optimization algorithms offered by GIRO

### Two objective functions for a real life Split Delivery Vehicle Routing Problem

International Conference on Industrial Engineering and Systems Management IESM 2011 May 25 - May 27 METZ - FRANCE Two objective functions for a real life Split Delivery Vehicle Routing Problem Marc Uldry

### LECTURE - 3 RESOURCE AND WORKFORCE SCHEDULING IN SERVICES

LECTURE - 3 RESOURCE AND WORKFORCE SCHEDULING IN SERVICES Learning objective To explain various work shift scheduling methods for service sector. 8.9 Workforce Management Workforce management deals in

### VEHICLE ROUTING AND SCHEDULING PROBLEMS: A CASE STUDY OF FOOD DISTRIBUTION IN GREATER BANGKOK. Kuladej Panapinun and Peerayuth Charnsethikul.

1 VEHICLE ROUTING AND SCHEDULING PROBLEMS: A CASE STUDY OF FOOD DISTRIBUTION IN GREATER BANGKOK By Kuladej Panapinun and Peerayuth Charnsethikul Abstract Vehicle routing problem (VRP) and its extension

### The Trip Scheduling Problem

The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems

### IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

### Vehicle Routing and Scheduling. Martin Savelsbergh The Logistics Institute Georgia Institute of Technology

Vehicle Routing and Scheduling Martin Savelsbergh The Logistics Institute Georgia Institute of Technology Vehicle Routing and Scheduling Part I: Basic Models and Algorithms Introduction Freight routing

### Research Paper Business Analytics. Applications for the Vehicle Routing Problem. Jelmer Blok

Research Paper Business Analytics Applications for the Vehicle Routing Problem Jelmer Blok Applications for the Vehicle Routing Problem Jelmer Blok Research Paper Vrije Universiteit Amsterdam Faculteit

### 7.1 Modelling the transportation problem

Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources

### Learning Objectives. Required Resources. Tasks. Deliverables

Fleet Modeling 10 Purpose This activity introduces you to the Vehicle Routing Problem (VRP) and fleet modeling through the use of a previously developed model. Using the model, you will explore the relationships

### Coordinating Supply Chains: a Bilevel Programming. Approach

Coordinating Supply Chains: a Bilevel Programming Approach Ton G. de Kok, Gabriella Muratore IEIS, Technische Universiteit, 5600 MB Eindhoven, The Netherlands April 2009 Abstract In this paper we formulate

### Optimal Vehicle Routing and Scheduling with Precedence Constraints and Location Choice

Optimal Vehicle Routing and Scheduling with Precedence Constraints and Location Choice G. Ayorkor Korsah, Anthony Stentz, M. Bernardine Dias, and Imran Fanaswala Abstract To realize the vision of intelligent

### Management of Software Projects with GAs

MIC05: The Sixth Metaheuristics International Conference 1152-1 Management of Software Projects with GAs Enrique Alba J. Francisco Chicano Departamento de Lenguajes y Ciencias de la Computación, Universidad

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### What is Linear Programming?

Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

### Solving the Vehicle Routing Problem with Multiple Trips by Adaptive Memory Programming

Solving the Vehicle Routing Problem with Multiple Trips by Adaptive Memory Programming Alfredo Olivera and Omar Viera Universidad de la República Montevideo, Uruguay ICIL 05, Montevideo, Uruguay, February

### Re-optimization of Rolling Stock Rotations

Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany RALF BORNDÖRFER 1, JULIKA MEHRGARDT 1, MARKUS REUTHER 1, THOMAS SCHLECHTE 1, KERSTIN WAAS 2 Re-optimization

### Modeling and Solving the Capacitated Vehicle Routing Problem on Trees

in The Vehicle Routing Problem: Latest Advances and New Challenges Modeling and Solving the Capacitated Vehicle Routing Problem on Trees Bala Chandran 1 and S. Raghavan 2 1 Department of Industrial Engineering

### Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

### Load Balancing of Telecommunication Networks based on Multiple Spanning Trees

Load Balancing of Telecommunication Networks based on Multiple Spanning Trees Dorabella Santos Amaro de Sousa Filipe Alvelos Instituto de Telecomunicações 3810-193 Aveiro, Portugal dorabella@av.it.pt Instituto

### Solution of Linear Systems

Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start

### Information Security and Risk Management

Information Security and Risk Management by Lawrence D. Bodin Professor Emeritus of Decision and Information Technology Robert H. Smith School of Business University of Maryland College Park, MD 20742

### Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

### Methodology. Figure 1 Case study area

Maximising Efficiency in Domestic Waste Collection through Improved Fleet Management Fraser N McLeod and Tom J Cherrett Transportation Research Group, School of Civil Engineering and the Environment, University

### HOME HEALTH CARE LOGISTICS PLANNING

HOME HEALTH CARE LOGISTICS PLANNING A Thesis Presented to The Academic Faculty by Ashlea R. Bennett In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the H. Milton Stewart

### Waste Collection Vehicle Routing Problem Considering Similarity Pattern of Trashcan

International Journal of Applied Operational Research Vol. 3, o. 3, pp. 105-111, Summer 2013 Journal homepage: www.ijorlu.ir Waste Collection Vehicle Routing Problem Considering Similarity Pattern of Trashcan

### A Scatter Search Algorithm for the Split Delivery Vehicle Routing Problem

A Scatter Search Algorithm for the Split Delivery Vehicle Routing Problem Campos,V., Corberán, A., Mota, E. Dep. Estadística i Investigació Operativa. Universitat de València. Spain Corresponding author:

### A Set-Partitioning-Based Model for the Stochastic Vehicle Routing Problem

A Set-Partitioning-Based Model for the Stochastic Vehicle Routing Problem Clara Novoa Department of Engineering and Technology Texas State University 601 University Drive San Marcos, TX 78666 cn17@txstate.edu

### This paper introduces a new method for shift scheduling in multiskill call centers. The method consists of

MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 10, No. 3, Summer 2008, pp. 411 420 issn 1523-4614 eissn 1526-5498 08 1003 0411 informs doi 10.1287/msom.1070.0172 2008 INFORMS Simple Methods for Shift

### Reciprocal Cost Allocations for Many Support Departments Using Spreadsheet Matrix Functions

Reciprocal Cost Allocations for Many Support Departments Using Spreadsheet Matrix Functions Dennis Togo University of New Mexico The reciprocal method for allocating costs of support departments is the

### Fowler Distributing Company Logistics Network Routing Analysis. Anthony Vatterott, Brian Vincenz, Tiffany Bowen. Executive Summary

Fowler Distributing Company Logistics Network Routing Analysis Anthony Vatterott, Brian Vincenz, Tiffany Bowen Executive Summary Our firm was contracted by Fowler Distributing Company to provide analysis

### Master s degree thesis

Master s degree thesis LOG950 Logistics Title: Collecting, dewatering and depositing sewage - logistics challenges Author(s): Lin keyong Number of pages including this page: 32 Molde, Date: 24.05.2012

### Unit 1. Today I am going to discuss about Transportation problem. First question that comes in our mind is what is a transportation problem?

Unit 1 Lesson 14: Transportation Models Learning Objective : What is a Transportation Problem? How can we convert a transportation problem into a linear programming problem? How to form a Transportation

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Software Metrics & Software Metrology. Alain Abran. Chapter 4 Quantification and Measurement are Not the Same!

Software Metrics & Software Metrology Alain Abran Chapter 4 Quantification and Measurement are Not the Same! 1 Agenda This chapter covers: The difference between a number & an analysis model. The Measurement

### OPTIMIZED H O M E C A R E S C H E D U L I N G AND ROUTING

OPTIMIZED H O M E C A R E S C H E D U L I N G AND ROUTING A White Paper for Home Care Executives and Operations Managers ALGORITHM-BASED OPTIMIZATION FOR SOLVING THE SCHEDULING AND ROUTING PROBLEMS IN

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Coordination in vehicle routing

Coordination in vehicle routing Catherine Rivers Mathematics Massey University New Zealand 00.@compuserve.com Abstract A coordination point is a place that exists in space and time for the transfer of

### Routing in Line Planning for Public Transport

Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany MARC E. PFETSCH RALF BORNDÖRFER Routing in Line Planning for Public Transport Supported by the DFG Research

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

### OFTECHNOLO..1_GY: X; 0 t;0~~~~~~~~~~~~~~ : ;. i : wokn. f t,0l'' f:fad 02 'L'S0 00';0000f ::: 000';0' :-:::

... -. -. _,.,...,..-. - - - - soft,, LI,,, '-'J''

### Scheduling and Routing Milk from Farm to Processors by a Cooperative

Journal of Agribusiness 22,2(Fall 2004):93S106 2004 Agricultural Economics Association of Georgia Scheduling and Routing Milk from Farm to Processors by a Cooperative Peerapon Prasertsri and Richard L.

### M. Sugumaran / (IJCSIT) International Journal of Computer Science and Information Technologies, Vol. 2 (3), 2011, 1001-1006

A Design of Centralized Meeting Scheduler with Distance Metrics M. Sugumaran Department of Computer Science and Engineering,Pondicherry Engineering College, Puducherry, India. Abstract Meeting scheduling

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Incremental bus service design: combining limited-stop and local bus services

Public Transp (2013) 5:53 78 DOI 10.1007/s12469-013-0067-7 ORIGINAL PAPER Incremental bus service design: combining limited-stop and local bus services Virot Chiraphadhanakul Cynthia Barnhart Published

### Performance of networks containing both MaxNet and SumNet links

Performance of networks containing both MaxNet and SumNet links Lachlan L. H. Andrew and Bartek P. Wydrowski Abstract Both MaxNet and SumNet are distributed congestion control architectures suitable for

### 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

### A Constraint Programming based Column Generation Approach to Nurse Rostering Problems

Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,

### Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach

MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical

### The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.

ADVANCED SUBSIDIARY GCE UNIT 4736/01 MATHEMATICS Decision Mathematics 1 THURSDAY 14 JUNE 2007 Afternoon Additional Materials: Answer Booklet (8 pages) List of Formulae (MF1) Time: 1 hour 30 minutes INSTRUCTIONS

### EVALUATION OF OPTIMIZATION ALGORITHMS FOR IMPROVEMENT OF A TRANSPORTATION COMPANIES (IN- HOUSE) VEHICLE ROUTING SYSTEM

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Industrial and Management Systems Engineering -- Dissertations and Student Research Industrial and Management Systems Engineering

### Solving a New Mathematical Model for a Periodic Vehicle Routing Problem by Particle Swarm Optimization

Transportation Research Institute Iran University of Science and Technology Ministry of Science, Research and Technology Transportation Research Journal 1 (2012) 77-87 TRANSPORTATION RESEARCH JOURNAL www.trijournal.ir

### Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

### Multi-layer MPLS Network Design: the Impact of Statistical Multiplexing

Multi-layer MPLS Network Design: the Impact of Statistical Multiplexing Pietro Belotti, Antonio Capone, Giuliana Carello, Federico Malucelli Tepper School of Business, Carnegie Mellon University, Pittsburgh

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

Load Balancing Backtracking, branch & bound and alpha-beta pruning: how to assign work to idle processes without much communication? Additionally for alpha-beta pruning: implementing the young-brothers-wait

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### Mathematics of Cryptography Part I

CHAPTER 2 Mathematics of Cryptography Part I (Solution to Odd-Numbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.

### Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

### Fleet Size and Mix Optimization for Paratransit Services

Fleet Size and Mix Optimization for Paratransit Services Liping Fu and Gary Ishkhanov Most paratransit agencies use a mix of different types of vehicles ranging from small sedans to large converted vans

### A MIXED INTEGER PROGRAMMING FOR A VEHICLE ROUTING PROBLEM WITH TIME WINDOWS: A CASE STUDY OF A THAI SEASONING COMPANY. Abstract

A MIXED INTEGER PROGRAMMING FOR A VEHICLE ROUTING PROBLEM WITH TIME WINDOWS: A CASE STUDY OF A THAI SEASONING COMPANY Supanat Suwansusamran 1 and Pornthipa Ongunaru 2 Department of Agro-Industrial Technology,

### Appendix: Simple Methods for Shift Scheduling in Multi-Skill Call Centers

MSOM.1070.0172 Appendix: Simple Methods for Shift Scheduling in Multi-Skill Call Centers In Bhulai et al. (2006) we presented a method for computing optimal schedules, separately, after the optimal staffing

### Comparison of Pavement Network Management Tools based on Linear and Non-Linear Optimization Methods

Comparison of Pavement Network Management Tools based on Linear and Non-Linear Optimization Methods Lijun Gao, Eddie Y. Chou, and Shuo Wang () Department of Civil Engineering, University of Toledo, 0 W.

### Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

### Vehicle Routing: Transforming the Problem. Richard Eglese Lancaster University Management School Lancaster, U.K.

Vehicle Routing: Transforming the Problem Richard Eglese Lancaster University Management School Lancaster, U.K. Transforming the Problem 1. Modelling the problem 2. Formulating the problem 3. Changing

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Project and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi

Project and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture - 15 Limited Resource Allocation Today we are going to be talking about

### APPENDIX G OPERATIONS AND MAINTENANCE COST ESTIMATION MODEL

DRAFT APPENDIX G OPERATIONS AND MAINTENANCE COST ESTIMATION MODEL Operations and Maintenance Cost Methodology and Results The Colorado Department of Transportation is conducting an Advanced Guideway System

### A Quantitative Decision Support Framework for Optimal Railway Capacity Planning

A Quantitative Decision Support Framework for Optimal Railway Capacity Planning Y.C. Lai, C.P.L. Barkan University of Illinois at Urbana-Champaign, Urbana, USA Abstract Railways around the world are facing

### Optimal Cheque Production: A Case Study

Blo UNIVERSITÀ DI SALERNO Dipartimento di Matematica e Informatica D.M.I. Via Ponte don Melillo 84084 Fisciano (SA) Italy Optimal Cheque Production: A Case Study Raffaele Cerulli, Renato De Leone, Monica

### Network Models 8.1 THE GENERAL NETWORK-FLOW PROBLEM

Network Models 8 There are several kinds of linear-programming models that exhibit a special structure that can be exploited in the construction of efficient algorithms for their solution. The motivation

### USING EXCEL SOLVER IN OPTIMIZATION PROBLEMS

USING EXCEL SOLVER IN OPTIMIZATION PROBLEMS Leslie Chandrakantha John Jay College of Criminal Justice of CUNY Mathematics and Computer Science Department 445 West 59 th Street, New York, NY 10019 lchandra@jjay.cuny.edu

### Optimising Patient Transportation in Hospitals

Optimising Patient Transportation in Hospitals Thomas Hanne 1 Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany, hanne@itwm.fhg.de 1 Introduction

### Make Better Decisions with Optimization

ABSTRACT Paper SAS1785-2015 Make Better Decisions with Optimization David R. Duling, SAS Institute Inc. Automated decision making systems are now found everywhere, from your bank to your government to

### 4.6 Linear Programming duality

4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

Submitted to Transportation Science manuscript (Please, provide the mansucript number!) Collaboration for Truckload Carriers Okan Örsan Özener H. Milton Stewart School of Industrial and Systems Engineering,

### Chapter 2: Systems of Linear Equations and Matrices:

At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

### Generating Personnel Schedules in an Industrial Setting Using a Tabu Search Algorithm

Generating Personnel Schedules in an Industrial Setting Using a Tabu Search Algorithm Pascal Tellier 1 and George White 2 1 PrairieFyre Software Inc., 555 Legget Dr., Kanata K2K 2X3, Canada pascal@prairiefyre.com

### Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

### . P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

### Social Media Mining. Network Measures

Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

### The application of linear programming to management accounting

The application of linear programming to management accounting Solutions to Chapter 26 questions Question 26.16 (a) M F Contribution per unit 96 110 Litres of material P required 8 10 Contribution per

### Adaptive Memory Programming for the Vehicle Routing Problem with Multiple Trips

Adaptive Memory Programming for the Vehicle Routing Problem with Multiple Trips Alfredo Olivera, Omar Viera Instituto de Computación, Facultad de Ingeniería, Universidad de la República, Herrera y Reissig

Load Planning for Less-than-truckload Carriers Martin Savelsbergh Centre for Optimal Planning and Operations School of Mathematical and Physical Sciences University of Newcastle Optimisation in Industry,

### Special cases in Transportation Problems

Unit 1 Lecture 18 Special cases in Transportation Problems Learning Objectives: Special cases in Transportation Problems Multiple Optimum Solution Unbalanced Transportation Problem Degeneracy in the Transportation

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Scheduling Algorithm with Optimization of Employee Satisfaction

Washington University in St. Louis Scheduling Algorithm with Optimization of Employee Satisfaction by Philip I. Thomas Senior Design Project http : //students.cec.wustl.edu/ pit1/ Advised By Associate

### OPTIMAL VEHICLE TIMETABLING, ROUTING AND DRIVER DUTY SCHEDULING CASE STUDY: BUS DEPOT PROBLEM

OPTIMAL VEHICLE TIMETABLING, ROUTING AND DRIVER DUTY SCHEDULING CASE STUDY: BUS DEPOT PROBLEM Mr Tushar T. Rasal Department of Electronics & Telecommunication Engineering Rajarambapu Institute of Technology