Obtaining the optimal fleet mix: A case study about towing tractors at airports

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Obtaining the optimal fleet mix: A case study about towing tractors at airports$ Jia Yan Du a, Jens O. Brunner b,n, Rainer Kolisch a a b

TUM School of Management, Technische Universität München, Arcisstr. 21, 80333 Munich, Germany Faculty of Business and Economics, Universität Augsburg, Universitätsstr. 16, 86159 Augsburg, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 7 August 2014 Accepted 15 November 2015

Planes do not have a reverse gear. Hence, they need to be towed by tractors when leaving the gate. Towing tractors differ with respect to investment as well as variable costs and plane type compatibility. We propose a model which addresses the problem of a cost minimal fleet composition to support towing service providers in their strategic investment decisions. The model takes into account a maximum lifetime, a minimum duration of use, an overhaul option and a sell option. In a case study with a major European airport (our cooperating airport) we generate a multi-period fleet investment schedule. Furthermore, we introduce a 4-step approach for demand aggregation based on flight schedule information. We analyze the impact of demand variation, flight schedule disruptions and cost structure on the optimal buy, overhaul and sell policy. The scenario analyses demonstrate the robustness of the investment schedule with respect to these factors. Ignoring the existing fleet, a green field scenario reveals saving potentials of more than 5% when applying this model. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Airport operations management Turnaround processes Fleet composition problem

1. Introduction Planes do not have a reverse gear. Hence, they need assistance to leave the gate. Furthermore, over long distances it is often more economical and ecological to use towing tractors (see [2]). Towing can be distinguished between (i) push-back: the fully boarded plane is pushed backwards from the gate to the taxiway; (ii) repositioning: the empty plane is towed from one parking position to another; and (iii) maintenance towing: the empty plane is towed to the hangar area for maintenance or repairs. Towing tractors differ with respect to technical compatibility with plane types, investment costs and variable costs. From the perspective of a towing service provider, there are two key questions: 1. What is the cost optimal assignment of towing jobs to towing tractors in daily operations? The towing service provider is responsible for carrying out all towing jobs on time. The assignment of tractors to towing jobs is part of their daily operations. Today most towing service providers apply manual planning tools, often resulting in inefficient schedules. The ☆

This manuscript was processed by Associate Editor Salazar-Gonzalez Corresponding author. E-mail addresses: [email protected] (J.Y. Du), [email protected] (J.O. Brunner), [email protected] (R. Kolisch). n

assignment significantly impacts service quality as well as operating costs. In the short-term the available fleet for the assignment is given by the existing tractors. This operative planning problem is covered in Du et al. [7]. 2. What is the cost optimal fleet composition and respective (dis-) investment strategy? On a strategic level the towing service provider is responsible for deciding on the fleet size and mix and thereby determining in each period (typically of 6 month length) how many tractors are to be bought, overhauled or sold. This decision impacts investment costs, operating costs, as well as the service level. This paper addresses the strategic planning problem. We introduce a model that generates a cost optimal schedule for a heterogeneous set of towing tractors considering a long-term horizon of e.g. 10 years with a period length of 6 months. We formulate the model using an extended formulation, which is solved by standard column generation technique (see [6]). We allow the fleet size and mix to change from period to period. The model includes aspects like a selling option, a general overhaul option, minimum duration of use, maximum lifetime and the technical compatibility of tractor types with plane types. For the strategic problem addressed in this paper, a “schedule” refers to an investment schedule which determines in which period to buy, overhaul and sell tractors of different types. In the literature determining the number of vehicles for a homogeneous fleet is referred to as Fleet Sizing Problem (FSP),

http://dx.doi.org/10.1016/j.omega.2015.11.005 0305-0483/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: Du JY, et al. Obtaining the optimal fleet mix: A case study about towing tractors at airports. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.11.005i

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while a Fleet Composition Problem (FCP) addresses the problem of deciding on the fleet size and mix simultaneously for a heterogeneous set of vehicles (e.g. see [8]). FSP and FCP literature can be categorized in those considering routing (e.g., see [15]) and those ignoring routing. Our proposed model does not include routing aspects since we focus on a long-term strategic perspective. At a strategic level, demand, costs and revenue uncertainties related to fleet operations are high, thus taking into account routing aspects on a detailed level is ineffective (see [10]). Kirby [11] and Wyatt [18] are among the first to address the FSP. Kirby [11] investigates the wagon fleet size of a railway system. He concludes that the fraction of days to hire external vehicles should be equal to the ratio of costs for external vehicles and fixed costs of internal vehicles. Wyatt [18] considers a fleet of barges. He extends the idea of Kirby [11] by adding variable costs. Papers investigating a heterogeneous fleet are Gould [9], Loxton et al. [12] and Redmer et al. [14]. In contrast to this work, their fleet composition is determined for a single period or is constant in all periods. New [13], Schick and Stroup [16], Etezadi and Beasley [8], Couillard and Martel [5], Wu et al. [17] and Burt et al. [4] examine a planning horizon of multiple periods and allow fleet composition to change over time. New [13] presents a linear programming model which minimizes the operating costs of an airline fleet by deciding on the timing of investment and disposal of planes. Schick and Stroup [16] propose a model to address the multi-year aircraft fleet planning problem which takes passenger demand requirements, a minimum and maximum flight frequency and aircraft balance equations into account. The authors discuss the application of their model in a real life environment. Etezadi and Beasley [8] propose a mixed integer program to determine the optimal fleet composition of vehicles which serve several customers from a central depot. Their model minimizes the fixed and variable costs of own and hired vehicles, while ensuring a sufficient number of vehicles in each period to cover the distance to and capacity for all customers. Couillard and Martel [5] introduce a stochastic programming model to tackle the FCP for road carriers. The model determines the cost minimal purchase, sell and rental policy for a set of heterogeneous trucks, while demand is subject to seasonal fluctuations. It considers among others the age of vehicles in the fleet as well as tax allowances for owning a vehicle. Wu et al. [17] apply the FCP to the specifics of the truck-rental industry. The authors introduce a linear programming model which decides on truck investment and divestment, demand allocation and repositioning of empty trucks. The solution procedure applies Benders decomposition and Lagrangian relaxation in a two stage approach. It can solve instances with 3 tractor types, 60 periods and 25 locations within 12 h. The work of Burt et al. [4] investigates the FCP for the mining industry. The proposed integer program determines the optimal buy and sell policy for trucks and loaders used in a mining location. A unique aspect of this model is the consideration of compatibility between trucks and loaders. In a case study the authors determine the optimal solution for a problem with eight trucks, 20 loaders and 13 periods within 2.5 h. Although the discussed papers are relevant, none of these papers capture a general overhaul option and a minimum duration of use. To the best of our knowledge, the towing fleet investment decision has not been investigated yet in the FCP literature. New [13], Etezadi and Beasley [8], Couillard and Martel [5], Wu et al. [17] and Burt et al. [4] come closest to this work. A general overhaul option and the minimum duration of use are not included in any of the models. Furthermore, technical compatibility is in most cases not taken into account. Yet, these aspects are essential when determining the optimal investment strategy in a real-world towing setting at airports. Our work closes the gap and contributes to the FCP literature by introducing a model which takes

into account a maximum lifetime, a minimum duration of use, an overhaul option and a sell option. Furthermore, our main contributions are a 4-step approach for demand aggregation and demonstrating the application of the model in an extensive case study. The remainder of this paper is organized as follows: in the next section we introduce the problem and explain the mathematical formulation and the solution approach. Section 3 presents an approach to aggregate demand using flight schedule information and describes how the existing fleet can be incorporated in the model. In Section 4 we demonstrate how the model can be applied in a real-world setting. For this, we determine the schedule at our cooperating airport. Additional scenario analyses are conducted to investigate the impact of demand and costs deviations. We conclude with a summary of the main findings and outline directions for future research in Section 5.

2. Model and solution approach The presented model generates a cost optimal multi-period schedule for a set of heterogeneous towing tractors. It considers a planning horizon of j T j periods. The model determines for each tractor type b the number of required tractors in each period t in order to satisfy a demand DM d;t of each demand pattern d in period t. A demand pattern is an aggregation of simultaneous towing jobs taking into account plane type and overlapping tractor type compatibility. The demand for one period is expressed by a set of demand patterns (see Section 3.1). To fulfill demand, a tractor type b has to be technically compatible with demand pattern d, i.e. CP b;d ¼ 1. The model takes into account the existing fleet. NEb denotes the number of existing tractors of type b. The fleet size and mix is adjusted from period to period by buying new tractors, overhauling or selling existing ones. A general overhaul is required, if a tractor is used beyond its maximum duration of use DU. A general overhaul extends a tractor's lifetime by additional AD periods. A tractor can be sold on the market before reaching its maximum lifetime DU (without general overhaul) or DU þ AD (with general overhaul). However, a tractor has a minimum duration of use of MU periods, before it can be sold. MU does not reflect a technical feature of a tractor, but rather is set by the management. Buying, using, overhauling and selling a tractor in period t is associated with investment costs ICt, variable costs VCt, overhaul costs OCt, and sales revenue SRt, respectively. In the case of a planning horizon of up to 10 years, costs and revenue are time-dependent by, amongst others, taking into account discount rates. Therefore, all cost and revenue parameters are time-indexed. Both cost changes and the discount rate are incorporated in the cost data, and do not explicitly appear in the model. The parameters DU, AD, MU, ICt, VCt, OCt and SRt are tractor type specific and assume different values for each tractor type b. We formulate the model using an extended formulation and propose a Column Generation Heuristic (CGH) as solution procedure. We do not present the compact mixed-integer linear programming model for the problem. In simple terms, column generation decomposes the problem into a Master Problem (MP) and a Subproblem (SP), which generates feasible columns (i.e. schedules). A feasible column a A AðbÞ represents one schedule for a specific tractor type b. The schedule defines in which periods the tractor is in use and accordingly when to buy, overhaul and sell the tractor. AðbÞ is the set of all schedules associated with tractor type b. Each schedule a A AðbÞ is associated with total schedule costs of TC b;a that are a function of IC t ; VC t ; OC t ; and SRt. MP determines the fleet size and mix by selecting the schedules to follow. It minimizes the costs while ensuring demand satisfaction. Only a subset of all feasible schedules are considered

Please cite this article as: Du JY, et al. Obtaining the optimal fleet mix: A case study about towing tractors at airports. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.11.005i

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(we term the problem as Restricted Master Problem (RMP)) and new columns are added iteratively. The procedure starts with a small subset of columns and solves the Linear Programming (LP) relaxation of RMP. By inserting the dual value information of RMP constraints into the objective function of SP, the most promising column is generated. A column with negative reduced costs is added to RMP and RMP is reoptimized. The procedure terminates with a valid lower bound for the problem (i.e. no absent column with negative reduced costs exists). In a second step, all generated columns which have been inserted in RMP thus far are taken and RMP is solved as Integer Program (IP) to generate a feasible solution, i.e. an upper bound. Master problem: the notation and mathematical formulation of RMP are as follows: Sets: B AðbÞ D T

set set set set

of of of of

tractor types (index b) schedules associated with tractor type b (index a) demand patterns (index d) periods (index t)

Parameters:

3

Constraints (1c) take the existing fleet into account. We create one additional SP for each existing tractor and adapt decision variables and parameter settings (see Section 3.3), i.e. j Bj is the number of general tractor types plus the number of existing tractors. Constraints (1c) enforce NEb number of schedules of tractor type b to be selected. NEb is greater than or equal to 1 for SP representing existing tractor types and NEb is 0 for all other SP. Variable definitions are given in (1d) and (1e). The dual solution of RMP is obtained by relaxing the integrality conditions and solving RMP with a subset of columns. Let δd;t Z0 denote the dual values of constraints (1b), then δt Z 0 is defined as X δt ¼ δd;t 8 t A T : ð2Þ dAD

And let δ~ b Z 0 denote the dual values of constraints (1c). Then the reduced costs of column a associated with tractor type b is ! X ~ c b;a ¼ TC b;a  δt  X b;a;t þ δ b ð3Þ t AT

with TC b;a representing the total costs of schedule a for tractor type b defined as X X TC b;a ¼ VC t  X b;a;t þ IC t  Y buy þOcost b;a;t b;a;t t AT

TC b;a CW CP b;d

costs of schedule a associated with tractor type b costs associated with auxiliary variable wd;t 1, if tractor type b is compatible with at least one plane type associated with demand pattern d, 0 otherwise 1, if schedule a for tractor type b covers period t, 0 otherwise demand of demand pattern d in period t number of existing tractors of tractor type b

X b;a;t DM d;t NEb Variables

λb;a

Minimize

X X

TC b;a  λb;a þ CW 

X

X

b A B: CP b;d ¼ 1

a A AðbÞ

X

XX

X b;a;t  λb;a þ wd;t Z DMd;t

λb;a Z NEb

wd;t

ð1aÞ

8 d A D; t A T

ð1bÞ

d A Dt A T

b A B a A AðbÞ

s:t:

XX

t A T t~ A T

t AT

SRt~  U b;a;t;t~ :

ð4Þ

; Ocost X b;a;t ; Y buy b;a;t and U b;a;t;t~ represent the variable solution values b;a;t in SP. A detailed description of the cost components is given in the explanation of SP's objective function (5a). Subproblem (b): One SP is created for each tractor type b and each existing tractor. Each SP generates schedules for tractor type b. The following additional notation is used to formulate SP(b): Parameters

number of type b tractors assigned to schedule a uncovered demand associated with demand pattern d and period t

wd;t



VCt ICt SRt

variable costs in period t investment costs in period t sales revenue per remaining use period, if tractor is sold in period t general overhaul costs in period t maximum duration of use without general overhaul maximum additional duration of use after a general overhaul minimum duration of use before tractor can be sold

OCt DU AD MU

8bAB

ð1cÞ

Variables

λb;a Z0 and integer 8 bA B; a A AðbÞ

ð1dÞ

xt

wd;t Z 0 and integer

ð1eÞ

ybuy t yov ysell t ut;t~

a A AðbÞ

8 d A D; t A T

The objective function (1a) of the master problem minimizes the total costs of the selected schedules. The first sum adds the costs of all schedules TC b;a which are selected. Variable λb;a denotes the number of tractors associated with type b which are bought, overhauled and sold according to schedule a. The (auxiliary) variables wd;t in the second term count the uncovered demand for pattern d in period t. We use the variables to initialize RMP. They guarantee feasibility in the course of the column generation procedure. The use of one not available tractor for one period is penalized with costs CW. CW is set to a value higher than the most expensive column costs, i.e. CW 4 TC a;b 8 bA B; a A AðbÞ. Demand constraints (1b) ensure that the demand in each period is fulfilled for each demand pattern, i.e. there must be sufficient numbers of compatible tractors to satisfy demand.

ocost t

1, if tractor is used in period t, 0 otherwise (we set x0 ¼ 0; x j T j ¼ 0) 1, if tractor is bought in period t, 0 otherwise 1, if tractor is overhauled, 0 otherwise 1, if tractor is sold in period t, 0 otherwise remaining lifetime if tractor is bought in period t and sold in period t~ costs of general overhaul if tractor is bought in period t

Minimize

X

VC t  xt þ

t AT

X

IC t  ybuy þ t

tAT



XX t A T t~ A T

s:t:

r1 xt  1 þybuy t

xt  xt  1 ybuy r0 t

8t AT 8t AT

X

ocost t

t AT

SRt~  ut;t~ 

X

!

δt  xt þ δ~ b

ð5aÞ

tAT

ð5bÞ ð5cÞ

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4

 xt  1 þ ysell t r0 xt þysell t r1

8t AT

8t AT

ð5eÞ

 xt þ xt  1  ysell t r0 yov r

X

ð5dÞ

8t AT

ð5f Þ

ybuy t

ð5gÞ

t AT

X t~ A T

t~  ysell  t~

X

t  ybuy Z ðDU þ1Þ  yov t

ð5hÞ

tAT

Z OC t þ DU  ðybuy þ yov  1Þ ocost t t

8 t A f1; …; j T j  DU g

ð5iÞ

ut;t~ rðDU þ ADÞ  ybuy t

8 t; t~ A T

ð5jÞ

ut;t~ rðDU þ ADÞ  ysell t~

8 t; t~ A T

ð5kÞ

ut;t~ rAD  yov þ DU ðt~  ysell  t  ybuy Þ t t~ Þ þ t~  ð1  ybuy t X

8 t; t~ A T

ð5lÞ

ybuy r1 t

ð5mÞ

tAT

xt~ Z ybuy t

  8 t A f1; …; j T j  1g; t~ A t; …; minft þ MU 1; j T j 1g ð5nÞ

1 1  xt þ DU þ AD Z yov þ ybuy t

8 t A f1; …; j T j 1  AD  DU g ð5oÞ

1  xt þ DU Z ybuy  yov t ; yov ; ysell xt ; ybuy t t A f0; 1g ocost ; ut;t~ Z0 t

8 t; t~ A T

8 t A f1; …; j T j  1 DU g 8t AT

ð5pÞ ð5qÞ ð5rÞ

The objective function (5a) minimizes the reduced costs of a new schedule. It consists of five terms that are discussed below. The first term takes into account the variable costs of all periods in which a tractor is used. The second term considers the investment costs ICt of a tractor bought in period t while the third term calculates the overhaul costs ocost due to a general overhaul in period t t. Selling a tractor results in additional earnings that are handled in the fourth term. Those earnings depend on the remaining lifetime ut;t~ and the sales revenue per remaining use period SRt~ when the tractor is sold in period t~ . We assume that all tractors are sold at the end of the planning horizon. Whereas the first four terms consider the real value (i.e. cost including earnings for selling) of a new schedule, the fifth term handles the dual information (i.e. dual values δt and δ~ b obtained from solving the relaxed RMP). Constraints (5b), (5c) and (5n) detect a shift of the x variable from 0 to 1 and thereby determine the buying period. These constraints are the linearization of ybuy ¼ xt  ð1  xt  1 Þ. Accordt ingly, constraints (5d), (5e) and (5f) determine the selling period ¼ xt  1  ð1 xt Þ. and linearize ysell t Constraints (5g), (5i) and (5h) set the rules for a general overhaul: a general overhaul can take place if the tractor has been bought previously (5g) and before the tractor is being sold (5h). We force yov ¼ 0, if the tractor is sold before the maximum duration of use DU is reached. Constraints (5i) determine the overhaul costs. A general overhaul takes place at the end of the tractor's lifetime, i.e. in period t þDU if the tractor has been bought in period t. Therefore, ocost is forced to OC t þ DU if ybuy ¼ 1 and yov ¼ 1. t t Here, OC t þ DU denote the costs of a general overhaul if this overhaul takes place in period tþDU. ocost is set to 0 in all other periods t since costs are minimized in the objective function.

Constraints (5j), (5k) and (5l) track the remaining lifetime of a tractor. The remaining lifetime depends on the period t in which the tractor is bought and the period t~ in which the tractor is sold. ut;t~ is enforced to 0 for all periods a tractors is not bought (5j) or sold (5k). In the periods in which the tractor is bought or sold ut;t~ is limited to an upper bound of DU þAD. For the period t in which the tractor is bought and the period t~ in which the tractor is sold, ut;t~ is set to the remaining lifetime: AD þ DU  t~ þ t (with general overhaul) or DU  t~ þ t (without general overhaul) (5l). Constraint (5m) ensures that the tractor is bought not more than once. Constraints (5n) enforce the use of a tractor once it was bought. The variable x is set to 1 for the period a tractor is bought and the following periods until the minimum usage duration MU or the end of the planning horizon is reached. For example, if a tractor is bought in period 1 and the minimum duration of use MU is 4 periods, x takes the value of 1 for periods 1, 2, 3 and 4. If the tractor is bought in period 8 and j T j ¼ 11, then x is set to 1 for periods 8, 9 and 10. Note, the last period is added only to sell remaining tractors, i.e. DM d;j T j ¼ 0, 8 d A D and consequently x j T j ¼ 0. Constraints (5o) and (5p) ensure that the maximum duration of use is not exceeded. In constraints (5o) the maximum duration of use equals the initial lifetime DU of a tractor plus the additional duration of use AD in case of a general overhaul. The constraints set the variable x in period t þDUþ AD to 0 if a general overhaul takes place in period t þDU. For example, if DU ¼AD ¼ 4 and the tractor is bought in period 1 (ybuy ¼ 1), then constraints (5o) 1 enforce x9 to take the value 0. In the second case no general overhaul takes place (5p), therefore the maximum duration of use corresponds to the initial lifetime DU of the tractors. Variable definitions are given in (5q)–(5r). Table 1 gives an example of a schedule a for tractor type b. The table shows the values of the decision variables and the total schedule costs TC b;a in the periods 1–10. The maximum lifetime DU is 4 periods and the additional lifetime after a general overhaul is 4 periods. According to the schedule, the tractor is used in periods 2–7. Variable ybuy is set to 1 when there is a shift from 0 to 1 in the x-variables, here ybuy ¼ 1. Accordingly, ysell is set to 1 if 2 there is a shift from 1 to 0 in the x-variables, i.e. ysell 8 ¼ 1. The maximum lifetime is reached and requires a general overhaul, i.e. yov ¼ 1. The remaining lifetime of the tractor, which is bought in period 2 and sold in period 8, is 2 periods, i.e. u2;8 ¼ 2, ut;t~ ¼ 0 in all other periods t; t~ . The solution of SP(b) is a new schedule (column) a given as 2 3 TC b;a 6! 7 6X 7 4 b;a 5 1b where TC b;a is the total costs of schedule a associated with tractor ! type b. X b;a is a vector with j T j elements indicating in which periods the tractor is used, or in other words containing the values of the decision variable xt of SP(b), i.e. X b;a;t ¼ xt 8 t A T . 1b is a unit vector with length j Bj and value 1 at position b and 0 else. To find Table 1 Example schedule a for tractor type b. t

1

2

3

4

5

6

7

8

9

10

x

0 0

1 1

1 0

1 0

1 0

1 0

1 0

0 0

0 0

0 0

ysell

0 0

0 0

0 0

0 0

0 0

1 0

0 0

0 1

0 0

0 0

yov u2;8 TC b;a

1 2 1,305,289

ybuy yov

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a feasible solution, all columns that have been generated when solving the LP relaxation of RMP are used and RMP is solved as IP.

 Case 2: The compatibility structure of one plane type is a subset

3. Demand and fleet related input data This section addresses the generation of appropriate and realistic input data (for future periods) to apply the model presented in Section 2. In the following, we introduce a procedure for demand aggregation based on a given flight schedule and present an example of demand forecasting. Moreover, we explain how the existing fleet can be incorporated in the model. 3.1. Demand pattern generation Constraints (1b) in MP ensure in each period a sufficient number of compatible tractors to satisfy demand. The demand for a period is expressed by a set of demand patterns. One period equals a winter or summer flight schedule, i.e. 6 months. In the following, we explain the four steps of demand pattern generation for one period from a given flight schedule. Step 1: Select a representative peak day in the period: Within a summer or winter flight schedule the departure times, parking positions and plane types are similar each day. Non-peak days differ from peak-days in that not all flights are scheduled while scheduled flights usually depart from the same gate at the same time. Step 2: Calculate the time window of tractor occupancy for each towing job of the selected day: The occupancy time comprises travel time, waiting time and processing time. The upper part of Fig. 1 visualizes the outcome of step 2. The horizontal axis refers to the time of the day, each row refers to one towing job. The bars show the occupation time of a tractor for each job. Step 3: Determine the number of simultaneous jobs for each time interval and plane type and derive the maximum of the day: The lower part of Fig. 1 shows for each plane type (rows) and each time interval (columns) the number of simultaneous jobs. The last column at the right displays the maximum for each plane type of a day. In this example, the maximum for plane type A is 3. The model generates one demand constraint for each plane type and period. For example, the demand constraint for the first row ensures that the number of tractors compatible with plane type A is greater or equal to 3. Step 4: Ensure the aggregated demand of plane types is satisfied for overlapping tractor compatibilities: In some cases it is not sufficient to ensure demand satisfaction for each plane type separately. One tractor might be used for several jobs at the same time if tractor compatibility overlaps. There are three cases:

 Case 1: The tractor compatibilities are disjoint sets. In Table 2 plane type A is compatible with tractor types 1 and 2, while plane type B is compatible with tractor type 3. Here one constraint per plane type is sufficient, i.e. no additional constraints are required. This results in two relevant demand constraints.

5



of the compatibility structure of another plane type. In Table 3 plane type A is compatible with all tractor types, and plane type B is only compatible with tractor type 2. In this case one additional demand constraint is required to ensure the number of tractors compatible with plane types A or B (here tractor types 1, 2 and 3) to be greater or equal to the sum of the maximum number of simultaneous jobs for plane types A and B (here 4). The constraint for plane type A becomes redundant. This results in two relevant demand patterns: B and A þB. Case 3: There is an overlap of tractor compatibility without subset structure (see Table 4). Here plane type A is compatible with tractor types 1 and 2 and plane type B is compatible with 2 and 3, i.e. both plane types are compatible with 2. This results in three relevant demand patterns: A, B and A þ B.

3.2. Demand forecasting A high quality demand forecasting as input is essential for a reliable schedule as output. Demand, and thus demand patterns, are primarily influenced by three factors: (i) the total number of towing jobs per day, (ii) the plane type mix and (iii) the number of towing jobs at each time of the day (in this paper called “temporal distribution”). The forecasting of the number of towing jobs for the case study in Section 4 is based on the following approach and assumptions. The forecast for the number of towing jobs (i.e. influencing factor i) is based on the forecast for the number of flight movements. We assume a constant ratio between the number of flight movements and the number of push-backs, repositionings and maintenance towings (0.8 , 0.11 and 0.09 for the winter schedule and 0.83, 0.09 and 0.08 for the summer schedule). The ratios are calculated from past data of our cooperating airport. The forecasts for the number of towing jobs per day are displayed in Table 5. The table gives the forecast of the number of towing jobs for one peak day in summer 2013–2022 and winter 2014–2023. Note that winter 2014 refers to winter 2013/2014. We assume an annual Table 2 Step 4 – compatibility structure case 1. Tractor type

1

2

Plane type A Plane type B

✓

✓

3

✓

Table 3 Step 4 – compatibility structure case 2. Tractor type

1

2

3

Plane type A Plane type B

✓

✓ ✓

✓

Fig. 1. Step 2 and step 3 – tractor occupancy time per job and maximum number of simultaneous jobs.

Please cite this article as: Du JY, et al. Obtaining the optimal fleet mix: A case study about towing tractors at airports. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.11.005i

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increase of 2% in flight movements and respectively the number of towing jobs is expected to increase by the same size per year. Assumptions on changes in the plane mix and temporal distribution of towing jobs (i.e. influencing factors ii and iii) are based on the standard flight schedule 2020 of our cooperating airport. The standard flight schedule 2020 does incorporate a change in the plane mix. Other relevant sources for the future development of the fleet mix are Airbus [1] and Boeing Commercial Airplanes [3]. Real processing times are used as input data for flight schedule summer 2013 and winter 2014. For flight schedule 2020 we take the average processing time per job type based on historic data. Furthermore, we assume the temporal distributions of maintenance towings and repositionings to remain the same as in summer 2013 and winter 2014. Based on these assumptions the temporal distribution of towing jobs during a day is given in Fig. 2 (summer flight schedule 2013), Fig. 3 (winter flight schedule 2014) and Fig. 4 (flight schedule 2020). This information is generated for each plane type. We assume a linear growth of the temporal distribution and change in plane type mix from summer 2013 and winter 2014 towards 2020 and a constant distribution thereafter. Since the distribution 2020 is not differentiated into winter and summer, we assume the same distribution for both seasons. This assumption seems reasonable since the temporal distribution of the schedule summer 2013 and winter 2014 resemble each other (see Figs. 2 and 3). By combining (i) total number of jobs per day, (ii) plane type mix and (iii) temporal distribution, all relevant input data are available to generate the demand patterns for the forecasting periods. 3.3. Consideration of the existing fleet An implementable schedule takes into account the existing fleet. We distinguish between three tractor categories. The pool of tractors which can be bought on the market is category 1. For each tractor type, we create one SP. Category 2 are the pool of existing tractors without decision options. These are primarily tractors with a remaining lifetime of one period. We insert these tractors directly as columns in MP and ensure their selection. Category 3 are existing tractors with decision options on overhaul or selling. We create one additional SP for each and adapt parameter and decision variable settings. For instance, if a tractor has been bought ¼ 1, i.e. the tractor has to be bought in in summer 2012, we set ybuy 1 the first period of the planning horizon (in the case study summer 2013). Furthermore, we decrease the lifetime DU and investment costs ICt for those two periods, which are not in the model's planning horizon. Constraints (1c) in MP ensure the selection of NE number of columns from this SP in the final solution. Table 4 Step 4 – compatibility structure case 3. Tractor type

1

2

3

Plane type A Plane type B

✓

✓ ✓

✓

4. Case study This section illustrates the application of the model for our cooperating airport. To alleviate the end of horizon effect, we consider a planning period of 30 years (i.e. j T j ¼ 60) but investigate only the first 10 years in more detail. In total, we consider 13 different tractor types. Two of them, tractor types T12 and T13, cannot be overhauled (i.e. ADT12 ¼ 0 and ADT13 ¼ 0), since they are so-called towbarless tractors which in contrast to towbar tractors, for technical reasons, do not have an overhaul option. Furthermore, we consider the existing fleet which results in 9 additional types and thus subproblems (see Section 3.3). We start by setting up a basic scenario in Section 4.1. In our sensitivity analysis we consider demand and waiting time scenarios (see Section 4.2), cost scenarios (see Section 4.3), and management scenarios (see Section 4.4). Finally, we investigate a green field scenario in Section 4.5. An overview of all 25 test instances with summary statistics is given in Table 6. Columns 1–7 give input statistics for each instance. Column 1 gives the ID and column 2 the number of tractor types, i.e. number of subproblems. The third column shows the selling price as the percentage of investment cost while column 4 indicates the change of overhaul cost in percent compared to the basic scenario. The minimum duration of use is shown in column 5. Column 6 gives the change in demand as a percentage whereas column 7 shows the waiting time considered. The final 5 columns give general output statistics. In particular, column 8 shows the final IP value, column 9 gives the relative gap, column 10 the total runtime, column 11 the number of tractors bought in the entire planning period, and column 12 the average number of usage periods. The first row shows the statistics for the basic scenario. The next four rows consider the waiting time and demand instances whereas rows 6–19 show results for different cost scenarios (here we vary the selling price and overhaul cost). The output for the management scenarios is given in rows 20–24. The last row gives the statistics for the green field case. IBM ILOG CPLEX Optimization Studio 12.5.1 is used to code and solve the RMP and the SPs. All computations are performed on a 2.7 GHz PC (Intel(R) Core(TM) i7-4600U CPU) with 16 GB RAM running under the Windows 7 operating system. For instance, for the basic scenario with a problem size of 22 tractor types (existing and new), 60 periods and 76 demand patterns, the CGH obtains a near optimal solution (with a optimality gap of 0.006%) within 109 min (see row 1 in Table 6). In general, all instances could be solved to near optimality (gap is less than 0.01%). Total runtime varies between 48 and 156 min with an average of 110 min. 4.1. Basic scenario The parameters in the basic scenario are derived from realworld data and are given in Table 8. Tractors are linearly depreciated over the maximum duration of use. However, when selling the tractor the received revenue is only a given percentage of the book value. For the base scenario this percentage is 70%. As an example, if a tractor with acquisition cost of 100 (in k euros) and a maximum duration of use of 10 years is sold after 6 years, the book value is 40 and the selling price is 0.7  40 ¼28. Dividing the sales revenue by the remaining use periods gives SRt as used in the objective function (5a) of the subproblem. Table 7 provides the sales revenue and the sales revenue per remaining usage period

Table 5 Forecasting of numbers of towing jobs per day.

Summer Winter

2013

2014

2015

2016

2017

2018

2019

2020

2021

2022

2023

554

566 552

579 564

592 576

605 589

617 601

630 614

643 626

656 638

668 651

663

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Fig. 2. Demand curve of a peak day in summer 2013.

Fig. 3. Demand curve of a peak day in winter 2014.

Fig. 4. Demand curve of a peak day in 2020.

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Table 6 Input and output statistics for all 25 test instances. ID

# Trct Selling price (%)

Change overhaul cost

MU Demand change Waiting time (min)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 13

– – – – – – – – – – – – – – –  20%  10% þ 10% þ 20% – – – – – –

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1 2 4 8 10 6

70 70 70 70 70 100 90 80 60 50 40 30 20 10 0 70 70 70 70 70 70 70 70 70 70

–  10% þ 10% – – – – – – – – – – – – – – – – – – – – – –

10 10 10 0 15 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

Table 7 Absolute sales revenue and sales revenue per remaining usage period for a maximum life time of 10 years. Usage periods

1

2

3

4

5

6

7

8

9 10

Remaining lifetime periods Sales revenue [k euro] Sales revenue per remaining lifetime period [k euro]

9 8 7 6 5 4 3 2 1 0 63 56 49 42 35 28 21 14 7 0 7 7 7 7 7 7 7 7 7 0

Table 8 Parameter settings and input data for the basic scenario. Selling price of book value SPt General overhaul costs OCt Minimum duration of use MU Maximum duration of use DU Additional lifetime after overhaul AD Waiting time per job

70% of investment costs 60–80% of the investment 6 periods 10 periods 10 periods 10 min

for all periods of the example. In our case study, we varied the selling price as percentage of the book value as well as other parameters (see Table 8). The variable cost per hour is fixed and does not change over time. Consequently, the operating costs (average hours of use within a period year multiplied by variable cost per hour) are fixed too. Furthermore, time dependent operating costs are not relevant, since the pushback company has a service contract with a service provider and pays for each minute the motor is running, independent of the age of the tractor. However, our model is capable to consider such cases. In the basic scenario, 60 tractors are bought, 13 tractors are sold and 6 tractors are overhauled. In the optimal solution for the first 10 years (i.e. 20 periods), a tractor is used on average 6.1 periods. Fig. 5 visualizes the number of tractors to be bought and sold in periods 1–20. Fig. 6 shows the number of tractors in use per period and tractor type and Fig. 7 displays the number of tractors to be used per period differentiating with respect to the categories of existing versus new tractors (see Section 3.3).

IP soln

Gap (%) Time (min)

# Trct bought Avg. usage (periods)

63,789,578 57,753,428 70,994,738 42,301,748 93,460,298 62,045,318 62,836,338 63,355,718 64,105,848 64,344,028 64,521,018 64,661,748 64,790,348 64,919,448 65,041,338 63,119,778 63,490,558 63,955,718 64,074,548 63,059,828 63,208,088 63,505,478 63,916,778 64,958,888 61,125,370

0.006 0.018 0.051 0.000 0.000 0.000 0.002 0.006 0.021 0.039 0.038 0.023 0.000 0.000 0.001 0.013 0.015 0.018 0.006 0.014 0.021 0.020 0.025 0.000 0.027

144 132 157 86 225 165 142 144 142 142 136 136 136 135 135 143 143 144 145 154 145 144 142 132 127

109 102 103 106 64 115 129 156 116 120 98 102 110 106 91 135 119 130 133 121 124 138 108 48 60

8.0 7.8 7.9 7.7 7.7 7.0 8.1 8.0 8.1 8.1 8.5 8.5 8.5 8.5 8.5 8.1 8.0 8.0 8.0 7.4 7.9 8.0 8.1 8.8 6.9

From the figures, two conclusions can be drawn: first, there is a clear preference for certain tractor types, namely T10, T12 and T13. The fleet mix is dominated by these three tractor types starting from period 7. From period 13 onwards, the fleet consists of only these three tractor types. In particular, T10 and T12 are characterized by high flexibility in terms of technical compatibility, while T13 has comparatively low investment and variable costs. Second, the current number of existing tractors is too high: in the first period, the number of tractors used is predetermined by the existing fleet. With a reduction in the existing fleet size due to aging, the total fleet size decreases in periods 2 and 3. As demand increases the fleet size grows again in period 4. Demand with respect to number of jobs is lower in winter than summer (see Table 5). However, processing time per job is longer during winter, thus total tractor occupation time is higher during winter. The zigzag-pattern in periods 1–14 (winter 2020) results from this seasonal variation (winter vs. summer). The fleet size stabilizes after period 14, since we assume one temporal distribution for both seasons from 2020 onwards. 4.2. Demand and waiting time scenarios Demand increases by 2% per year in the basic scenario. Furthermore, an average waiting time of 10 min is added to each job, based on historical data. Both factors impact the fleet size. We analyze in the following the impact of an increase and a decrease in demand by 10% (scenarios Dþ 10% and D  10%), an increase and a decrease of the average waiting time of 15 min (WT15) and of 0 min (WT0). Waiting time scenarios: Fig. 8 compares the waiting time scenarios with the basic scenario. The bars show the total costs TC of the schedules in percentage of the basic scenario costs (right vertical axis). The lines show the number of tractors to be bought, overhauled and sold (left vertical axis). Waiting time itself is an indicator of the robustness of a schedule regarding disruptions in the flight schedule and daily operations. The fleet size decreases considerably when ignoring waiting time (WT0). However, without buffer time there is a high risk of push-back delays due to

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Fig. 5. Number of tractors bought and sold per period (basic scenario).

Fig. 6. Number of tractors used per period and tractor type (basic scenario).

Fig. 7. New vs. existing number of tractors used per period (basic scenario).

Fig. 8. Waiting time scenarios vs. basic scenario.

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Fig. 9. Demand scenarios vs. basic scenario.

Fig. 10. Delta of D  10% scenario and basic scenario.

flight schedule disruptions. Increasing the average waiting time from 10 min to 15 min (WT15) creates a larger buffer for disruptions. The comparison of WT0 with the basic scenario can also be interpreted from the following perspective: The towing service provider faces about 34% higher investment costs due to tractor occupation from schedule disruptions in daily operations. Demand scenarios: Fig. 9 shows the results of the demand scenarios. An increase or decrease of demand by 10% is roughly equivalent to a cost increase or decrease of 10%. A demand increase does not expose the towing service provider to any risks, since new tractors can be bought anytime. However, a demand decrease might lead to a suboptimal fleet, if part of the investment schedule has already been realized. The greater the differences between the various scenarios, the higher the risk of a suboptimal decision. The later in the planning horizon the differences occur, the higher the chance of revising the schedule without losing optimality. Fig. 10 shows the difference of number of tractors to be bought (dark gray bars) and sold (light gray bars) per period between the scenario D  10% and the basic scenario. A positive number in the chart means more tractors are bought or sold in the D  10% scenario. In the basic scenario 4 tractors more are bought whereas the number of sold tractors is the same in both scenarios. Looking at the first 10 periods (i.e. 5 years), the net difference for buying and selling is almost zero. In comparison, the next five years exhibit different decisions, i.e. buying fewer tractors in the low demand scenario D 10%. So, buying or selling more tractors if demand does not develop as expected are decisions that easily can be

carried out. Thus, the schedule in the case study is rather robust with respect to demand variations, while waiting time or schedule disruptions have a greater influence on the optimal fleet size. 4.3. Cost scenarios In this section we investigate how the ratio between investment costs and selling prices, and the ratio between investment costs and general overhaul costs influence the schedule. Selling price scenarios: in the selling price scenarios (S100%– S0%), we vary the revenues for selling tractors. The percentage number in the scenario label indicates the selling price that can be realized on the market in percentage of the initial investment costs. In scenario S100% we assume that a tractor can be sold to the market without any loss of value, while scenario S40% assumes a loss of 60% in value if a tractor is sold. Scenario S0% reflects the scenario of not having a selling option at all. Fig. 11 summarizes the results for the selling price scenarios. In the basic scenario we assume that the selling price equals 70% of the initial investment costs (i.e. a 30% loss of value). Decreasing the selling price has limited impact on the schedule and costs (see scenarios S60–S0% in Fig. 11). Compared to the basic scenario, the total costs increases at most by 2%. In contrast to the negligible changes in scenarios S60–S0%, an increase in selling prices (scenarios S80–S100%) does change buying and selling behavior considerably. In scenario S100% in which we assume that tractors can be sold to the market without any loss of value, the number of tractors to be sold almost triples, and accordingly the number of tractors to be bought

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Fig. 11. Selling price scenarios vs. basic scenario.

Fig. 12. Overhaul cost scenarios vs. basic scenario.

increases by about 15%. Without loss of value when selling a tractor, the fleet more frequently adapts to better fit changing demand. Overhaul costs scenarios: Analogously to the selling price, we vary the general overhaul costs in scenarios OV  20% up to OV þ20%. Here the percentage number in the scenario label indicates an increase or reduction of the general overhaul costs compared to the basic scenario. In the basic scenario the general overhaul costs are between 60% and 80% of the investment costs for the different tractor types. In scenario OV  10% the overhaul costs are set to 50–70% of the investment costs. The scenario results are displayed in Fig. 12. As expected, the number of overhauls increases slightly with decreasing overhaul costs, while total costs remain almost constant.

shows that setting MU to different values, i.e. 1, 2, 4, 8 or 10 periods, affects total costs by less than 2% (see Fig. 13). High flexibility decreases costs by more than 1% which amounts to almost a quarter of a million euros in our planning horizon of 30 years. However, allowing vehicles to be sold after one period does not seem reasonable from a fleet management effort perspective. On the other side, low flexibility with MU ¼10 increases cost by about 1.8% or 1.16 million euros. Based on our results, the current policy with MU¼ 6 seems to be a good balance between changing the fleet too often and total costs. It is noticeable, that with MU¼ 10 there are more overhauls than tractors sold in the first 10 years. An explanation is that after an overhaul the tractor can be sold right away giving more flexibility in making decisions. 4.5. Green field scenario

4.4. Fleet management scenarios In the fleet management scenarios, we investigate the impact of the fleet management policy on the schedule. The minimum duration of use MU does not reflect a technical feature of a tractor, but rather a management decision. A small MU value allows more flexibility to adjust the fleet composition more frequently, while a high MU value results in less fleet management effort and greater stability in daily operations since schedulers and drivers do not have to adapt to a new fleet that often. In the basic scenario the minimum duration of use is set to 6 periods (this value has been set by management of our cooperating airport). The analysis

In the green field scenario we ignore the existing fleet and assume that the fleet is built from scratch. Compared to the basic scenario, the total costs in the green field scenario decreases by 4.2% (see Fig. 14). This equals the savings potential which can be achieved in the long run. Fewer tractors are bought in the green field scenario compared to the basic scenario. The reasons are those tractors in the existing fleet with a remaining lifetime of 1 or 2 periods. Fig. 15 shows that without an existing fleet, tractor types T10, T12 and T13 dominate the fleet composition from the first period on. In the entire planning horizon, the fleet mix consists of only these three tractor types. The spikes in periods 6–14 can be

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Fig. 13. Fleet management scenarios vs. basic scenario.

Fig. 14. Green field scenario vs. basic scenario.

Fig. 15. Number of tractors used per period and tractor type (green field scenario).

explained due to different demand in winter and summer where summer demand is lower. This behavior vanishes over time since the demand in winter and summer converges.

5. Conclusion This paper addresses the fleet composition problem of towing tractors at airports at a strategic level. The set-covering

formulation derives a multi-period schedule for a set of heterogeneous towing tractors. The model optimizes fleet size and mix by determining the time of buying, overhauling and selling tractors. The model takes into account restrictions such as technical compatibility of tractor types with plane types, a minimum and a maximum duration of use. The model incorporates an existing fleet. The inclusion of aspects such as an overhaul option better captures investment decisions in real-world situations. Furthermore, we introduce a 4-step approach to aggregate demand using

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flight schedule information. The proposed column generation heuristic solves all scenario instances to near optimality with an average runtime of 110 min. We illustrate the application of the model for a major European airport (our cooperating airport) in a case study. Using the model we derive optimal schedules for a basic scenario and a number of additional scenarios. The results demonstrate the robustness of the schedule of the basic scenario towards changes in demand, flight disruptions and varying cost parameters. About 5% saving potentials are identified if the model is applied in the long-run. The fleet composition model might be adapted to other application areas, such as other vehicles at airports (e.g. de-icing trucks, passenger transport buses, towable passenger boarding stairs), as well as road vehicles (e.g. street buses, trucks). Moreover, the model might be extended by incorporating temporary unavailability of tractors due to maintenance cycles or a general overhaul. Also, variable general overhaul periods and lifetime dependent variable costs might be considered in a model extension.

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Please cite this article as: Du JY, et al. Obtaining the optimal fleet mix: A case study about towing tractors at airports. Omega (2015), http://dx.doi.org/10.1016/j.omega.2015.11.005i