Strategic vehicle fleet management – a joint solution of make-or-buy, composition and replacement problems

Purpose – The purpose of this paper is to develop an original model and a solution procedure for solving jointly three main strategic fleet management problems (fleet composition, replacement and make-or-buy), taking into account interdependencies between them. Design/methodology/approach – The three main strategic fleet management problems were analyzed in detail to identify interdependencies between them, mathematically modeled in terms of integer nonlinear programing (INLP) and solved using evolutionary based method of a solver compatible with a spreadsheet. Findings – There are no optimization methods combining the analyzed problems, but it is possible to mathematicallymodelthemjointlyandsolvetogetherusingasolvercompatiblewithaspreadsheetobtainingasolution/fleetmanagementstrategyansweringthequestions:Keepcurrentlyexploitedvehiclesinafleetorremovethem?Ifkeep,howoftentoreplacethem?Ifremovethenwhen?Howmanyperspective/newvehicles,ofwhattypes,brandneworusedonesandwhenshouldbeputintoafleet?Therelativelylargescaleinstanceofproblem(50vehicles)wassolvedbasedonareal-lifedata.Theobtainedresultsoccurredtobebetter/cheaperby10%thanthetworeferencesolutions – random and do-nothing ones. Originality/value – The methodology of developing optimal fleet management strategy by solving jointly three mainstrategic fleetmanagement problemsisproposedallowingfor the reductionofthe fleet exploitation costs by adjusting fleet size, types of exploited vehicles and their exploitation periods.


Introduction
Vehicles constitute the main resource used to meet transportation requirements both in-house (transportation activities taken by nontransportation companies) and external (transportation services delivered by common carriers). A whole of vehicles (transportation means) used by a given company to satisfy its in-house or outside transportation requirements is called a fleet. An appropriate fleet management is a key factor to a successful management of all transportation activities in a company. In the paper, a strategic level of the fleet management is considered. The considerations presented in the paper are given to road fleets of vehicles. However, the discussed fleet management problems and the proposed solution method can be applied to the other transportation modes as well.
Basic definitions of the analyzed problems are as follows: (1) Make-or-buydetermination of whether it is more advantageous to make a particular item/to do a particular activity in-house or to buy it from a supplier taking into account both qualitative and quantitative factors (Nikolarakos and Georgopoulus, 2001;Ravindran et al., 2018).
(2) Fleet sizingmatching transportation supply and demand by defining such a number of vehicles in a fleet that satisfies a complete fulfillment of incoming transportation orders (demand satisfaction, lost sales costs minimization) and, at the same time, allows to avoid high fixed costs associated with fleet underutilization (supply surplus) (Zak et al., 2008).
(4) Fleet compositiondefining both the fleet size and the fleet mix of vehicles in a fleet (Etezadi and Beasley, 1983).
(5) Fleet replacementdeciding on how long to exploit particular vehicles in a fleet or when to dispose/replace them and by what types of a brand new vehicles or used ones in order to avoid too high ownership or exploitation costs, including also a selection of vehicles' investment/acquisition option (e.g. outright purchase, credit, leasing or rental) (Redmer, 2016).
The most crucial and common factor influencing all the strategic fleet management problems is the demand for transportation services. Level, seasonal changes, trend, but also character of the demand result in particular types of transportation requirements to be fulfilled. Thus, the demand can be of different types according to the specific features of loads, distances, routes or locations of destination points/customers, their orders and many others. For example resulting in (Redmer, 2015): (1) Transportation of commodities that require or not special treatment while moving or handling them (e.g. general freights vs. dangerous goods, loos materials, temperature controlled environment shipments); (2) transportation of heavy and/or oversized loads; (3) local, regional, domestic or international shipments (short-and long-distance haulage); (4) urban, suburban or rural deliveries; (5) less than truck-load (LTL) or full truck-load (FTL) shipments.
The demand for transportation services can be defined by a number of kilometers, tonkilometers, tones, cubic-meters, pallets, liters or any other measure of the amount of loads (or passengers if applicable) to be transported/transports to be done within a predefined time period (the period of analysis). As a result, different (universal, specialized or special) types of vehicles are necessary to transport particular types of loads. However, the range of load types being within transportation capabilities of particular types of vehicles is limited. It depends on both, vehicle/load characteristics and vehicle maximum productivity (e.g. mileage). JQME 28,2 All the above-mentioned features of the demand (level and seasonal changes of particular demand types) can lead to an oversized fleet or to an unmet demand (transportation requirements not fulfilled by vehicles in a fleet) or even both concurrently. The unmet demand usually cannot be back ordered and has to be outsourced or lost. Moreover, some long-term changes of the demand can force fleet size increase or reduction that can be obtained by appropriate fleet replacement decisions. Planned fleet size changes can shorten or lengthen exploitation periods of particular vehicles in a fleet (Redmer, 2015).
All these observations are fundamental for the proposed joint solution method for the three main strategic fleet management problems. The combination of those three problems (i.e. makeor-buy, composition covering size and mix, and replacement) and the joint solution of them, taking into account interdependencies between them, constitute the main contribution of the paper.
The reminder of the paper is organized as follows. In the next section a literature review is presented to answer the question whether there are any optimization methods combining the three analyzed fleet management problems. Then, in Section 3, the integer and nonlinear mathematical/optimization model of the combined strategic fleet management problem (CSFMP) is described in detail. To solve the model, a spreadsheet/solver-based solution method is proposed in Section 4. A real-life implementation of the model and the proposed solution method is presented in Section 5, including obtained results. Finally, in Section 6, the research is concluded with a summary of results and an outlook to further research. The paper is completed with extensive references.

Literature review
One of the chronologically first types of optimization methods applied to fleet management problems was linear (including integer, mixed integer and binary) mathematical programming -LP. There are numerous and diverse applications of the LP to the fleet management including the following: the fleet sizing problem -FSP (Dantzig and Fulkerson, 1954;Kirby, 1959;Wyatt, 1961;Mole, 1975;Ceder and Stern, 1981;List et al., 2003a bi-objective, stochastic model solved with a robust optimization technique; Balac et al., 2020a mixed-integer linear programming (MILP) applied to a fleet of pooled automated vehicles, AVs; Zhang et al., 2020an MILP applied to a fleet of autonomous electric vehicles, AEVs); the fleet composition problem -FCP (Gould, 1969;Mole, 1975;Etezadi and Beasley, 1983;Bojovic et al., 2010problem solved with the Fuzzy Analytic Hierarchy Process method); the replacement problem -RP (Suzuki and Pautsch, 2005;Figliozzi et al., 2011;Boudart and Figliozzi, 2012;Parthanadee et al., 2012;Li et al., 2015;Buyuktahtakin and Hartman, 2016 an MIP formulation of the parallel RP under economies of scale and technological change; Ngo et al., 2018an integer linear programming (ILP) model solved using a branch-and-bound algorithm; Emiliano et al., 2020an integer programming model integrating both budgetary and environmental constraints); the vehicle assignment problem -VAP (Rushmeier et al., 1997;Ziarati et al., 1999problem solved with a customized branch-and-cut algorithm); the vehicle routing problem -VRP (Taillard, 1999 -HFFVRP); the mixed FCP/VRP (Golden et al., 1984 -FSMVRP;Vis et al., 2005 -FSMVRPTW solved with simulation) and the make-or-buy problem -MoB (Klincewicz et al., 1990;Stojanovic et al., 2011). Important extensions of the LP applications to the fleet management are nonlinear and goal mathematical programming. Nonlinear methods are applied to the FSP (Hall et al., 2001) and the RP (Preinreich, 1940;Glasser, 1969;Grinyer and Toole, 1972;Jardine, 1973;Christer and Goodbody, 1980;Jardine and Buzacott, 1985;Spooner, 1989;Navon and Maor, 1995;Mathew et al., 2010problem solved with genetic algorithm and branch-and-bound method) and the mixed FSP/RP (Navon and Maor, 1995). Whereas goal programming is applied to the mixed FCP/VRP (Calvete et al., 2007 -FSMVRPTWmixed integer goal programing combined with the enumeration-followed-by-optimization technique).
There are also other types of methods having more niche applications to the fleet management. One of them is queuing theory applied to the FSP (Parikh, 1977;Du and Hall, 1997;Koo et al., 2005). The second is simulation applied to the FSP (Koo et al., 2005 stochastic model combined with a queuing theory as well; Fagnant and Kockelman, 2018an agent-based microsimulation model applied to a fleet of shared autonomous/fully-automated vehicles, SAVs) and the FCP (Petering, 2011large scale simulation technique). The third is network models applied to the FSP (Beaujon and Turnquist, 1991;Sherali and Tuncbilek, 1997;Bojovi c, 2002), the FCP (Wu et al., 2005) and the RP (Vemuganti et al., 1989). Such types of methods are used rarely as optimal control theory applied to the FSP (Bojovi c, 2002); inventory control models applied to the FSP (Du and Hall, 1997) and neural networks applied to the VAP (Vukadinovic et al., 1999a, b).
In the past, methods based on the deficit function concept were also widely applied to the FSP (Salzborn, 1974;Gertsbach and Gurevich, 1977;Ceder and Stern, 1981).
Heuristics are another important type of methods that had their application to fleet management problems over decades. There are a lot of heuristic methods usually dedicated to particular fleet management problems, especially to the VRP. The selected examples of the applications are as follows: (1) savings algorithm applied to the VRP (Clarke and Wright, 1964) and the FCP/VRP (Bodin et al., 1983 -FSMVRP;Golden et al., 1984 -FSMVRP;Desrochers and Verhoog, 1991 -FSMVRP), (2) sweep algorithm applied to the FCP/VRP (Renaud and Boctor, 2002 -FSMVRP); (3) giant TSP-tour (also known as tour partitioning or route-first, cluster-second algorithm) applied to the FCP/VRP (Golden et al., 1984 -FSMVRP); (4) construction algorithms applied to the FSP  a route-first, clustersecond algorithm based on greedy insertion) and the FCP/VRP (Liu and Shen, 1999 -FSMVRPTW); (5) multilevel composite algorithm applied to the FCP/VRP (Salhi and Sari, 1997 -FSMVRPMD or MDVFMmulti depot vehicle fleet mix problem).
And finally, a multiple objective decision making/aid -MCDM/MCDA finds its application in the fleet management area including the RP (Gopalaswamy et al., 1993;Redmer et al., 2000), the VSP (Singh and Saxena, 2003;Ghoseiri et al., 2004), the MoB (Min, 1998) and the transportation problem -TP (Singh and Saxena, 2003). Nowadays fuzzy, stochastic, random approaches or mix of them can be pointed out as a main direction of the development of optimization methods applied to the fleet management problems. They are mostly applied to the FSP (Milenkovic and Bojovic, 2013 - Only a few researches can be found in the literature presenting some attempts to combine explicitly main strategic fleet management problems and solve them jointly. One of the examples uses the econometric global model to solve the RP combined with the FSP (Raposo et al., 2017). However, the FSP is solved analyzing availability of vehicles and its dependence on maintenance and maintenance costs. Thus, these factors help to determine the size of the reserve fleet only to guarantee expected availability of the whole fleet. The second example (Li et al., 2018) is the ILP model combining the RP with the FSP/FCP and also with the VRP (the route assignment). And the last element, i.e. the VRP, makes the model inapplicable to unscheduled transports, e.g. freight ones. The model skips the MoB problem too. There are also two other and the last examples of researches combining the RP with the FSP (the fleet mix only). The first and chronologically earlier research (Feng and Figliozzi, 2013) proposes a deterministic, integer programming model that decides on the number of vehicles of a given age and type used in a given year (however their sum, thus, the fleet size remains constant through particular years of the planning horizon), the number of vehicles of a given age and type salvaged at the end of a given year and the number of new vehicles of a given type purchased at the beginning of a given year. However, it is required that all the demand will be fully satisfied by the analyzed fleet. Thus, the use of outside vehicles (MoB problem) is skipped. The second research (Ahani et al., 2016) extends the previous one by the use of the portfolio theory to reduce total risk (associated with an uncertainty of selected parameters) by diversification of the fleet mix aiming to achieve an optimal value of total cost of ownership (TCO). However, mentioned above drawbacks remain valid. The both researches take into consideration diesel engine vehicles, DVs, and electric vehicles, EVs.
Summarizing, there are no optimization methods combining all the main strategic fleet management problems, i.e. FCP, RP and MoB, solving them together, even though the strong interdependencies between them exist. The most frequent combination (mix) of problems is to solve together the FCP and the VRP being the combination of completely different problems, i.e. strategic and operational ones. This combination, however, typical for scheduled transport systems, seldom if ever occurs in non-scheduled transport ones (e.g. freight road transport). Moreover, the application of very popular and modern solution methods which are heuristics and metaheuristics available in commonly used spreadsheets (e.g. MS Excel) is limited when Strategic vehicle fleet management solving strategic fleet management problems, especially solving them jointly. Also from the practical point of view, the use of spreadsheet based problem modeling and solving (using solvers) technics significantly reduces the effort of managers/analysts in terms of required expertise (e.g. programming/modelling skills) in comparison with the other.

Mathematical model of the optimization problem
The CSFMP encompassing composition (size and mix), replacement and MoB subproblems for a fleet consisting of O company's "own" (currently exploited) vehicles i (i 5 1, . . ., O) and N potential (perspective) vehicles i (i 5 O þ 1, . . ., O þ N), that can be acquired in the future, is considered. Vehicles in a fleet are referred to as a company's "own" independently of their form of ownership (financing). The suggested value of N is not lower than value of O to allow for the replacement of a whole fleet if necessary. A fleet serves/fulfills demand for the transportation services of different types k (k 5 1, . . ., K).
The proposed model is a single criterion, nonlinear (due to Equations (2)-(4) which are of if type), deterministic, static and discrete (integer) mathematical representation of the CSFMP.

Assumptions
(1) The demand for the transportation services can be of different types k, according to the specific features of transported loads, traveled distances, carried out routes, etc.
(2) The types of demand k that can be served/fulfilled by particular company's "own" or perspective vehicles i are limited/defined in advance.
(3) The transportation capacity (e.g. annual mileage) of particular vehicles i is distributed between particular demand types k proportionally to the share of the demand type k in all (a sum of) demand types that can be served/fulfilled by vehicle i.
(4) The total capacity utilization index of vehicle i is calculated as a weighted average of the total demand of type k divided by the total transportation capacity of all vehicles i assigned, according to the assumption (3), to the demand of this type to serve/fulfill it (reduced to value of 1, i.e. 100% if higher). The weights used to calculate the weighted average are the share of the demand type k in all the demand types that can be served/fulfilled by vehicle i.
(5) The unmet demand (not served/fulfilled by vehicles being in a fleet) is outsourced (option "make" in the MoB problem).
(6) Exploitation costs of nonengine vehicles (trailers, semi-trailers) are added to the exploitation costs of engine vehicles (trucks, truck-tractors) as a percent of their own costs.
(7) The currently exploited vehicles i already being in a fleet (company's "own" vehicles) are considered to be individual (particular) ones, whereas perspective vehicles to be included into a fleet ("new" ones) are considered at their type level only (including the number of vehicles of a given type in a fleet).
(8) Perspective/"new" vehicles can be added to a fleet as a brand new (with age 5 0) or as a second hand/used ones (age > 0)that is a result of the optimization process (decision variable). "New" refers only to the fact that they are new in a given fleet.
(9) As a result, input data (parameters of the optimization/mathematical model) for company's "own" vehicles are defined for particular vehicles, whereas for "new" vehicles input data are defined for their potential types only. It is a result of the optimization process of which type (according to the value of a decision variable VT i ) JQME 28,2 will be particular "new" vehicle i put into a fleet and exactly which input data will be assigned to it (data characterizing the type assigned to vehicle i).
(10) Parameters of the model (input data) are divided into two groups. Constant parameters (marked by a prime) that depend on the vehicle i age a i only and recalculated (temporary) parameters that depend on the vehicle i age in particular time periods of analysis ja ij and are subject to change with time.
(11) Company's "own" vehicles i if not sold by the end of the planning horizon J are replaced by vehicles at the same acquisition age as they have been at the beginning of their exploitation.
(12) Company's "own" vehicles i if sold by the end of the planning horizon J cannot be put into exploitation again.
(13) "New" vehicles i put into a fleet are replaced by the vehicles at the same acquisition age as they have been at the beginning of their exploitation.
(14) "New" vehicles i put into a fleet cannot be sold/removed from the exploitation until the end of the planning horizon H.
(15) The length of a single period of analysis j has to be predefined, e.g. one month. It is worth mentioning that the total planning horizon H is used for the mathematical purposes only. It is connected with the replacement problem to assure comparability of different, possible to apply replacement periods. Such long planning horizons are typical for the replacement problem (Sousa and Guimarães, 1997).

Decision variables
There are five following types of decision variables: (1) AT iacquisition time of a "new" (perspective) vehicle i; AT i ∈ C0, JD, AT i 5 0 indicates that vehicle i is not acquired, otherwise it is included into a fleet at the beginning of period j 5 AT i ; (2) ST isell time of a company's "own" (currently exploited) vehicle i; ST i ∈ C0, JD, ST i 5 0 indicates that vehicle i is not removed from a fleet, otherwise it is removed at the end of period j 5 ST i ; (3) AA iacquisition age of a "new" vehicle i; AA i ∈ C0, L -1D, age a ij of vehicle i in its first exploitation period j (j 5 AT i ) equals to AA i þ 1 (acquisition age of company's "own" vehicles is predefined for each vehicle individually); Strategic vehicle fleet management (4) RA ireplacement age of vehicle i; for a company's "own" vehicle i (i ≤ O), RA i ∈ CA i (0) þ 1, LD, whereas for a "new" vehicle i (i > O), RA i ∈ CAA i þ 1, LD, thus depends on AA i ; vehicle i is replaced at the end of period j 5 RA i ; (5) VT itype of a "new" vehicle i, one of the T possible types; VT i ∈ C1, TD (types of company's "own" vehicles are predefined for each vehicle individually).
One should note that variables AT i are used only for "new" vehicles, i > O, whereas variables ST i describe company's "own" vehicles only, i ≤ O. Additionally, acquisition ages AA i , vehicle types VT i and vehicle age in period j 5 0, A i (0) are predefined for company's "own" vehicles and are not subject to optimization. As far as "new" vehicles (i > O) are considered, there is a relationship between variables AA i and RA i . The values of variables RA i have to be higher than or at least equal to the values of variables AA i (RA i ≥ AA i , for given i). Hence, it is crucial for the solution procedure to calculate values of RA i taking into account values of AA i .
Since decision variables ST i and RA i concern company's "own" vehicles, whereas variables AT i , AA i , RA i and VT i concern "new" vehicles, the total number of all decision variables can be calculated as 2 $ O þ 4 $ N.

Criterion function
The goal of the optimization is to determine: (1) The strategy of selling or replacing O company's "own" vehicles. It is represented by variables ST i (sell time) and RA i (replacement age) for each owned vehicle i (i 5 1, . . ., O).
(2) The strategy of acquiring (including decisions on vehicles' age and type) and replacing N "new" vehicles. It is represented by variables AT i (acquisition time), AA i (acquisition age), RA i (replacement age) and VT i (vehicle type) for each potential "new" vehicle i (i 5 O þ 1, . . ., O þ N).
(3) The MoB strategy that is indirectly derived from the two above strategies, since the all demand unsatisfied due to the limited transportation capacity of company's "own" vehicles (a fleet) is subcontracted to outside transportation companies.
The VRP, as the operational management problem, is not addressed in the proposed optimization procedure that encompasses strategic fleet management problems only. However, it has to be stressed that in some transport systems, i.e. scheduled ones as mentioned above, there is a bi-or even tri-directional relationship between operational, tactical and strategic fleet management problems. Thus, the applicability of the proposed model can be (but does not have to) partially limited to non-scheduled transport systems only. The objective function is to minimize the total discounted cost, TDC, defined as follows: (1) JQME 28,2 where The TDC consists of the following three cost components (all of them discounted using DF(. . .) factor): (1) The first term is the total cost in the total planning horizon H, of a make option, taking into account all company's "own" O and "new" N vehicles exploitation costs; fixed (including road taxes, insurance, environmental taxes, licenses e.g. TIR carnet, drivers' salarya fixed part, independent of millage and ownership cost as well) and variable (including all O&M costs as fuel, maintenancespare parts and labor, tires, single/one-time licenses, parking fees, phone calls, road tolls, drivers' salarya variable part, depending on working/driving time and/or millage) (the first line of Equation (1)-the first two sums).
(2) The second term is the total cost in the total planning horizon H, of a buy option, taking into account costs of buying transportation services in a market to fulfill particular demand types k not satisfied within a make option (the first line of Equation (1)-the first and the third sums).
(3) The third term is the penalty cost added to the TDC if the budget constraint is not met in a given fiscal year l (when the real, necessary budget B real (l) exceeds a maximum, available budget B max (l)). Thus, all the solutions of the problem having values of TDC The TDC of satisfying the whole demand for transportation services within horizon H, including costs of an in-house transportationthe make option, and an outside transportationthe buy option (monetary unitsm.u./H) FC i (j) fixed cost of vehicle i in period j, including the ownership and utilization cost components (m.u./ period of analysis) VC i (j) variable utilization cost (operating and maintenance -O&M) of vehicle i in period j (m.u./vehicle productivity unit) P i (j) maximum productivity (mileage, tonnage carried, amount of loads transported) of vehicle i in period j (km, tkm, tone, m 3 , pallet, liter, . . ./period of analysis) VUI i (j) vehicle productivity utilization index of vehicle i in period j (À) EURO i coefficient of variable cost VC i (j) of vehicle i, representing the emission-based (the European Emission Standards or any other) road tolling (À) current calendar year, at the moment of analysis (À) ba ij "binary age" of vehicle i in period j denoting if vehicle i is exploited/in a fleet in period j (value 1) or not (value 0); ba ij 5 sign(a ij ) (À) acf i additional costs factor of vehicle i (when an engine vehicle is combined with a non-engine vehicle, such as a trailer, semitrailer) (À) TP k unit transportation market price for satisfying demand of type k (m.u./vehicle productivity unit) UD k (j) unmet (5outsourced) demand of type k in period j (the buy option) (km, tkm, tone, m 3 , pallet, liter, . . ./period of analysis) BO(l) budget overrun in fiscal year l; l 5 1, . . ., J/12, . . ., H/12 (m.u./fiscal year) DF(j) discount factor in period j, based on interest rate r j in period j (À) Strategic vehicle fleet management decision variables within predefined boundaries can be treated as feasible regardless of the budget constraint (a weak one). However, it forces the solution procedure to search for the solutions that fulfill budget constraint, have no penalties added to the TDC (the second line of Equation 1). (1), such as FC i (j), VC i (j) and P i (j) relate to vehicle i in period j and depend on its age. The age a ij of owned vehicle i (i 5 1, . . ., O) in period j ≥ 0 is defined/calculated as follows:

Many of the terms/parameters in Equation
whereas, the age a ij of a "new" vehicle i (i 5 O þ 1, . . ., O þ N) in period j ≥ 0 is defined as follows: Similarly, many of the terms/parameters in Equation (1), concerning "new" vehicles i (i 5 O þ 1, . . ., O þ N), such as FC i (j), VC i (j), P i (j), EURO i and acf i relate to/depend on their types VT i (the first three of the listed ones also apart from period j, and thus the vehicles' age). These terms in Equation (1), i.e. parameters of the model characterizing "new" vehicles i are defined not for particular vehicles (as for owned ones), but for their potential types onlysee assumption (9). For example, value of the FC i (j) is defined as follows (the other parameters similarly): The most crucial element of Equation (1), allowing for interchangeability of vehicles when satisfying particular demand types k, is VUI i (j) index that joins vehicles' productivity P i (j) with the demand types k by the productivity range PR ik parameter. The VUI i (j) index, being a weighted average, is calculated as follows: where: The VUI i (j) consists of the following two components: total demand of all types in period j being appropriate, according to the PR ik parameter, to be satisfied by vehicle i (km, tkm, tone, m 3 , pallet, liter, . . ./period of analysis) SMD k (j) supply meeting demand type k in period j, calculated as the total productivity of all vehicles i being able, according to the PR ik parameter, to satisfy demand type k taking into account their engagement to satisfy other demand typesinterchangeability of vehicles with demand types k (km, tkm, tone, m 3 , pallet, liter, . . ./period of analysis) PR ik productivity range of vehicle i in relation to demand of type k denoting if vehicle i is suitable to satisfy demand of type k (value 1) or not (value 0); PR ik ∈ {0, 1} (À) D k (j) demand of type k in period j, expressed in the same unit measure as P i (j) (km, tkm, tone, m 3 , pallet, liter, . . ./period of analysis) other notations as in Equation (1) JQME 28,2 (1) The first term under the sum represents mathematically the weight and physically the part of vehicle's i total productivity assigned to satisfy demand of type k calculated as a value of demand k divided by the total demand for services of vehicle i (the sum of weights for particular vehicles i equals to 1), (2) The second term under the sum is the utilization index when satisfying demand of type k, calculated as a value of demand k divided by the total supply (vehicles' productivity) meeting demand of type k and if the demand exceeds the supply equal to 1.
Whereas, TD i (j) and SMD k (j) parameters are calculated as follows: where notations as in Equation (1) and Equation (5).

Constraints
Due to construction of the decision variables and the criterion function, in the TDC model there are two types of constraints only. They are an integer constraint for all the decision variables and their minimum and maximum values. Due to construction of the criterion function -TDC no budget constraint is required. Budget limitations are taken into account in the model as the penalty cost added to the TDC always if the presumed budget is exceeded.

Spreadsheet/solver based solution method
The defined previous section model of the CSFMP has been implemented in the MS Excel 2016 spreadsheet. A compatible MS Excel spreadsheet professional solver named Evolver developed by the Palisade Corporation has been applied to solve the problem/model. The Evolver is equipped with a so-called "engine" (solution procedure) based on a genetic algorithm combined with a local search procedure that improves generated solutions locally. The genetic algorithm based "engine" of the solver has been used taking into account the problem/model characteristics. At first it is the nonlinear, combinatorial optimization problem with well-defined objective function, however, practically with no constraints. It is recognized that if the space to be searched is not so well understood and relatively unstructured but a genetic algorithm representation of that space can be developed, then genetic algorithms provide a powerful search heuristic for large, complex spaces (De Jong, 1990). In the analyzed problem/model, the well-defined objective function constitutes a clear way to evaluate fitness and the search space is not well-constrained, that is why the genetic algorithm based "engine" solver has been applied.
The spreadsheet is organized as follows:

Input data
Particular parameters of the model are given in separate tables/sheets (see Figure 1). The dimensions of particular input data tables depend on the parameters they contain. As it was mentioned in Section 3.1, parameters of the model are divided into two groups. Constant parameters (marked by a prime) that depend on the vehicle i age a i and recalculated (temporary) parameters that depend on the vehicle i age in particular time periods of analysis Strategic vehicle fleet management ja ij that is subject to change with time of exploitation. All the input data are constant parameters and are recalculated into temporary ones, according to the following scheme: (1) for i ≤ O (company's "own" vehicles): e.g. vehicle productivity P i (j) 5 P 0 i (a i 5 a ij ); (2) for i > O ("new" vehicles of type VT i ): P i (j) 5 P 0 OþVTi (a i 5 a ij ). For example, if there is a company's "own" (i ≤ O) vehicle i 5 e.g. 2 being in the period of analysis j 5 e.g. 3 at the age of a ij 5 a i 5 2 j 5 3 5 a 23 5 e.g. 24 months, its recalculated (temporary) parameters in the period of analysis 3, e.g. productivity P i (j) 5 P i 5 2 (j 5 3) 5 P 2 (3), equals to the constant productivity P 0 i (a i 5 a ij ) 5 P 0 i 5 2 (a i 5 2 5 a i 5 2 j 5 3 ) 5 P 0 2 (a 2 5 a 23 5 24) 5 P 0 2 (24) taken from the input data table. The constant parameters constitute the input data, whereas the elements of the criterion function, Equation (1), are temporary parameters only.

Decision variables and constraints
The decision variables have been implemented in a spreadsheet as a vector containing 2 $ O þ 4 $ N cells. With every variable (cell) there are associated two parameters/ constraintsits minimum and maximum values. Some of these constraints (those for RA i ; i ≥ O) are dynamic constraints having right-hand side (RHS) values calculated on the basis of the value of the other decision variables (AA i ) (see Figure 2). All the decision variables and their minimum and maximum values are integers.

Calculations
During the optimization process, in every iteration, all the elements of the criterion function depending on the age of vehicles (temporary parameters) are calculated based on the decision variables' values and the input data (constant parameters of the model) (see Figure 3). As a result, current value of the criterion function is calculated.

Solver
The solver (in this case the Evolver) minimizes value of the function in a cell (in this case KP4) being the criterion function -TDC calculated based on the values of the decision variables (in this case cells B5:KO5 -300 cells) and constraining these values to the integer ones only. All the other constraints (minimum and maximum values of the decision variables) come down to a one cell containing logical function giving value of 1 if all the values of the decision variables are in the assumed ranges and 0 otherwise. That is why there is only one, hard constraint for the cell containing mentioned logical function (see Figure 4).    Table 1.

Numerical example
Moreover, there are K 5 5 types of the domestic transport demand to be served, as shown in Table 2.
To transport together general freights and perishable goods (temperature controlled loads) special isotherm containers/boxes are assumed to be utilized.
Due to the specific features of the particular vehicle types and the specific features of the particular demand types, there are some rigid limitations when assigning vehicles to transportation tasks (demand types). As a result, particular vehicles can serve selected demand types only.  There are also many other data characterizing the analyzed problem, including acquisition and salvage/market residual values (RV) of particular vehicles according to their age, monthly productivity (mileage) of particular vehicles according to their age, fixed and variable exploitation costs of particular vehicles according to their age, prices in the market for transportation services satisfying particular types of the demand, cost of capital when investing in the fleet, the tax rate, company's budget available to invest in the fleet in particular fiscal years and the other.
The following time frames are considered: a 3-year long decision planning horizon (J 5 36 months), a 20-year long data planning horizon (L 5 240 months) and a 40-year long total planning horizon (L 5 480 months).

Optimization process
The solution algorithm (the "engine") used by the solver (in this case the Evolver) was a metaheuristic approach (a genetic algorithm combined with a local search procedure). As a result, the optimization process was asymptotic (taking into account improvement of the criterion functionsee Figure 5) and obtained (solutions) approximate results.
It took 63 min and 2,850 trials, i.e. the number of all the generated and examined solutions to find an optimal, however still approximate one.

Optimization results
Since the optimization problem (the CSFMP) defined in the paper is novel, there is no possibility to assess the accuracy of its model and the performance of the proposed solution method in relation to the other approaches. Therefore, it was decided to compare the obtained optimal 3-year long fleet management plan (the solution denoted as an "optimal" in Table 3) with two reference solutions: (1) Randomvalues of particular decision variables drown randomly within assumed feasible ranges; (2) Do-nothingrepresenting status quo, i.e. no changes in the fleet (exploitation of the current fleet over the next three years).   Table 3.
Based on Table 3, it can be concluded that the worst approach is to do nothing, make no changes in the fleet structure for the next three years. The do-nothing option turns out to be even worse than the random one, giving a little bit higher costs. The optimal solution is more than 10% better/cheaper when taking into account the total and unit transportation costs, even though the utilization of the vehicles decreased (from 98% or 96%-90%), it turned out to 3 Ageat the moment of acquisition in years (average / min / max) -/-/-4/0/19 5/0/12 Vehicles versatilityability to serve particular demand types 45% 47% 59% Average vehicles utilization index 98% 96% 90% Make-or-buy (% of total kilometers 5 % of total demand) option make 67% 75% 83% option buy 33% 25% 17% Costs * Total discounted (within the decision planning horizon) In total 100% 98% 89% Company's "own" vehicles (the make option) 100% 115% 116% Outside vehicles (the buy option) 100% 75% 51%

Unit discounted
In total 100% 98% 89% Company's "own" vehicles (the make option) 100% 104% 95% Outside vehicles (the buy option) 100% 96% 94% Note(s): *In comparison to the do-nothing option Table 3. The results of the optimizationthe different solution types JQME 28,2 be profitable after all. Moreover, the optimal solution leads to the significantly increased fleet size (from 50 or 57 to 68 vehicles) and to the significant reduction (from 33% or 25%-17%) of the transports outsourced (a one-third or double reduction of the share of the buy option in the total number of kilometers covered). It can be also observed that the average age of the vehicles in the fleet within the decision planning horizon J has remained almost constant (about 7-8 years). And finally, the brand new and the second hand vehicles have been put into the fleet concurrently. However, only 4 of the 43 vehicles put into the fleet have been brand new. The second hand vehicles were of age up to 12 years, while the average age at the moment of acquisition was 5 years.

Conclusions
The presented CSFMP solution method (the model and the solution procedure) allows for defining the fleet management strategy in transportation and nontransportation companies and institutions exploiting small and large fleets of vehicles. The strategy results in the reduction of the total costs of transportation and the improvement of a technical and an economic condition of a fleet.
The proposed model of the analyzed problem is relatively simple and allows for an intuitive/natural representation of a solution. Particular decision variables are associated with strategic decisions to be made with regard to particular currently exploited and perspective vehicles. For the currently exploited vehicles, the decisions are keep them in or remove from a fleet? If keep, how often to replace them? If remove then when? For the perspective vehicles: how many vehicles, of what types, brand new or used ones and when should be put into a fleet? The crucial issue is not how to mathematically model the joint problem, but how to control an optimization process to obtain expected feasible solutions. From this point of view, an objective function applied in the optimization process is of utmost importance. The function proposed in this research is sensitive to changes in company's "own" vehicles parameters as well as parameters of "new"/perspective vehicles, transportation prices in a market for services fulfilling different types of transportation demand and also budget limitations influencing the whole process. All these aspects have been taken into account in the proposed criterion function.
Worth mentioning is that the proposed method deals with the different types of vehicles serving the different types of transportation demand. In the other words, the method deals with interchangeability of vehicles when serving partially particular types of demand.
The main contribution of the paper to the literature/research is that particular and different strategic fleet management problems are solved jointly. It allows taking into account all the interdependencies between them.
The further research will be focused on searching for better, specialized (heuristic) solution algorithms assuming that the goal is to find not only fast and effective ones but also robust and, thus, effectively dealing with instances of the problem having different properties.