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MCDM 2006, Chania, Greece, June 19-23, 2006 FLIGHT AND MAINTENANCE PLANNING OF MILITARY AIRCRAFT FOR

MAXIMUM FLEET AVAILABILITY: A BIOBJECTIVE MODEL

George Kozanidis1 and Athanasios Skipis Systems Optimization Laboratory

Dept. of Mechanical & Industrial Engineering University of Thessaly

Pedion Areos, 38334 Volos, Greece E-mail: {gkoz; atskipis}@uth.gr

Keywords: fleet availability, flight and maintenance planning, mixed integer linear program, biobjective model, military application

Summary: Every aircraft, military or civilian, must be grounded for maintenance after it has completed a certain number of flight hours since its last maintenance check. Flight and maintenance planning of military aircraft addresses the problem of deciding which available aircraft to fly and for how long, and which grounded aircraft to perform maintenance operations on, in a set of aircraft that comprise a combat unit. The objective is to achieve maximum availability of the unit over the planning horizon. In this work, we develop a biobjective optimization model of the flight and maintenance planning problem, and we illustrate its application and solution on a real life instance drawn from the Hellenic Air Force. We formulate the flight and maintenance planning problem as a mixed integer linear program, with two objectives: total number of available aircraft and total residual flight time. The residual flight time of an available aircraft is defined as the total remaining time that this aircraft can fly, until it has to be grounded for maintenance check. For the solution of the problem we apply the weighted sums approach and lexicographic optimization. By comparing and analyzing the solutions obtained, we get insight into the behavior of the model. We conclude with a discussion based on these results and suggestions on how the model can be extended in the future.

1. Introduction

Flight and maintenance planning (FMP) of military aircraft addresses the problem of deciding which available aircraft to fly and for how long and which grounded aircraft to perform maintenance operations on, in a set of aircraft that comprise a combat unit. The objective is to achieve maximum availability of the unit over the planning horizon. The large number of parameters involved increases the complexity of the problem and the time required to reach an optimal solution. In this work, we develop a biobjective optimization model for the FMP problem and we illustrate its application on a real life instance drawn from the Hellenic Air Force (HAF). We formulate the FMP problem as a mixed integer linear program with two objectives, both of which depict the readiness of the unit to respond to external threats. We apply several approaches for the solution of the problem which enable us to explore the behavior of the model. Even though this is a military application, the model can be applied to several non-military ones, such as planning for fire-fighting aircraft, rescue choppers, etc.

1 George Kozanidis is a recipient of a research grant from the State Scholarships Foundation, which partially supported this work

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A significant part of the total operational budget of a fleet is spent for maintaining the aircraft that belong to it. The safety standards used by the Air Force in different countries are usually similar, since these standards are often determined by the aircraft manufacturer. Each aircraft has to undergo a regular maintenance check as soon as it completes a certain number of flight hours since its last maintenance check. There are also restrictions regarding the calendar time and number of takeoffs, but these are rarely used in practice, because the flight time restrictions usually apply sooner. The HAF, which provided the application that motivated this study, is supported by a three level maintenance program as follows: • 1st level maintenance (organizational level): This maintenance check is performed on site and

includes inspection, repair and parts replacement. • 2nd level maintenance (intermediate level): This check is performed on site and includes more

thorough inspection, repair, and parts replacement than the 1st level maintenance. • 3rd level maintenance - Manufacturer’s maintenance (depot level): This check is performed in

special facilities by specially trained professionals. It includes more thorough repair and parts replacement than the other two levels.

Several aircraft planning and scheduling problems have been investigated in the past, although the vast majority of the published works are on scheduling rather than planning. Additionally, several authors have presented reviews of models and methods for airline operations related problems. Arguello et al. (1997) studied models and methods for dynamic management of airline operations in case of irregular situations. Gopalan and Talluri (1998) surveyed models and solution techniques for various airline problems that include fleet assignment and maintenance routing decisions. Barnhart et al. (2003) presented an overview of several important areas of operations research applications in the air transport industry as well as a brief summary of the state of the art. Even though the literature dealing with airline operations is quite rich, none of the published works deals with the type of problem that we address in this paper. To the best of our knowledge, no such work has been published to date, perhaps because most of the publicly reported research in this area has been directed towards problems in the commercial airline industry, which have different objectives and requirements than problems in the Air Force. We believe that the FMP problem is an important practical problem that demands special attention, due to the significance of the related issues and the serious impact that the involved decisions can have on national security. The remainder of this paper is structured as follows. In Section 2, we describe in detail the FMP problem, and we present the mathematical model that was developed for its solution. In Section 3, we illustrate the application of the model on a real-life instance drawn from the HAF and we present a discussion that provides insight into the model. Finally, in Section 4 we conclude this work, by summarizing the results obtained and proposing possible future extensions. 2. Model Development 2.1 Problem Description The problem that we study arose as an operations management problem in a typical Combat Wing of the HAF. This wing consists of three squadrons, each of which serves as the base for several aircraft of various types. In what follows, we use the term "wing" to refer to all the squadrons considered together. At the beginning of each planning horizon, the wing command issues the flight requirements for each squadron and period combination. These requirements determine the total time that all the aircraft in each squadron should fly during each time period of the horizon. Separate requirements are issued for each aircraft type, because different aircraft types have different flight capabilities and maintenance needs. For this reason, the model introduced in this paper was developed for use on a specific aircraft type. Of course, the same model can be applied repeatedly until all plans have been issued, if more than one

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aircraft types are involved. The requirements issued by the wing command contain target values from which only small deviations are permitted. For each specific aircraft, we define its residual flight time as the total remaining time that this aircraft can fly until it has to undergo a maintenance check. The residual flight time of an aircraft is positive if and only if this aircraft is available to fly. At any time, the total residual flight time of a squadron is equal to the sum of the residual flight times of all the aircraft that belong to this squadron. The total residual flight time of the wing is equal to the sum of the residual flight times of all squadrons. Clearly, there exist many possible combinations of individual aircraft residual flight times that can result in the same total squadron or wing residual flight time. Similarly, we define the residual maintenance time of a non-available aircraft as the total remaining time that this aircraft needs in order to complete its maintenance check. The residual maintenance time of an aircraft is positive if and only if this aircraft is undergoing a maintenance check (and is therefore not available to fly). For the maintenance needs of the wing, there exists a maintenance station, responsible for providing maintenance services to the aircraft of the wing. This station has certain space and time capacity capabilities. Given the flight requirements for each squadron and period combination, and the physical constraints that stem from the capacity of the maintenance station, the objective is to issue a flight and maintenance plan for each individual aircraft in each squadron of the wing so that some appropriate measure of effectiveness is optimized. As already mentioned, maximizing the readiness to respond to external threats is the most appropriate measure of effectiveness for this application. This readiness depends on the total number of available aircraft, and on the total residual flight time of all available aircraft. Therefore, the two objectives of the model depict aircraft and residual flight time availability, respectively. Consider the 2-dimensional graph shown in Figure 1. The vertical axis represents residual flight time measured in some appropriate units, and the horizontal axis represents the indices of the available aircraft in increasing order of their residual flight times, 1 being the index of the aircraft with the smallest and N being the index of the aircraft with the largest residual time (N is the total number of available aircraft). Consider also the line segment connecting the origin and the point with coordinates (N, Y), where Y is the maximum time that an aircraft can fly between two consecutive maintenance checks. By mapping each aircraft on this graph, we can visualize the total availability of the squadron or of the wing, whichever of the two the graph refers to. To describe the smoothness of the distribution of the total residual flight time of all aircraft, we use a "total deviation index". This index is equal to the sum of the vertical distances (deviations) of each point mapping a single aircraft from the line segment that connects the origin with point (N, Y). The smaller this sum is, the smoother the distribution of the total residual flight time. Ideally, the total deviation index is equal to zero, in which case all points lie on this line segment. When issuing the individual aircraft plans, the intention is to keep each point close to this line segment, in order to keep the value of the total deviation index as small as possible. To date, this technique has been used to issue the flight and maintenance plans of individual aircraft in the Combat Wing that we studied. The intuition behind the utilization of the graph described above is straightforward. By providing a wide range of different residual flight times, we establish a smooth sequence that determines the order in which the aircraft should visit the maintenance station. This in turn prevents bottlenecks in the maintenance station and ensures a smooth utilization of the maintenance station. More importantly, it ensures a fairly constant level of aircraft availability.

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Figure 1: Visual representation of aircraft residual flight times 2.2 Problem Formulation In this section, we present the mathematical model that we developed for the FMP problem described above. We use the following notation: Sets: M : set of squadrons, indexed by m, Nm : set of aircraft in squadron m, indexed by n, T : length of the planning horizon, indexed by t. Decision Variables: z1 : minimum total number of available aircraft over all periods, z2 : minimum total residual flight time over all periods, amnt : binary decision variable that takes the value 1 if aircraft n of squadron m is available in period t, and 0 otherwise, ymnt : residual flight time of aircraft n of squadron m at the beginning of period t, xmnt : flight time of aircraft n of squadron m during period t, gmnt : residual maintenance time of aircraft n of squadron m at the beginning of period t, hmnt : maintenance time of aircraft n of squadron m during period t, dmnt : binary decision variable that takes the value 1 if aircraft n of squadron m exits the maintenance station at the beginning of period t, and 0 otherwise, fmnt : binary decision variable that takes the value 1 if aircraft n of squadron m enters the maintenance station at the beginning of period t, and 0 otherwise, qt, pmnt, rmnt : auxiliary binary decision variables. Parameters: Smt : required flight time of squadron m during period t, Bt : time capacity of the maintenance station during period t, G : residual maintenance time of an aircraft immediately after it enters the maintenance station, Y : residual flight time of an aircraft immediately after it exits the maintenance station, C : maximum number of aircraft that the maintenance station can handle simultaneously, A1mn : state of aircraft n of squadron m at the first period of the planning horizon (= amn1), Y1mn : residual flight time of aircraft n of squadron m at the first period of the planning horizon (= ymn1), G1mn : residual maintenance time of aircraft n of squadron m at the first period of the planning horizon (= gmn1), Xmax : maximum time that an aircraft can fly during a single time period, Ymin : minimum residual flight time of an available aircraft,

Y

1 2 3 4 N A/F

residual flight times

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Gmin : minimum residual maintenance time of a non-available aircraft, L, U : real numbers denoting the maximum deviation from the value of Smt that can be tolerated, K : a sufficiently large number. Based on the above notation, we present next the formulation of the FMP problem. In order to make easier the comprehension of this formulation, each time we introduce a set of constraints we include a description that explains its purpose.

The objective function (1) maximizes z1, which, by constraint set (3), denotes the minimum total number of available aircraft over all periods. Similarly, the objective function (2) maximizes z2, which, by constraint set (4), denotes the minimum total residual flight time over all periods. The availability of the first period is fixed; therefore it is not included in these objective functions. In order to account for a smooth continuity of the model over the next horizon, the expressions for z1 and z2 are also extended to the first period of that horizon, i.e. period T + 1. This is necessary in other constraints of the model too.

Constraint set (5) is used to update the residual flight time of each aircraft at the beginning of the next period, based on its residual flight time at the beginning of the previous period and the time that it flew during that period. Binary variable d takes the value 1 only when the corresponding aircraft exits the maintenance station. This way, an aircraft that exits the maintenance station is available to fly again with the maximum residual flight time, Y. Similarly, constraint set (8) is used to update the residual maintenance time of each aircraft at the beginning of the next period, based on its residual maintenance time at the beginning of the previous period and the time that it received maintenance during that period. Binary variable f takes the value 1 only when the corresponding aircraft enters the maintenance station for service. This way, an aircraft that enters the maintenance station is ready to receive maintenance with the maximum residual maintenance time, G.

Max z1 (1)

Max z2 (2)

s.t. 11 1

, 2,..., 1mNM

mntm n

z a t T= =

≤ = +∑∑ (3)

21 1

, 2,..., 1mNM

mntm n

z y t T= =

≤ = +∑∑ (4)

ymnt+1 = ymnt − xmnt + dmnt+1Y, m =1,…,M, n =1,…,Nm, t =1,..,T (5)

dmnt+1 > amnt+1 − amnt, m =1,…,M, n =1,…,Nm, t =1,..,T (6)

amnt+1 − amnt + 1.1(1−dmnt+1) > 0.1, m =1,…,M, n =1,…,Nm, t =1,..,T (7)

gmnt+1 = gmnt − hmnt + fmnt+1G, m =1,…,M, n =1,…,Nm, t =1,..,T (8)

fmnt+1 > amnt − amnt+1, m =1,…,M, n =1,…,Nm, t =1,..,T (9)

amnt − amnt+1 + 1.1(1−fmnt+1) > 0.1, m =1,…,M, n =1,…,Nm, t =1,..,T (10)

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Constraint sets (6), (7), (9) and (10) ensure that variables d and f take correct values, based on the values of variables a. Consider the nth aircraft of the mth squadron. Then, (amnt,amnt+1) can take any of the values (0,1), (0,0), (1,0) and (1,1) and the difference (amnt+1 - amnt) is equal to 1, 0, -1 and 0, respectively. Variable dmnt+1 should take the value 1 when (amnt,amnt+1) = (0,1) and this is ensured by constraint set (6). In any other case, dmnt+1 should be 0 and this is ensured by constraint set (7). Similarly, variable fmnt+1 should take the value 1 when (amnt,amnt+1) = (1,0) and this is ensured by constraint set (9). In any other case, fmnt+1 should be 0 and this is ensured by constraint set (10).

Constraint set (11) ensures that the flight requirements for each squadron and period combination are met. Variables L and U define an interval [LSmt, USmt], in which the actual squadron flight time for the associated period should lie. For example, when L = 0.95 and U = 1.05, a maximum deviation of 5% from the target values of the flight requirements is permitted. Constraint sets (12) and (13) ensure that the time and space capacity constraints of the maintenance station are not violated in any period. Constraint sets (14) and (15) are introduced to ensure that the maintenance will not idle whenever there is at least one aircraft waiting for service. With the introduction of the auxiliary binary variable qt, it is ensured that the total maintenance time provided by the station in period t will either be equal to the total time capacity of the station during this period or to the total maintenance requirements of this period, whichever of these two is smaller.

Constraint sets (16) and (17) are introduced to ensure that an aircraft’s availability terminates as soon as its residual flight time drops to zero. If ymnt > 0, the auxiliary binary decision variable pmnt in constraint (16) is forced to zero. In this case, constraint (17) forces amnt+1 to zero if ymnt = xmnt, since this means that the residual flight time of this aircraft drops to zero during period t. Similarly, constraint sets (18) and (19) ensure that an aircraft becomes available as soon as its residual maintenance time drops to zero. If gmnt > 0, the auxiliary decision variable rmnt in constraint (18) is forced to zero. In this case, constraint (19) forces amnt+1 to 1 if gmnt = hmnt, since this means that the residual maintenance time of this aircraft drops to zero during period t.

1, 1,..., , 1,...,

mN

mt mnt mtn

LS x US m M t T=

≤ ≤ = =∑ (11)

1 1, 1,...,

mNM

mnt tm n

h B t T= =

≤ =∑∑ (12)

1 1(1 ) , 2,..., 1

mNM

mntm n

a C t T= =

− ≤ = +∑∑ (13)

1 1(1 ), 1,...,

mNM

t mnt tm n

B h K q t T= =

≤ + − =∑∑ (14)

1 1 1 1, 1,...,

m mN NM M

mnt mnt tm n m n

g h Kq t T= = = =

≤ + =∑∑ ∑∑ (15)

ymnt + K pmnt < K, m =1,…,M, n =1,…,Nm, t =1,..,T (16)

amnt+1 ≤ (ymnt − xmnt)K + K pmnt, m =1,…,M, n =1,…,Nm, t =1,..,T (17)

gmnt + K rmnt < K, m =1,…,M, n =1,…,Nm, t =1,..,T (18)

1-amnt+1 ≤ (gmnt − hmnt)K + K rmnt, m =1,…,M, n =1,…,Nm, t =1,..,T (19)

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Constraint set (20) states that the residual flight time of an aircraft cannot exceed the maximum value, Y, and ensures that it will be zero whenever this aircraft is not available. Similarly, constraint set (21) states that the residual maintenance time of an aircraft cannot exceed the maximum value, G, and ensures that it will be zero whenever this aircraft is available. Constraint set (22) imposes an upper bound on the maximum time that an aircraft can fly during a single time period. Such a restriction is usually present due to technical reasons. Constraint sets (23) and (24) impose lower bounds on the residual flight and maintenance time of each aircraft. They are introduced to eliminate the situation in which an aircraft has a negligible residual flight or maintenance time. Constraint set (25) ensures that the total time that an aircraft flies during a single period does not exceed its residual flight time at the beginning of the same period. Similarly, constraint set (26) ensures that the total time that the maintenance crew works on a particular aircraft during a single period does not exceed the residual maintenance time of this aircraft at the beginning of the same period.

Finally, constraint sets (27), (28) and (29) are used to initialize the state of the system at the first period of the planning horizon. When an aircraft exits or enters the maintenance station at the first period of the planning horizon, its residual flight or maintenance time is directly updated; therefore variables dmn1 and fmn1 are never used. 3. Application of the Model In this section we illustrate the application of the model in a problem instance drawn from a Combat Wing of the HAF with 3 squadrons and 8 aircraft in each squadron. The planning horizon is 6 monthly periods. At the beginning of the planning horizon there are 3 aircraft at the maintenance station, one from each squadron. Their residual maintenance times are g131 = 320, g221 = 200 and g331 = 100 hours, respectively. Table 1 presents the required flight times for each squadron and period combination, Table 2 presents the time capacity of the maintenance station for each time period, and Table 3 presents the

ymnt ≤ amntY, m =1,…,M, n =1,…,Nm, t = 2,...,T + 1 (20)

gmnt ≤ (1-amnt)G, m =1,…,M, n =1,…,Nm, t = 2,...,T + 1 (21)

xmnt ≤ amntXmax, m =1,…,M, n =1,…,Nm, t =1,..,T (22)

ymnt ≥ amntYmin, m =1,…,M, n =1,…,Nm, t = 2,...,T + 1 (23)

gmnt ≥ (1 − amnt)Gmin, m =1,…,M, n =1,…,Nm, t = 2,...,T + 1 (24)

xmnt ≤ ymnt, m =1,…,M, n =1,…,Nm, t =1,..,T (25)

hmnt ≤ gmnt, m =1,…,M, n =1,…,Nm, t =1,..,T (26)

amn1 = A1mn, m =1,…,M, n =1,…,Nm (27)

ymn1 =Y1mn, m =1,…,M, n =1,…,Nm (28)

gmn1 = G1mn, m =1,…,M, n =1,…,Nm (29)

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residual flight times of the aircraft in the first period of the planning horizon. The time capacity of the maintenance station may vary from period to period, due to personnel vacations, holidays, etc. The values of the other problem parameters are G = 320 hours, Y = 300 hours, C = 3, Xmax = 50 hours, Ymin = 0.1 hours, Gmin = 0.1 hours, L = 0.9 and U = 1.1.

Table 1: Required flight times (Smt) in hours

Table 2: Time capacity of the maintenance station in hours

Table 3: Aircraft residual flight times at the beginning of the planning horizon (ymn1) in hours The first method that we apply for the solution of the problem is the weighted sums approach (see Steuer, 1986). We consider strictly positive weights w1 and w2 such that w1 + w2 = 1. After multiplying the first objective with 150 in order to scale it, we transform the problem into a single criterion one with objective Max 150w1z1 + w2z2. Table 4 presents the solutions obtained for values of w1 and w2 that vary between 0.05 and 0.95, with a step of 0.15.

Table 4: Solutions obtained using the weighted sums approach

The seven solutions shown in Table 4 are distinct, although they provide us with only two possible choices for the values of the two objectives, (z1, z2) = (22, 2800) and (z1, z2) = (23, 2311). This is a main characteristic of the problem, since a slight change of the optimal flight and maintenance plan can produce a different solution with same objective function values. After comparing the solutions for which (z1, z2) = (22, 2800), we select the one with w1 = 0.2, because it gives better results for the minimum availability of each squadron over all periods. For the same reason, out of the two solutions for which (z1, z2) = (23, 2311), we select the one with w1 = 0.95. In order to make our final decision, we perform a side by side comparison between these two solutions. The results are shown in Table 5. The upper half of this table shows how the number of available aircraft of the wing and of each squadron separately varies from period to period according to these two solutions, and the lower half shows similar results for the residual flight time.

t 1 2 3 4 5 6

m 1 150 130 150 140 150 160 2 140 150 140 160 150 140 3 140 160 150 140 160 130

t 1 2 3 4 5 6

Bt 480 470 480 460 490 450

n 1 2 3 4 5 6 7 8

m 1 100 123 0 263 90 300 40 10 2 133 0 150 34 10 150 114 218 3 250 150 0 140 100 50 10 152

w1 w2 z1 z2 0.05 0.95 22 2800 0.2 0.8 22 2800

0.35 0.65 22 2800 0.5 0.5 22 2800

0.65 0.35 22 2800 0.8 0.2 23 2311

0.95 0.05 23 2311

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Table 5: Comparison of the solutions with w1 = 0.2 and w1 = 0.95

We observe in Table 5 that, with respect to z1, the solution with w1 = 0.95 gives better results in five out of the six periods of the planning horizon, i.e. in periods 2-6. Note that the availability of the first period of the planning horizon is fixed; therefore it was not included in the objective functions. On the other hand, while the residual flight time availability of the wing in period 2 is the same in both solutions, in the solution with w1 = 0.2 it is higher by approximately 300 hours in periods 3-5 and by approximately 500 hours in periods 6 and 7. Moreover, with very few exceptions, the residual flight time availability of each squadron in each time period is significantly higher in the solution with w1 = 0.2. Therefore, we conclude by selecting that solution over the one with w1 = 0.95. The maximum possible values for z1 and z2 are 23 and 2800, respectively. These values are obtained when the problem is solved with a single objective, z1 or z2. Table 6 presents the two solutions obtained when lexicographic optimization (see Steuer, 1986) is applied for the solution of the problem. When z2 is maximized subject to the constraint that z1 is no less than 23, the optimal solution is 2311. When z1 is maximized subject to the constraint that z2 is no less than 2800, the optimal solution is 22. Of course, both these solutions are efficient (see Steuer, 1986). Interesting is the fact that this pair of solutions looks very similar to the pair of solutions that were compared in the final step of the weighted sums approach (i.e., in Table 5). Using a similar reasoning as above, between the two solutions compared in Table 6, we promote the one with z1 = 22 and z2 = 2800. Although there do not exist many differences between this solution and the one selected using the weighted sums approach, we select that one over the one obtained using lexicographic optimization, mainly because it exhibits a more constant level of availability. The above analysis motivated us to consider a four-objective version of the model, which besides the minimum availability of the wing, maximizes the minimum availability of each squadron too. Hence, we considered two additional objective functions, z3 and z4, expressed as follows:

w1 = 0.2 w1 = 0.95 t m = 1 m = 2 m = 3 wing m = 1 m = 2 m = 3 wing 1 7 7 7 21 7 7 7 21 2 7 8 7 22 8 7 8 23 3 7 7 8 22 8 7 8 23 4 7 8 7 22 8 8 7 23 5 7 8 7 22 7 8 8 23 6 7 8 7 22 8 7 8 23 7 7 8 8 23 7 8 8 23

1 926 809 852 2587 926 809 852 2587 2 791 983 1026 2800 1091 683 1026 2800 3 974 848 1182 3004 974 848 882 2704 4 839 1022 1047 2908 839 1022 747 2608 5 1007 872 921 2800 713 878 921 2512 6 1172 707 1045 2924 878 743 777 2398 7 1022 581 1202 2805 734 917 660 2311

Max z3 (30)

Max z4 (31)

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Table 6: The two solutions obtained using lexicographic optimization In these expressions, z3 denotes the minimum number of available aircraft of any squadron over all periods, and z4 denotes the minimum residual flight time of any squadron over all periods. For the solution of this problem, we considered the weighted sums approach, with various combinations of the weights, in which w1 = w2 and w3 = w4. For scaling purposes, we multiplied the first objective with 150, the third objective with 3(150) = 450, and the fourth objective with 3. This way, the single objective that was obtained was Max 150w1z1 + w2z2 + 450w3z3 + 3w4z4. The results are shown in Table 7. We observe in this table that all solutions obtained give the same exact values for the four objectives. Moreover, while the values of z1, z2 and z3 are the same with those of the solution selected for the biobjective version of the problem, the value of z4 is improved (791 over 581).

Table 7: Solutions obtained for the four criteria problem with the weighted sums approach The model of the FMP problem that we introduced is a mixed integer linear program, and the computational effort needed to reach an optimal solution depends strongly on the size and the parameters of the specific problem instance. In many cases, the large number of constraints may lead to the optimal solution very fast, due to the fact that the feasible region determined by these constraints is small. In other cases, this may not occur, due to the fact that a larger feasible region exists and many different combinations of the decision variables have to be compared. For this reason, the size of the problem

s.t. 31

, 1,..., , 2,..., 1mN

mntn

z a m M t T=

≤ = = +∑ (32)

41

, 1,..., , 2,..., 1mN

mntn

z y m M t T=

≤ = = +∑ (33)

z1 = 23, z2 = 2311 z1 = 22, z2 = 2800 t m = 1 m = 2 m = 3 wing m = 1 m = 2 m = 3 wing 1 7 7 7 21 7 7 7 21 2 8 7 8 23 7 8 7 22 3 7 8 8 23 8 7 7 22 4 8 7 8 23 7 7 8 22 5 8 7 8 23 6 8 8 22 6 8 8 7 23 7 8 8 23 7 8 8 8 24 7 7 8 22

1 926 809 852 2587 926 809 852 2587 2 1091 683 1026 2800 791 983 1026 2800 3 974 848 882 2704 974 848 1175.9 2997.9 4 1139 722 747 2608 833.1 722 1340.9 2896 5 1013 878 621 2512 707.1 878 1214.9 2800 6 878 1043 477 2398 1156.6 713 1038.9 2908.5 7 734 917 660 2311 1312.6 587 900.4 2800

w1 w2 w3 w4 z1 z2 z3 z4 0.05 0.05 0.45 0.45 22 2800 7 791 0.1 0.1 0.4 0.4 22 2800 7 791

0.15 0.15 0.35 0.35 22 2800 7 791 0.2 0.2 0.3 0.3 22 2800 7 791

0.25 0.25 0.25 0.25 22 2800 7 791 0.3 0.3 0.2 0.2 22 2800 7 791

0.35 0.35 0.15 0.15 22 2800 7 791 0.4 0.4 0.1 0.1 22 2800 7 791

0.45 0.45 0.05 0.05 22 2800 7 791

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alone is not indicative of the time needed to reach an optimal solution. Of course, an increase in the size of the problem results in general in greater computational effort, since the number of decision variables and constraints increases significantly. The time needed to solve each of the problems presented above was typically less than half a minute on a Pentium IV/2.5 GHz processor, using version 9.1 of AMPL (see Fourer et al. 2002) as the mixed integer optimization software, with solver CPLEX and default values. Various experiments that were conducted showed that besides the size of the problem, the total computational effort also depends on the range in which the problem parameters take their values. For the solution of larger problems, one can either implement the heuristic discussed shortly in Section 2.1, or break the planning horizon into smaller ones and apply the model repeatedly. This latter option however does not always produce solutions of high quality. 4. Conclusions In this work, we introduced a mixed integer biobjective optimization model for military aircraft flight and maintenance planning, and illustrated its application on a real instance drawn from the Hellenic Air Force. The two objectives of the model maximize the readiness of a combat unit to respond to external threats. The model can be easily extended to include additional aspects of the problem that may arise in different situations, such as the rotation of the aircraft that are used to satisfy the flight requirements, the incorporation of special maintenance checks and parts removal, the observance of the daily schedule of the pilots, etc. We believe that future research should be more intensively directed towards finding analytical tools for similar problems, due to the importance of the related issues and the serious impact that the involved decisions can have on national security. References Arguello, M.F., Bard, J.F. and Yu, G. (1997) “Models and methods for managing airline irregular operations aircraft routing” Operations Research in the Airline Industry ed. G. Yu, pp. 1-45. Kluwer Academic Publishers, Boston. Barnhart, C., Belobaba, P. and Odoni, A.R. (2003) “Applications of operations research in the air transport industry” Transportation Science, 37(4), 368-391. Fourer, R., Gay, D.M. and Kernighan, B.W. (2002) AMPL: A Modeling Language for Mathematical Programming, Duxbury Press. Gopalan, R. and Talluri, K.T. (1998) “Mathematical models in airline schedule planning: A survey” Annals of Operations Research, 76, 155-185. Steuer, R. E. (1986) Multiple Criteria Optimization: theory, computation and application, New York: Wiley.