Barry StannardBarry Stannard

Sajjad ZahirSajjad ZahirFaculty of Management University of LethbridgeFaculty of Management University of Lethbridge

E.S. RosenbloomE.S. RosenbloomI.H. Asper School of Business University of ManitobaI.H. Asper School of Business University of Manitoba

DECISION MODEL -DECISION MODEL -MILITARY AIRLIFT MILITARY AIRLIFT

RESOURCE ALLOCATIONRESOURCE ALLOCATION

Presentation OutlinePresentation Outline

IntroductionIntroduction Problem ScenarioProblem Scenario Formulation of the Mathematical ModelFormulation of the Mathematical Model Assessing Relative Importance of each MissionAssessing Relative Importance of each Mission An Example and Discussion of the ResultsAn Example and Discussion of the Results ConclusionsConclusions Future DevelopmentFuture Development

IntroductionIntroduction

Capacity Planning ProblemCapacity Planning Problem nn variable length airlift mission requests (tasks) from variable length airlift mission requests (tasks) from

several users several users with many priority levelswith many priority levels constrained assignment toconstrained assignment to m m airframes (parallel airframes (parallel

machines)machines) Solution combinesSolution combines

sequential goal programming sequential goal programming analytic hierarchy processanalytic hierarchy process

Problem ScenarioProblem Scenario Several airlift users from gov’t departmentsSeveral airlift users from gov’t departments

Foreign affairs, DND …Foreign affairs, DND … Hundreds of airlift requests exceed capacityHundreds of airlift requests exceed capacity

date, time, load, origin, destination, numbers of date, time, load, origin, destination, numbers of airframes …airframes …

An ordered list of prioritiesAn ordered list of priorities one for emergency and code 1 VIPsone for emergency and code 1 VIPs others for deployed peacekeeping forces, scheduled others for deployed peacekeeping forces, scheduled

fights …fights … Current output is non-optimized Gantt chartCurrent output is non-optimized Gantt chart

Formulation of Mathematical ModelFormulation of Mathematical Model What should the model handle? What should the model handle?

priority of the missionpriority of the mission number of taskable airframes & overtaskingnumber of taskable airframes & overtasking user flying hours budget user flying hours budget total fleet flying hours limittotal fleet flying hours limit linked airlift missionslinked airlift missions level of importance of mission to userlevel of importance of mission to user degree of mission training value to airlift system degree of mission training value to airlift system degree of effective use of specific airframedegree of effective use of specific airframe

Formulation of Mathematical ModelFormulation of Mathematical ModelMax

n

j ij iji

: Z = C XLP

(1)

. .s t

id i d d d

i

a X + A M - A P = TAC for d = 1,…,D (2)

1

D

d

d

A P OT

(3)

d dA P HM (4) Mr EMr

i i i i r r r

i Lr i ELr

H X + H X +U M - U P = F

for r = 1,…,R

(5)

1

n

i i

i

H X + FM - FP = YFR

(6)

21 22 23 24 25 4MY X X X X X (7)

21 22 23 24 25MY X X X X X M (8)

jiiji

Z C X for j = 2,…,K (9)

, iX Y = 0, 1 (10)

, , , 0d d r rA M A P U M U P (11)

AHPAHPAssessing Relative Mission ImportanceAssessing Relative Mission Importance

U serC a teg o ry

M issio nV a lu e

T ra in in gV a lu e

E ffec tiv eU se

M issio n 1

M issio n 2

M issio n ... n

M issio n 1

M issio n 2

M issio n ... n

M issio n 1

M issio n 2

M issio n ... n

AHPAHPAssessing Relative Mission ImportanceAssessing Relative Mission Importance

Table 1: Pairwise comparison of the criteria Table 2: Pairwise intensity comparisons for TV

Criteria TV UC EU

TV 1 3 7

UC 1/3 1 4

EU 1/7 1/4 1

TV H V L

H 1 3 5

V 1/3 1 3

L 1/5 1/3 1

Table 3: Pairwise intensity comparisons for CU Table 4: Pairwise intensity comparisons for EU

CU A B C

A 1 5 9

B 1/5 1 5

C 1/9 1/5 1

EU H V L

H 1 3 5

V 1/3 1 3

L 1/5 1/3 1

Mission Aggregate ValueMission Aggregate Value from Expert Choice from Expert Choice

An ExampleAn Example

Modeled representative set of 33 mission requestsModeled representative set of 33 mission requests from users 2, 4, 5, 6, and 7 simulating planning inputsfrom users 2, 4, 5, 6, and 7 simulating planning inputs taskable airframes (TACtaskable airframes (TACdd) set equal to 6) set equal to 6

overtasking enabledovertasking enabled overtasking set to a limit of 2 days for the periodovertasking set to a limit of 2 days for the period user hours budget set for each useruser hours budget set for each user total fleet hours (YFR) set for the periodtotal fleet hours (YFR) set for the period linked missions 17 & 19 modeledlinked missions 17 & 19 modeled linked missions 21, 22, 23, 24, 25 modeled linked missions 21, 22, 23, 24, 25 modeled

at least four must be accepted or all must be declinedat least four must be accepted or all must be declined

Formulation – Final Iteration of Run 4Formulation – Final Iteration of Run 41) MAX Z = 0.0309X4 + 0.0313X8 + 0.0264X9 + 0.0534X10 +

0.0426X11+ 0.0581X14

SUBJECT TO2) X1 + X7 + X15 + X16 + X20 + A1M ‑ A1P = 6

3) X1 + X7 + X8 + X15 + X16 + X20 + A2M ‑ A2P = 6

4) X2 + X7 + X8 + X9 + X16 + X17 + X20 + A3M ‑ A3P = 6

5) X2 + X7 + X8 + X12 + X14 + X17 + X20 + A4M ‑ A4P = 6

6) X2 + X3 + X10 + X12 + X14 + X17 + X20 + A5M ‑ A5P = 6

7) X2 + X3 + X10 + X12 + X14 + X20 + X32 + A6M ‑ A6P = 6

8) X2 + X3 + X10 + X12 + X14 + X20 + X32 + A7M ‑ A7P = 6

9) X2 + X3 + X10 + X12 + X14 + X20 + X32 + A7M ‑ A8P = 6

10) X2 + X11 + X12 + X14 + X20 + X32 + A9M ‑ A9P = 6

11) X2 + X11 + X12 + X14 + X20 + A10M ‑ A10P = 6

12) X11 + X20 + X22 + X23 + X24 + X25 + A11M ‑ A11P = 6

13) X4 + X20 + X21 + X22 + X23 + X24 + X25 + A12M ‑ A12P = 6

14) X4 + X20 + X21 + X22 + X23 + X24 + X25 + A13M ‑ A13P = 6

15) X4 + X21 + X22 + X23 + X24 + X25 + A14M ‑ A14P = 6

16) X4 + X21 + X22 + X23 + X24 + X25 + A15M ‑ A15P = 6

17) X4 + X21 + X22 + X23 + X24 + X25 + A16M ‑ A16P = 6

18) X4 + X21 + X22 + X23 + X24 + X25 + A17M ‑ A17P = 6

19) X21 + X22 + X23 + X24 + X25 + A18M ‑ A18P = 6

20) X5 + X21 + X22 + X23 + X24 + X25 + A19M ‑ A19P = 6

21) X5 + X21 + X22 + X23 + X24 + X25 + A20M ‑ A20P = 6

22) X5 + X19 + X22 + X23 + X24 + X25 + A21M ‑ A21P = 6

23) X5 + X6 + X18 + X19 + A22M ‑ A22P = 6

24) X6 + X18 + X19 + X26 + X27 + X28 + X29 + A23M ‑ A23P = 6

25) X6 + X18 + X26 + X27 + X28 + X29 + X30 + X33 + A24M ‑ A24P = 6

26) X6 + X18 + X26 + X27 + X28 + X29 + X30 + X33 + A25M ‑ A25P = 6

27) X6 + X26 + X27 + X28 + X29 + X30 + X33 + A26M ‑ A26P = 6

28) X13 + X26 + X27 + X28 + X29 + X30 + X31 + A27M ‑ A27P = 6

29) X13 + X26 + X27 + X30 + X31 + A28M ‑ A28P = 6

30) X13 + X26 + X31 + A29M ‑ A29P = 6

31) X13 + X26 + X31 + A30M ‑ A30P = 6

32) A1P + A2P + A3P + A4P + A5P + A6P + A7P + A8P + A9P + A10P +

A11P + A12P + A13P + A14P + A15P + A16P + A17P + A18P +

A19P + A20P + A21P + A22P + A23P + A24P + A25P + A26P +

A27P + A28P + A29P + A30P = 2

33) A1P <= 0

34) A3P <= 0

35) A2P <= 0

36) A4P <= 0

37) A5P <= 0

38) A6P <= 0

39) A7P <= 0

40) A8P <= 0

41) A9P <= 0

42) A10P<= 0

43) A11P <= 0

44) A12P <= 0

45) A13P <= 0

46) A14P <= 0

47) A15P <= 0

48) A16P <= 0

49) A17P <= 0

50) A18P <= 0

51) A19P <= 0

52) A20P <= 0

53) A21P <= 0

54) A22P <= 0

55) A23P <= 0

56) A24P <= 1

57) A25P <= 1

58) A26P <= 0

59) A27P <= 0

60) A28P <= 0

61) A29P <= 0

62) A30P <= 0

63) 12 X1 + 41 X2 + 29 X3 + 18 X4 + 10 X5 + 14 X6 + U2M ‑ U2P = 112

64) 24 X7 + 19 X8 + 7 X9 + 20 X10 + 22 X11 + 35 X12 + 8 X13 + 45 X14 +

24 X32 + U4M ‑ U4P = 162

65) 14 X15 + 20 X16 + 11 X17 + 28 X18 + 16 X19 + 90 X20 +

U5M ‑ U5P = 161

66) 60 X21 + 60 X22 + 60 X23 + 60 X24 + 60 X25 + 50 X26 + 41 X27 +

35 X28 + U6M ‑ U6P = 383

67) 36 X29 + 34 X30 + 22 X31 + 12 X33 + U7M ‑ U7P = 83

68) U2P <= 11

69) U4P <= 16

70) U5P <= 16

71) U6P <= 372) U7P <= 8

73) 12 X1 + 41 X2 + 29 X3 + 18 X4 + 10 X5 + 14 X6 + 24 X7 + 19 X8 +

7 X9 + 20 X10 + 22 X11 + 35 X12 + 8 X13 + 45 X14 + 14 X15 + 20 X16

+ 11 X17 + 28 X18 + 16 X19 + 90 X20 + 60 X21 + 60 X22 + 60 X23 +

60 X24 + 60 X25 + 50 X26 + 41 X27 + 35 X28 + 36 X29 + 34 X30 +

22 X31 + 24 X32 + 12 X33 + FM ‑ FP = 900

74) FP <= 10075) X17 ‑ X19 = 0

76) X21 + X22 + X23 + X24 + X25 + 1000 Y >= 4

77) X21 + X22 + X23 + X24 + X25 + 1000 Y <= 1000

78) 0.0313 X1 + 0.0412 X5 + 0.0683 X6 >= 0.1408

79) 0.0346 X15 + 0.0309 X16 + 0.0426 X17 + 0.0502 X18 + 0.0276 X19 +

0.0683 X20 >= 0.2233

80) 0.0346 X7 + 0.015 X13 + 0.0683 X26 + 0.0548 X27 + 0.0502 X28 +

0.0426 X29 + 0.0534 X30 + 0.0426 X31 + 0.0276 X32 +

0.0697 X33 >= 0.4162

81) 0.0459 X2 + 0.0581 X3 + 0.0459 X12 + 0.0697 X21 + 0.0697 X22 +

0.0697 X23 + 0.0697 X24 + 0.0697 X25 >= 0.4984

Model ResultsModel Results Initial Plan (Run 1, 31 requests):Initial Plan (Run 1, 31 requests):

Mission 4 not selected due to lack of user hoursMission 4 not selected due to lack of user hours Missions 14, 16, 24, 29, and 31 not selectedMissions 14, 16, 24, 29, and 31 not selected

lack of taskable airframeslack of taskable airframes Only four of linked missions 21 to 25 selectedOnly four of linked missions 21 to 25 selected

no airframes available on days 11 and 21no airframes available on days 11 and 21

Re-plan (Run 2, 32 requests):Re-plan (Run 2, 32 requests): Fifth airframe planned for days 12 to 20Fifth airframe planned for days 12 to 20

for linked missions 21 to 25for linked missions 21 to 25 Additional request (#32) from user 4 plannedAdditional request (#32) from user 4 planned

Model Results - ContinuedModel Results - Continued Re-plan (Run 3, 33 requests):Re-plan (Run 3, 33 requests):

User 7 still wants mission 31, agrees to start on the User 7 still wants mission 31, agrees to start on the 2727thth vs the 25 vs the 25thth as originally requested as originally requested

User 7 introduces mission 33 as an additional requestUser 7 introduces mission 33 as an additional request Problem: mission 33 accepted but mission 28 (user 6) Problem: mission 33 accepted but mission 28 (user 6)

now dropped (unacceptable)now dropped (unacceptable) Re-plan (Run 4, 33 requests):Re-plan (Run 4, 33 requests):

Solution. Overtask MOB (Wing) to provide and extra Solution. Overtask MOB (Wing) to provide and extra airframe for days 24 and 25.airframe for days 24 and 25.

Final Iteration of Run 4 confirms solutionFinal Iteration of Run 4 confirms solution

Abridged Summary of Excel RunsAbridged Summary of Excel Runs

MISSION REQUEST

DATES (INCL)

PRIORITY USER HOURS ATTRIBUTES AHP WORTH

RUN

1

RUN

2

RUN

3

RUN

4

1 1-2 1 2 12 LAV 0.0313 1 1 1 1

2 3-10 4 2 41 VAH 0.0459 1 1 1 1

14 4-10 5 4 45 HBH 0.0581 0 0 0 0

16 1-3 2 5 20 VBH 0.0309 0 0 0 0

20 1-13 2 5 90 HAL 0.0683 1 1 1 1

21 11-21 4 6 60 HAV 0.0697 1 1 1 1

22 11-21 4 6 60 HAV 0.0697 1 1 1 1

23 11-21 4 6 60 HAV 0.0697 1 1 1 1

24 11-21 4 6 60 HAV 0.0697 0 1 1 1

25 11-21

12-20 4 6 60 HAV 0.0697 1 1 1 1

28 23-27 3 6 35 HCV 0.0502 1 1 0 1

31 25-28

27-30 3 7 22 VAV 0.0426 0 1 1 1

32 6-9 3 4 24 VBV 0.0276 N/A 1 1 1

33 24-26 3 7 12 HAV 0.0697 N/A N/A 1 1

ConclusionsConclusions AHP and sequential goal programming work well for AHP and sequential goal programming work well for

aggregate capacity planningaggregate capacity planning of this type of this type model is flexible, computationally quick and handles:model is flexible, computationally quick and handles:

priority of the missionpriority of the mission number of taskable airframes & overtaskingnumber of taskable airframes & overtasking user flying hours budget user flying hours budget total fleet flying hours limittotal fleet flying hours limit linked airlift missionslinked airlift missions level of importance of mission to userlevel of importance of mission to user degree of mission training value to airlift system degree of mission training value to airlift system degree of effective use of specific airframe model handles all the degree of effective use of specific airframe model handles all the

Management can change the importance values by Management can change the importance values by changing decision criteria (Table 1)changing decision criteria (Table 1) enables transition from peace to other levels of tension (incl war)enables transition from peace to other levels of tension (incl war)

Future DevelopmentFuture Development

As a capacity planning model, we have a first step As a capacity planning model, we have a first step towards a Decision Support System that supports:towards a Decision Support System that supports: multiple fleets of airframes (C130s, Airbus, …)multiple fleets of airframes (C130s, Airbus, …)

our prototype model only used one fleetour prototype model only used one fleet multiple main operating bases (MOB) for each fleetmultiple main operating bases (MOB) for each fleet

single MOB per fleet in our modelsingle MOB per fleet in our model develop optimal mission assignment to specific MOBdevelop optimal mission assignment to specific MOB

sliding time windows for begin/end dates (automated)sliding time windows for begin/end dates (automated) currently done manually in our prototypecurrently done manually in our prototype

optimal base-level optimal base-level schedulingscheduling by airframe tail number by airframe tail number incorporate crew duty time limitationsincorporate crew duty time limitations

Questions?Questions?

Q & AQ & A