Chapter 3 The Uncooperative Relationship of a Bilevel Multi-Follower
4.5 The Solution Algorithm
Let {U(i):i1,2,,k} be a collection of k divisions/followers with a budget to be distributed, and let P(i)
p(i,j): j1,2,,n
be the collection of n proposals submitted by division i to request funding. Without loss of generality, we can assume that every division submits n proposals. Each proposal has input I(i,j) and output data O(i,j). The input data includes one’s own resources g(i,j) and the request for funding b(i,j); the output data includes implicit value vi(i,j) and explicit value ve(i,j). The inputs and the outputs for the proposals of the divisions are listed in table 4.1.Table 4.1 The Inputs and the Outputs for the Proposals of the Divisions
U p( ji, ) I( ji, ) O( ji, )
U(1)
) 1 , 1 (
p g(1,1) b(1,1) vi(1,1) ve(1,1) )
2 , 1 (
p g(1,2) b(1,2) vi(1,2) ve(1,2)
) , 1 ( n
p g( n1, ) b( n1, ) vi( n1, ) ve( n1, )
U(2)
) 1 , 2 (
p g(2,1) b(2,1) vi(2,1) ve(2,1) )
2 , 2 (
p g(2,2) b(2,2) vi(2,2) ve(2,2)
) , 2 ( n
p g( n2, ) b( n2, ) vi( n2, ) ve( n2, )
U(i)
) 1 , (i
p g(i,1) b(i,1) vi(i,1) ve(i,1) )
2 , (i
p g(i,2) b(i,2) vi(i,2) ve(i,2)
) , ( ni
p g( ni, ) b( ni, ) vi( ni, ) ve( ni, )
U(k)
) 1 , (k
p g(k,1) b(k,1) vi(k,1) ve(k,1) )
2 , (k
p g(k,2) b(k,2) vi(k,2) ve(k,2)
) , (k n
p g(k,n) b(k,n) vi(k,n) ve(k,n)
Preprocessing:
For all i1,2,...,kand j1,2,...,n. 1. Let SI .
2. Compute the input-output efficiency E( ji, ) using the GDEA mathematical program as shown below:
.
Selection Process:
1. Apply the theory of the grey relationship to obtain the grey relationship grade z’s among all the lower level DMUs.
Let S be the space of grey relation factors
And grey relational coefficient
grey relationship grade
.
otherwise, the end of comparison and the selection in the unit i is complete.
Next j is feasible.
7. Set x, let r0.5xL(m)andr0.5xL(i), algorithm of the BLMF-PC are written in the Appendix 3, 4.
Example 3:
The government will distribute $6,000 millions to the energy industry of four fields, which including energy saving, renewable energy, new energy and energy technology etc. The amount to be distributed to each department of the field is determined by the proposals submitted by each division. Each proposal includes the input data I(i,j) and the output data O(i,j). The input contains one’s own recourses g(i,j) and the request for funding b(i,j); the output includes implicit value vi(i,j) and explicit value ve(i,j). The data of inputs and outputs for the proposals are listed in table 4.2.
Table 4.2 The Data of Inputs and Outputs for the Proposals in Example 3 Fields p( ji, )
F(3) program for the Unit 1:
1
The above linear program GDEA using LINDO gives a set of optimal weights:
1=0.000001 2= 0.002357 1= 0.000061 2= 0.000211
Plug the values of , into Ej yj xjto obtain the efficiency of the Unit 1 for desired rankings. Repeat Step 2 to compute E( ji, ) for Unit 2, Unit 3, and Unit 4. The results showing the input-based efficiency of each proposal are listed in table 4.3.
Table 4.3 The Results Showing the Input-Output Based Efficiency of Each Proposal
Selection Process:
1. Apply the theory of the grey relationship to obtain the grey relationship grade z’s among all the lower level DMUs.
First, consider in unit 1 the input/output data of six proposals obtained by first selection stage:
)
to initialization j, j1,2,3,5,6,7
14
Let distinguishing coefficient 0.5, and the grey relational coefficient:
57
By definition of the grey relational grade
Repeat the selection process 1 and obtain the grey relational grade for Field 2, Field 3, and Field 4; see Table 4.4 below:
Table 4.4 The Results Showing the Grey Relational Grade of Each Proposal
Fields p1 p2 p3 p4 p5 p6 p7 p8 p9
3. Compute , (, ) . )
, (
) , ) ( ,
( i j SI
j i I
j i j vi i
e
Let {e(i,(j))} be a decreasing rearrangement of e( ji, ), i.e. e(i,(1))e(i,(2))e(i,(n)) for all (i, j)SI.
Table 4.5 Implicit Efficiencies for Each Project of Divisions with Proposal Ranking
Fields p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9
F(1)
vi 1 8160 941 1486 u 4761 3449 1809 u －
I 1 571 86 396 u 200 438 415 u －
e 1 14.29 10.94 3.75 u 23.80 7.87 4.36 u －
(2) (3) (6) u (1) (4) (5) u －
F(2)
vi 2 2992 2853 1210 5186 4900 743 4150 678 －
I 2 959 973 195 712 991 176 575 509 －
e 2 3.12 2.93 6.21 7.28 4.94 4.22 7.22 1.33 －
(6) (7) (3) (1) (4) (5) (2) (8) －
F(3)
vi 3 u 2255 1989 395 u 123 u 866 26
I 3 u 774 363 131 u 89 u 321 71
e 3 u 2.91 5.48 3.02 u 1.38 u 2.70 0.37
u (3) (1) (2) u (5) u (4) (6)
F(4)
vi 4 296 u 66 u 760 39 18 44 －
I 4 384 u 152 u 160 135 278 195 －
e 4 0.77 u 0.43 u 4.75 0.29 0.06 0.23 －
(2) u (3) u (1) (4) (6) (5) －
Also compute , (, ) . )
, (
) , ) (
,
( i j SI
j i I
j i j ve i
w
Let {w(i,(j))} be a decreasing rearrangement of w( ji, ), i.e. w(i,(1))w(i,(2))w(i,(n)) for all (i,j)SI.
Table 4.6 Explicit Efficiencies for Each Project of Divisions with Proposal Ranking
Fields p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9
F(1)
ve 1 863 74 3065 u 213 1531 2013 u －
I 1 571 86 396 u 200 438 415 u －
w 1 1.51 0.86 7.74 u 1.07 3.50 4.85 u －
(4) (6) (1) u (5) (3) (2) u －
F(2)
ve 2 7154 8325 162 9898 7276 300 7481 380 －
I 2 959 973 195 712 991 176 575 509 －
w 2 7.46 8.56 0.83 13.90 7.34 1.70 13.01 0.75 －
(4) (3) (7) (1) (5) (6) (2) (8) －
F(3)
ve 3 u 547 62 60 u 24 u 55 395
I 3 u 774 363 131 u 89 u 321 71
w 3 u 0.71 0.17 0.46 u 0.27 u 0.17 5.56
u (2) (6) (3) u (4) u (5) (1)
F(4)
ve 4 4373 u 4491 u 1797 1884 4123 2680 －
I 4 384 u 152 u 160 135 278 195 －
w 4 11.39 u 29.55 u 11.23 13.96 14.83 13.74 －
vo 1 (5) u (1) u (6) (3) (2) (4) －
4.
Between e(1,(3)) and w(1,(1)), w(1,(1))is chosen after computation and
Repeat the process 4; similarly, the rearrangement of each division’s projects in table 4.7 is obtained.
Table 4.7 The Rearrangement of Each Field’s Projects
F(1)
p p 5 p 1 p 3 p 7 p 6 p 2
vi 4761 8160 1486 1809 3449 941
z
ve 213 751 2146 1530 1209 64
V 4974 8911 3632 3339 4658 1005
b 142 424 313 294 321 65
L(1) 9.11 36.31 56.38 75.24 95.83 100
F(2)
p p 4 p 7 p 2 p 1 p 5 p 3 p 6 p 8
vi 5186 4150 2853 2992 4900 1210 743 678
z
ve 9898 7032 6743 5580 5894 123 228 274
V 15084 11182 9596 8572 10794 1333 971 952
b 569 455 808 767 794 146 140 487
L(2) 13.66 24.58 43.98 62.39 81.45 84.95 88.31 100
F(3)
p p 3 p 4 p 2 p 8 p 9 p 6
vi 1989 395 2255 866 26 123
z
ve 62 51 454 48 265 19
V 2051 446 2709 914 291 142
b 262 100 481 229 51 34
L(3) 22.64 31.29 72.86 92.65 97.06 100
F(4)
p p 3 p 7 p 5 p 1 p 8 p 6
vi 66 18 760 296 44 39
z
ve 4491 3463 1438 3717 2466 1677
V 4557 3481 2198 4013 2510 1716
b 114 254 111 282 168 112
L(4) 10.95 35.35 46.01 73.10 89.24 100
Thus, the decision variables are set to equal 1, i.e., the proposals are selected.
Solution
Table 4.8 The Decision Variables of the Lower Level for Each Unit’s Project
Fields p1 p2 p3 p4 p5 p6 p7 p8 p9
F(1) 1 1 1 0 1 0 1 0 －
F(2) 1 1 1 1 0 1 1 0 －
F(3) 0 1 1 1 0 1 0 0 1
F(4) 1 0 1 0 1 1 1 0 －
This final solution gives the level of satisfaction are
86 programming in figure 4.2 and 4.3.
Figure 4.2 Diagram of the Preprocessing Stage for BLMF-PC Source: Study
Let
Computing E(i,j)
.
Discard p(i,j) no
Select p(i,j) )}
Figure 4.3 Diagram of the Heuristic Algorithm for BLMF-PC Decision Variables
1
Feasible Solution
max ( ) min ( )
4.6 Summaries
This chapter extends the BLMF problems to the case when decision variables are partially cooperative. The definitions and the characteristics of BLMF-PC are discussed, and a multi-follower budget distribution model with partial cooperative variables is constructed.
In this model, the goal of the upper level DM is to minimize the level of satisfaction among the individual divisions upon their funding approvals, and the constraints are to ensure the output efficiency is no less than the input efficiency. Under the budget allocation policy of the upper level DM, the lower level DM pursues the largest value of each individual division in order to maximize the total value of the organization. So, the optimal decision is to maximize the explicit and the implicit values of each proposal.
The BLMF-PC budget allocation problems are solved using the concepts of GDEA for preprocessing the data of the projects from each division to guarantee the quality of funded projects; this avoids unnecessary distributions. The grey relational analysis and the heuristic algorithm are then applied for a budget distribution.
This extension models the hierarchical structure of the real world more precisely. This chapter mainly investigates the case with multiple lower level DMUs with partially cooperative variables in a bilevel program. The model has the following properties:
(a) It is a bilevel programming model.
(b) Lower level DM has multiple objectives.
(c) All DMUs are partially dependent with partial cooperative decision variables.
(d) Lower level decision variables are discrete.
The two-stage solution algorithm is developed: stage 1: GDEA, and stage 2: the grey relational analysis and the heuristic algorithm. The final solution is feasible and can be optimal or near optimal. Most importantly, a quick user guide and a source code for the grey relational analysis and the heuristic algorithm of the BLMF-PC are written in this dissertation, they can easier to solve these difficult problems.