Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 22
Long-Range Financial Planning –
A Linear-Programming Modeling Approach
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
Outline









22.1 Introduction
22.2 Carleton’s model
22.3 Brief discussion of data inputs
22.4 Objective-function development
22.5 The constraints
22.6 Analysis of overall results
22.7 Summary and conclusion
Appendix 22A. Carleton’s linear-programming model:
General Mills as a case study
Appendix 22B. General Mills’ actual key financial data
22.2
Carleton’s
model
22.2
Carleton’s
model
22.2
Carleton’s
model
22.2
Carleton’s
model
22.2 Carleton’s model
22.2 Carleton’s model
22.3 Brief discussion of data inputs
22.3 Brief discussion of data inputs
22.3 Brief discussion of data inputs
22.3 Brief discussion of data inputs
(Cont.)
22.4 Objective-function development
P0 T 1
Dt
PT


,
t
t
N 0 t 1 N 0 (1  K )
N T (1  K )
where
(22.1)
22.4 Objective-function development
Pt
E  N ( ),
Nt
''
t
N j 1
N j 1
1
Nj
N j 1

E nj 1
Pj 1

or
~n
E j 1
Pj 1
Nj
N j 1
(22.2)
,
(22.3)

Pj 1  E nj 1
Pj 1
.
(22.3a)
22.4 Objective-function development
N j  Pj 1 


or Pj  D j 

.
N j N j N j 1 (1  K )
N j 1  1  K 
Pj
Dj
Pj 1
Pj  D j 
Pj  D j 
~n
( Pj 1  E j 1 )
~n
D j 1  E j 1
(1  K )
(1  K )

D j 2
.
~n
 E j  2
(1  K ) 2
(22.4)
~
PT  ETn
 ... 
.
T j
(1  K )
Pj
P0
D0
D1


 ... 
.
j
N 0 N 0 N1 (1  K )
N j (1  K )
(22.5)
22.4 Objective-function development
~
Dj
D j 1  E jn1
P0
D0
D1


 ... 

 ...
N 0 N 0 N1 (1  K )
(1  K )
N j (1  K ) j
~
PT  ETn

.
(1  K ) T  j
(22.6)
P0
D0
D1
E1''
D2
Max




N 0 N 0 N 1 (1  K )
N 0 (1  K )(1  C ) N 0 (1  K ) 2
D3
E3''
E 2''



N 0 (1  K ) 2 (1  C ) N 0 (1  K ) 3 N 0 (1  K ) 3 (1  C )
D4
E 4''


4
N 0 (1  K )
N 0 (1  K ) 4 (1  C )
(22.7)
P3
E5''


5
N 0 (1  K )
N 0 (1  K ) 5 (1  C )
Max 0.018D1  0.0196 E1  0.015D 2  0,017 E 2  0.013D3
 0.0144 E 3  0.011D 4  0.0125E 4  0.015E 5
(22.7a)
22.5 The constraints
 Definitional
 Policy
constraints
constraints
22.5 The constraints
Fig. 22.1 Structure of the optimizing financial planning model. (From Carleton, W. T., C. L. Dick,
Jr., and D. H. Downes, "Financial policy models: Theory and Practice," Journal of Financial
and Quantitative Analysis (December 1973). Reprinted by permission.)
22.5 The constraints
AFCt  ATPt  Pfdivt  SAt .
(22.8)
Z
t


'
ATPt  (1   ) t  eAt  aAat   iz ( Lz ,0  Cz ,t )  it  DLs 
z 1
s 1


 B1B2 ( I1  eAt 1 )  (1   )( aAat  eAt ).
(22.9)
Because General Mills has no preferred stock or extraordinary items,
AFC = ATP:
22.5 The constraints
,
22.5 The
i1  0constraints
i2  0.085
371.32
 431.9 


L1   497.7 


570
.
8


 651.2 
,
0 
0 
 
~
P1  i2 L1  0
 
0 
0
 66.8 
 77.7 


L2   89.6 


102.7
117.2
 5.678 
6.6045


~
P2  i2 L2   7.616 


8.7295
 9.962 
22.5 The constraints
22.5 The constraints
L3, 0  C3,t
79
.
80
81
82
83
0
0
0
0
0
30
30
30
20
20
11.3 9.8 8.3 6.3 4.6
L3 
18.7 17.3 15.9 14.5 13.1
73.1 68.1 63.1 58.1 53.1
93.7 88.7 83.1 78.1 73.1
R3  7% 4.25% 8% 4.875% 8.875% 8%
22.5 The constraints
22.5 The constraints
To get the interest payment on long-term debt
22.5 The constraints
~ 3 ~
~L
~
~
AFC  (1   )     Pi  i'DL   B1  B2[ I1  eA ].
i 1


~
~
P2
~
P3'
~
DL
 311.9   5.678  17.04425
 DL1
 357.9  6.6045 16.04225
 DL 2

 
 



AFC  (1  0.51) 406.53   7.616   14.96225  (0.09)  DL 3

 
 



460
.
53
8
.
7295
13
.
46525
DL
4

 
 



519.34  9.962  12.24475
 DL 5
~
~
I
AL
 243.6 
 1613 
303.15
1856.6 




 (0.07)  (0.36)  329.1   (0.033) 2159.4




365
.
1
2488
.
5




383.15
 2854 
22.5 The constraints
 DL1 149.17 
 AFC1 
 DL 2 173.45 
 AFC 2

 



 AFC3  0.0441   DL 3  198.22 

 



05
.
226
4
DL
4
AFC

 



 DL 5 255.62
 AFC5
AFC1+0.00441DL1=149.17
(22.10a)
AFC2+0.00441DL2=173.45
(22.10b)
AFC3+0.00441DL3=198.22
(22.10c)
AFC4+0.00441DL4=226.05
(22.10d)
22.5 The constraints
t
 t   0,t    s' ( I s )
where
s 1
(22.11)
22.5 The constraints
   C0
I t  
 C1

  C0
1 
C1


  C0
At  1 
C1




t 1
A0
(22.12a)
t

 A0

(22.12b)
22.5 The constraints
Z
I t  AFCt 1  Dt 1   CLz ,t  DLt  DTLt 1  Etn
z 1
where
Etn
(22.13)
22.5 The constraints
Z
I t  AFC L  D L   CLz  DL  DLL  DTLL  E
z 1
 AFC0  105.79


AFC
1



AFC L  
AFC2


AFC3




AFC4
Z
AFC0  D0  DL1  E1  I t   CLz
z 1
22.5 The constraints
AFC0  (1   )[ 0   i z Lz ,0 ].
 0  245.2,
[ i z Lz ,0 ]  29.3.
D0  48.2.
~
~L
CL  LT  LT .
22.5 The constraints
696.82
 744.4 


~
LT   809.2 


 872.0 
 953.5 
641.63
696.82

~L 
LT .   744.4 


 809.2 
 872.0 
22.5 The constraints
55.19
47.58


CL.  64.80.


62.80
81.50
AFC0  (1  0.51)[ 254.2  29.3]  105.79.
105.79  48.2  DL1  E1  243.6  55.19,
DL1  E1  130.82
22.5 The constraints
Z
I  AFC L  D L   CLz  DL  DLL  DTLL  E ,
z 1
 243.6  105.79 48.2 55.19  DL1   0   E1 
303.15  AFC1   D  47.58  DL 2   DL1  E 

 
  1  
 
 
  2
 329.1    AFC 2    D2   64.80   DL3    DL 2   E3 
 

 
 
 
 
  
 365.1   AFC3   D3  62.80  DL 4   DL 3  E 4 
383.15  AFC 4   D4  81.50 488.4  DL 4  E5 
22.5 The constraints
DL1  E1  131.38.
(22.10e)
AFC1  D1  DL2  DL1  E2  255.7 (22.10f)
AFC2  D2  DL3  DL 2  E3  264.3(22.10g)
AFC3  D3  DL 4  DL3  E4  302.3 (22.10h)
AFC4  D4  DL 4  E5  182.15
(22.10i)
22.5 The constraints

t
Z
i ( L z , 0  C z ,t )  i
z 1 z
 5.678  17.04
6.6045 16.04
 

~ ~ 
P2  P3   7.616   14.96

 

8.7295 13.47 
 9.962  12.24
'
t

t
s 1
DL s
 X,
 DL1 
 DL 2 


(0.09)  DL 3 


 DL 4 
488.4
(22.14)
22.5 The constraints
.
 311.9 
 357.9 

~ 
  406.53


460.53
519.34
~ ~
~
~
  X [ P2  P3  0.09DL ].
22.5 The constraints
~ ~
~
~
  6.35[ P2  P3  0.09DL ].
~ ~
~
~
  6.35[ P2  P3 ]  0.5715DL ,
  5.678  17.04  
 311.9 
 DL1 


 357.9 



 DL2 
6.6045
16.04

  6.35  

   0.5715 
.
 406.53
 DL3 
  7.616  14.96  



 




 460.53
 DL4 
 8.7295  13.47  
22.5 The constraints
DL1.LE.293.33,*
(22.15a)
DL2 .LE.374.64,
(22.15b)
DL3 .LE.460.49,
(22.15c)
DL4 .LE.559.17,
(22.15d)
22.5 The constraints
Z
 (L
z 1
Z
z ,0
 C z ,t )   DL s  At ,
z 1
(22.16)
( Lz ,0  C z ,t )  Liabilitiest  DL0 ;
DL1 .LE.243.6
DL2  DL1 .LE.303.15
(.LE. " " )
(22.17a)
(.GE. "") (22.17b)
22.5 The constraints
DL3  DL2 .LE.329.1
(22.17c)
DL4 .GE.101.15
(22.17d)
Dt   t Dt 1  0.
(22.19)
22.5 The constraints
1
~n
Pt  E t t  Pt .

(22.19a)
Nt

,
N t 1
N t 1
1
1
 1 .
Nt

N t
1
 1 .
Nt

(22.19b)
22.5 The constraints
5
Z



'
AFC5  (1   )  5   i z ( Lz 0  C z 5 )  i5   DL s   B1 B2 ( I 5  e 4 )
z 1

 s 1

 (519.34  22.20  43.96)(0.49)
 (0.07)(0.36)[383.15  (0.033)( 2854)]
 234.087.
D3
D2
D4
P1  D1 


2
(1  K ) (1  k )
(1  k ) 3
E3n
E1n
E 2n
2340




4
(1  c) (1  c)(1  k ) (1  c)(1  k ) 2
(1  k )
E5n
E 4n


.
3
4
(1  c)(1  k )
(1  c)(1  k )
22.5 The constraints
E1
(  1) P1 
 0,

1 c
1
 0.0566 D1  0.0486 D2  0.0417 D3  0.0358D4
 1.174E1  0.0539E 2  0.0463E3  0.0387E 4  0.034E5
2340
1 


1
  71.9
4 
(1  0.165)  1.06 
(22.17f)
22.5 The constraints
0.0566 D2  0.0486 D3  0.0417 D4  1.1728E2
0.0539E3  0.0463E4  0.0387E5 .LE.83.8.
0.0566 D2  0.0486 D3  1.1728E2  0.0539E4  0.0463E5 .LE.97.6.
0.0566 D2  1.728E4  0.0539E5 .LE.113.69.
1.1728E5 .LE.132.44.
D1.GE.51.092 ( D0 is given)
D2  1.06 D1.GE.0,
D3  1.06 D2 .GE.0,
D4  1.06 D3 .GE.0.
22.5 The constraints
D5
 0.36;
AFC5


Z
5


'
AFC5  (1   )  5   iz ( Lz 0  Cz 5 )  i5  DLs 
z 1
s 1


 B 1 B2 ( I 5  e 4 )
22.5 The constraints
i2  0.085
b2  117.2
i2 b2  9.962
i3
b3
i3 b3
0.0425
20
0.085
0.08
4.3
0.344
0.04875
0.08875
13.1
53.1
0.638625
4.712625
0.08
73.7
5.886
i3 b3  12.44125
DL5  488.4
AFC5  (0.49)519.34  9.962  12.44125  (0.09)( 488.41)
 (0.07)(0.36)383.15  (0.033)( 2854)
 221.96  12.03  233.99,
D5  AFC5  (0.36)  84.24.
22.5 The constraints
D4.  79.74
(22.17o)
22.5 The constraints
22.5 The constraints
Dt  1 AFCt  0 (t  0, ,5),
Dt   2 AFCt  0
(t  0,
,5),
D1  0.75 AFC1 .LE.0
D1  0.15 AFC1.GE.0;
D2  0.75 AFC2 .LE.0
D2  0.15 AFC2.GE.0;
D3  0.75 AFC3 .LE.0
D3  0.15 AFC3.GE.0;
D4  0.75 AFC4 .LE.0
D4  0.15 AFC4.GE.0.
22.5 The constraints
D1   0 AFC1  D2   0 AFC2  D3   0 AFC3
 D4   0 AFC4  D5   0 AFC5  0
D1  0.4 AFC1  D2  0.4 AFC2  D3  0.4 AFC3
 D4  0.4 AFC4 .LE.9.36.
(22.17f)
22.5 The constraints
22.5 The constraints
22.5 The constraints
22.5 The constraints
22.6 Analysis of overall results
22.6 Analysis of overall results
22.7 Summary and conclusion
In this chapter, we have considered Carleton's linearprogramming model for financial planning. We have also
reviewed some concepts of basic finance and accounting.
Carleton's model obtains an optimal solution to the wealthmaximization problem and derives an appropriate financing
policy. The driving force behind the Carleton model is a
series of accounting constraints and firm policy constraints.
We have seen that the model relies on a series of estimates
of future factors. In making these estimates we have
reviewed our growth-estimation skills from Chapter 6.
In the next chapter, we will consider another type of
financial-planning model, the simultaneous-equation models.
Many of the concepts and goals of this chapter will carryover
to the next chapter. We will, of course, continue to expand
our horizons of knowledge and valuable tools.
NOTES
 4.

Dn 
1 

 AFCn 
1  Dn

g
AFCn  AFCn 
En
NOTES
 6.
5.678 + 17.04 + (131.38)(0.09) = 34.542
(1979)
6.605 + 16.04 + (225.18)(0.09) = 42.911
(1980)
7.616 + 14.96 + (297.65)(0.09) = 49.365
(1981)
8.730 + 13.47 + (406.89)(0.09) = 58.820
(1982)
9.962 + 12.24 + (488.40)(0.09) = 66.158
(1983)
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
PROBLEM SPECIFICATION
MPOS VERSION 4.0
NORTHWESTERN UNIVERSITY
MP0S
VERSION 4.0
MULTI-PURPOSE OPTIMIZATION SYSTEM
***** PROBLEM NUMBER 1 *****
MINIT VARIABLES
Dl D2 D3 D4 El E2 E3 E4 E5 AFC1 AFC2 AFC3 AFC4 DL1 DL2 DL3 DL4
MAXIMIZE
.018Dl－.0196El＋.015D2－.017E2＋.013D3－.0144E3＋.011D4－.0125E4－.015E5
CONSTRAINTS
1.
AFC1＋.0441DLl .EQ. 149.17
2.
AFC2＋.0441DL2 .EQ. 173.45
3.
AFC3＋.0441DL3 .EQ. 198.22
4.
AFC4＋.0441DL4. EQ. 226.05
5.
DL1＋E1 .EQ. 131.38
6.
AFC1－D1＋DL2－DL1＋E2 .EQ. 255.7
7.
AFC2－D2＋DL3－DL2＋E3 .EQ. 264.3
8.
AFC3－D3＋DL4－DL3＋E4 .EQ. 302.3
9.
－AFC4＋D4＋DL4－E5 .EQ. 182.15
10.
DL1 .LE. 284 .42
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
PROBLEM SPECIFICATION (Cont.)
11.
DL2 .LE. 374.1
12.
DL3 .LE. 460
13.
DL4 .LE. 558.7
14.
DL1 .LE. 243. 6
15.
DL2－DL1 .LE. 303.15
16.
DL3－DL2 .LE. 329.1
17.
DL4－DL3 .LE. 365.1
18.
DL4 .GE. 101.15
19.
－.0566D1－.0486D2－.0417D3－.0358D4＋1.1740El＋.0539E2＋.0463E3＋.0387E4 ＋.034E5 .LE. 71.8
20.
－.0566D2－.0486D3－.04 17D4＋.1728E2＋.0539E3＋.0463E4＋.0397E55 .LE. 83.8
21.
－.0566D3－.0486D4＋1.1728E3＋.0533E4＋.046E5 .LE. 97.6
22.
－.0566D4＋1.7280E4＋.0539E5 .LE. 113.69
23.
1.1728E5 .LE. 132.44
24.
Dl .GE. 51.092
25.
D2－1.06D1 .GE. 0
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
PROBLEM SPECIFICATION (Cont.)
26.
D3－1.06D2 .CE. 0
27.
D3－1.06D3 .GE. 0
28.
D4 .LE. 79.47
29.
D1－.75AFC1 .LE. 0
30.
D2－.75AFC2 .LE. 0
31.
D3－.75AFC3 .LE. 0
32.
D4－.75AFC4 .LE. 0
33.
Dl－. 15AFC1 .GE. 0
34.
D2－.15AFC2 .GE. 0
35.
D3－.15AFC3 .GE. 0
36.
D4－.15AFC4 .GE. 0
37.
Dl－.4AFCl＋D2－.4AFC2＋D3－.4AFC3＋D4－.4AFC4 .LE. 9.36
,
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
SOLUTION
MPOS VERSION 4.0
NORTHWESTERN UNIVERSITY
PROBLEM NUMBER
USING MINIT
SUMMARY OF RESULTS
VARIABLE NO.
VARIABLE
NAME
BASIC NON-BASIC
ACTIVITY LEVEL
OPPORTUNITY COST
1
Dl
B
51.0920000
--
2
D2
B
54.1575200
--
3
D3
B
57.4069712
--
4
D4
B
60.8513895
--
5
El
NB
--
.0015408
6
E2
B
69.6152957
--
7
E3
B
82.4681751
--
8
E4
B
65.3689022
--
9
E5
B
77.4902713
--
10
AFC1
B
143.3761420
--
11
AFC2
B
163.5195372
--
12
AFC3
B
185.0936187
--
ROW NO.
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
SOLUTION (Cont.)
VARIABLE NO.
VARIABLE
NAME
BASIC NON-BASIC
ACTIVITY LEVEL
OPPORTUNITY COST
ROW NO.
13
AFC4
B
208.1059384
--
14
DL1
B
131.3800000
--
15
DL2
B
225.1805623
--
16
DL3
B
297.6503700
--
17
DL4
B
406.8948203
--
18
--SLACK
B
153.0400000
--
( 10)
19
--SLACK
B
148.9194377
--
( 11)
20
--SLACK
B
162.3496300
--
( 12)
21
--SLACK
B
151.8051797
--
( 13)
22
--SLACK
B
112.2200000
--
( 14)
23
--SLACK
B
209.3494377
--
( 15)
24
--SLACK
B
256.6301923
--
( 16)
25
--SLACK
B
255.8555497
--
( 17)
26
--SLACK
B
305.7448203
--
( 18)
27
--SLACK
B
69.1612264
--
( 19)
28
--SLACK
NB
--
.0002527
( 20)
29
--SLACK
NB
--
.0018351
( 21)
30
--SLACK
NB
--
.0018840
( 22)
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
SOLUTION (Cont.)
VARIABLE NO.
VARIABLE
NAME
BASIC NON-BASIC
ACTIVITY LEVEL
OPPORTUNITY COST
31
--SLACK
B
41.5594098
--
( 23)
32
--SLACK
NB
--
-.0087826
( 24)
33
--SLACK
NB
--
-.0089493
( 25)
34
--SLACK
NB
--
-.0069790
( 26)
35
--SLACK
NB
--
-.0039896
( 27)
36
--SLACK
B
18.6686105
--
( 28)
37
--SLACK
B
56.4401065
--
( 29)
38
--SLACK
B
68.4821329
--
( 30)
39
--SLACK
B
8l.4132428
--
( 31)
40
--SLACK
B
95.2280643
--
( 32)
41
--SLACK
B
29.5855787
--
( 33)
42
--SLACK
B
29.6295894
--
( 34)
43
--SLACK
B
29.6429284
--
( 35)
ROW NO.
Appendix 22A. Carleton’s linear-programming
model: General Mills as a case study
SOLUTION (Cont.)
VARIABLE NO.
VARIABLE
NAME
BASIC NON-BASIC
ACTIVITY LEVEL
OPPORTUNITY COST
44
--SLACK
B
29.6354987
--
( 36)
45
--SLACK
B
65.8902139
--
( 37)
46
- -ARTIF
NB
--
.0172964
( 1)
47
--ARTIF
NB
--
.0165658
( 2)
48
--ARTIF
NB
--
.0158661
( 3)
49
--ARTIF
NB
--
.0151960
( 4)
50
--ARTIF
NB
--
-.0180592
( 5)
51
--ARTIF
NB
--
-.0172964
( 6)
52
--ARTIF
NB
--
-.0165658
( 7)
53
--APTIF
NB
--
-.0158661
( 8)
54
--ARTIF
NB
--
.0151960
( 9)
MAXIMUM VALUE OF THE OBJECTIVE FUNCTION =
-1,202792
CALCULATION TIME WAS .0670 SECONDS FOR 21 ITERATIONS.
ROW NO.
Appendix 22B. General Mills’ actual key
financial data
Appendix 22B. General Mills’ actual key
financial data