Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 21
Elementary Applications of Programming Techniques in
Working-Capital Management
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
Outline

21.1 Introduction

21.2Linear programming

21.3 Working-capital model and short-term financial planning

21.4 Goal programming

21.5 Programming approach to cash transfer and concentration

21.6 Summary and conclusion remarks

Appendix 21A. The simplex algorithm for solving eq. (21.8)

Appendix 21B. Mathematical formulation of goal programming
21.1 Introduction
21.2 Linear programming
Z  c1 x1  c2 x2  c3 x3  ...  cn xn
(Objective function),
a11 x1  a12 x2  a13 x3  ...  a1n xn  b1 ,
a21 x1  a22 x2  a23 x3  ...  a2 n xn  b2 ,
am1 x1  am 2 x2  am 3 x3  ...  amn xn  bm ,
xj  0
(j  1, 2, . . . , n).
(21.1)
21.3 Working-capital model and short-term
financial planning
 Questions
to be answered
 Model specification and its solution
 Which constraints are causing bottlenecks?
 How much more profit is being lost because
of constraints?
 How do the constraints affect the solution?
 Duality and shadow prices
 Short-term financial planning
21.3 Working-capital model and short-term
financial planning
TABLE 21.1
Toy
Machine Time (hours)
Assembly Time (hours)
Krazie Kube
Plastic Pistol
Race Car
6
3
4
5
2
3
21.3 Working-capital model and short-term
financial planning
21.3 Working-capital model and short-term
financial planning
Maxπ  x1  2 x2  3x3
(21.2)
6 x1  3x2  4 x3  100
(21.3)
5 x1  2 x2  3x3  60
(21.4)
x2  x3  5
(21.5)
21.3 Working-capital model and short-term
financial planning




Cash available  Marketable  Accounts
for operations
Securities
Receivable
1
Accounts  Loans
Payable
Outstanding



(21.6)
 200  60  10  10  10 x1  2 x2  3x3   0  50  1
0  70
10 x1  2 x2  3x3  100
xi  0
(i = 1, 2, 3)
(21.6a)
(21.7)
21.3 Working-capital model and short-term
financial planning
Maximize  = x1  2 x2  3x3
(21.8)
6 x1  3x2  4 x3  100,
5 x1  2 x2  3x3  60,
.
x2  x3  5
10 x1  2 x2  3x3  100
 x1, x2 , x3  0
21.3 Working-capital model and short-term
financial planning
21.3 Working-capital model and short-term
financial planning
n
Maximize z   c j x j
j 1
n
a x
j 1
ij
j
 b1
xj  0
(21.9)
(i = 1, 2, …, m),
(j = 1, 2, …, n).
21.3 Working-capital model and short-term
financial planning
m
Minimize y   bi ui
(21.10)
i 1
m
a u
i 1
ij i
 cj
ui  0
(j = 1, 2, …, n),
(i = 1, 2, …, m).
21.3 Working-capital model and short-term
financial planning
6u1  5u2  10u4  1
3u1  2u2  u3  2u4  2
4u1  3u2  u3  3u4  3
ui  0
(i = 1, 2, 3, 4)
min y  100u1  60u2  5u3  100u4
21.4 Goal programming
 Introduction
 Application
of GP to working-capital
management
 Summary
and remarks on goal programming
21.4 Goal programming

1 1

2

3
Minimize Y  Pd  P2 d  P2 d  P3d

4
(21.11a)

1
3 X  Y  d  15 (hours),
(21.11b)

2
X  d  10 (units of product X ),
(21.11c)
21.4 Goal programming

3
Y  d  6 (units of product Y ),
(21.11d)

4
10 X  8Y  d  7 (revenue),
(21.11e)

1

2

3

4
X ,Y , d , d , d , d  1
(21.11f)
21.4 Goal programming

 

 $5  units Y sold   $4.5  units Y sold 
for cash
for credit
 $7.5  units Z sold   $6.5  units Z sold 
for cash
for credit
 (0.1)($35)  ending inventory   (0.1)($45)  ending inventory 
of Y
of Z
 (0.05)  value of   (0.05)  ending cash 
loan
balance
Max profits  Revenue for less Any costs charged
the period
to the period
21.4 Goal programming
21.4 Goal programming
*Profit has a much higher priority than the working capital goal.
** The working capital goals have a much higher priority than the profit goal.
*** The priorities for all goals are similar.
21.4 Goal programming
21.5 Programming approach to cash transfer
and concentration
 Transfer
mechanisms
 Cash-transfer Scheduling: contemporary
practice
 Weekend timing and dual balances
 Limitations of the popular techniques
 Mathematical-programming formulation
 Relation of model formulation to current
practice
21.5 Programming approach to cash transfer
and concentration
21.5 Programming approach to cash transfer
and concentration
21.5 Programming approach to cash transfer
and concentration
TABLE 21.10
Managing about the target balance
21.5 Programming approach to cash transfer
and concentration
21.5 Programming approach to cash transfer
and concentration


TSC  CPT Number of  7  I  EXCESS - ADBAL 
transfers
(21.12)
1 7


TSC  CPT    i  7  I  EXCESS   AT
i i
7
i 1
i 1

 (21.13)
7
1 7
   ABALi  Required balance
 7  i1
(21.14)
21.5 Programming approach to cash transfer
and concentration
7
1 7
TCDB


ABALi  TAB + 

  i
7 i 1
 ECRDB  i 1
(21.15)
7
1 7
TCDB


ABAL

EXCESS
=
TAB
+

i

  i
7 i 1
 ECRDB  i 1
(21.16)
7
7
 T   AA
i 1
i
i 1
i
(21.17)
21.5 Programming approach to cash transfer
and concentration
LBALi  LMINi and ABALi  AMINi
(21.18)
Ti  LBALi 1   Transfers yet to clear   LMINi c
(21.19)
Ti  LBALi 1  Transfers yet to clear   LMINi  c
  Amount anticipated 
(21.20)
21.5
Programming
approach to
ash transfer
and
concentration
21.5
Programming
approach to
cash transfer
and
concentration
21.6 Summary and conclusion remarks
In this chapter, we have looked at a variety of
financial-management problems and their solution
through mathematical-programming techniques. As
we have seen, linear-programming and goalprogramming are very useful. We have also
considered certain working-capital problems,
including cash concentration and scheduling. In the
next chapter we will again be using our linearprogramming skills in long-range financial planning.
We will use our knowledge gained from this chapter,
in combination with other information, as inputs to
our financial-planning models.
Appendix 21A. The simplex algorithm for
solving eq. (21.8)
Appendix 21A. The simplex algorithm for
solving eq. (21.8)
 6  0  x1   3  4  x2   4  4  x3  1  0  x4
  0  0  x5   0  4  x6   0  0  x7  100  20
(21.A.2a)
 5  0  x1   2  3 x2   3  3 x3   0  0  x4
 1  0  x5   0  3 x6   0  0  x7  60  15 (21.A.2b)
10  0  x1   2  3 x2   3  3 x3   0  0  x4
  0  0  x5   0  3  x6  1  0  x7  100  15 (21.A.2c)
1  0  x1   2  3 x2   3  3 x3   0  0  x4
  0  0  x5   0  3  x6   0  0  x7
(21.A.2d)
Appendix 21A. The simplex algorithm for
solving eq. (21.8)
Appendix 21B. Mathematical formulation of goal
programming
Following is a list of definitions of all variables used in the GP formulation
of the working-capital problem:
Appendix 21B. Mathematical formulation of goal
programming
2 This appendix is reprinted from Sartoris, W. L., and M. L. Spruill, “Goal programming and working capital
management,” Financial Management (1974): 67-74, by permission of the authors and Financial
Management.
Appendix 21B. Mathematical formulation of goal
programming
These weights are defined in Table 21.6 for ach of the three sets of priorities.
Using these definitions, the GP problem is formulated as follows:
Minimize: P1 X 9  P2 X10  P3 X11  P4 X12  P5 X13  P6 X14  P7 X15
Subject to:
Appendix 21B. Mathematical formulation of goal
programming
Appendix 21B. Mathematical formulation of goal
programming
The following list defines the constraint given by each row in the
constraint matrix:
Row 1: Profit plus downside deviation = $2698.94;
Row 2: Time used in production at most 1000 hours;
Row3: At most 60 units of Y drawn from inventory;
Row 4: At most 30 units of Z drawn from inventory;
Row 5: At most 150 units of Y sold for cash;
Row 6: At most 100 units of Y sold on credit;
Row 7: At most 175 units of Z sold for cash;
Row 8: at most 250 units of Z sold on credit;
Row 9: Total cash goals 9;*
Appendix 21B. Mathematical formulation of goal
programming
Row 10: Inventory loan constraint;*3
Row 11: Current ratio goal;*†
Row 12: Quick ratio goal; †
Row 13: Constraint requiring cash to be nonnegative;
Row 14: Sales of Y for cash plus sales of Y for credit must be
greater than or equal to Y drawn from inventory;
Row 15: Sales of Z for cash plus sales of Z for credit must be
greater than or equal to Z dawn from inventory.
*The numbers on the right-hand side include not only the goal but also constants carried to right-hand side
of the equality from left-hand side.
† Both ratio goals have been linearized by multiplying right-hand side by denominator of ratio.
Appendix 21B. Mathematical formulation of goal
programming
Cash: X8 = 5X1 -36X2 +7.5X3 -47X4 -3.5(60- X5)
-4.5(30- X6)+0.95X7 = 75;
(21.B.1)
Current ratio:
X 8  40 X 2  52.5 X 4  35(60  X 5 )  45(30  X 6 )
2
150  X 7
(21.B. 2)
X 8  40 X 2  52.5 X 4
1
Quick ratio:
150  X 7
(21.B. 3)