Financial Analysis, Planning and Forecasting Theory and Application By Alice C. Lee San Francisco State University John C. Lee J.P. Morgan Chase Cheng F. Lee Rutgers University Chapter 21 Elementary Applications of Programming Techniques in Working-Capital Management

Outline  21.1Introduction  21.2Linear programming  21.3Working-capital model and short-term financial planning  21.4Goal programming  21.5Programming approach to cash transfer and concentration  21.6Summary 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 (Objective function), (21.1)

21.3Working-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

21.3Working-capital model and short-term financial planning TABLE 21.1 ToyMachine Time (hours)Assembly Time (hours) Krazie Kube Plastic Pistol Race Car

21.3Working-capital model and short-term financial planning

(21.2) (21.3) (21.4) (21.5)

21.3Working-capital model and short-term financial planning (21.6) (21.6a) (i = 1, 2, 3) (21.7)

21.3Working-capital model and short-term financial planning (21.8).

21.3Working-capital model and short-term financial planning

(21.9) (i = 1, 2, …, m), (j = 1, 2, …, n).

21.3Working-capital model and short-term financial planning (21.10) (j = 1, 2, …, n), (i = 1, 2, …, m).

21.3Working-capital model and short-term financial planning (i = 1, 2, 3, 4)

21.4Goal programming  Introduction  Application of GP to working-capital management  Summary and remarks on goal programming

21.4Goal programming (21.11a) (21.11b) (21.11c)

21.4Goal programming (21.11d) (21.11e) (21.11f)

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

21.5Programming 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

21.5Programming approach to cash transfer and concentration

TABLE Managing about the target balance

21.5Programming approach to cash transfer and concentration

(21.12) (21.13) (21.14)

21.5Programming approach to cash transfer and concentration (21.15) (21.16) (21.17)

21.5Programming approach to cash transfer and concentration (21.18) (21.19) (21.20)

21.5 Programming approach to ash transfer and concentration

21.5 Programming approach to cash transfer and concentration

21.6Summary 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 goal- programming 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 linear- programming 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)

(21.A.2a) (21.A.2b) (21.A.2c) (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: Subject to:

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 = $ ; 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: X 8 = 5X 1 -36X X 3 -47X (60- X 5 ) -4.5(30- X 6 )+0.95X 7 = 75; (21.B.1) Current ratio: (21.B. 2) Quick ratio: (21.B. 3)