from what if to what's best in dss

From What If to What's Best in DSS*

27

Asim ROY Arizona State University, Tempe, AZ 85287, USA

The author of IFPS/OPTIMUM presents some typical applications of optimization within decision suppor~ systems (DSS). They show that finding 'what's best' is gradually super- seding the 'what if' and 'goal seeking' type of analysis that is prevalent now. IFPS/OPTIMUM is now used by over 130 companies, including many of the Fortune 500 ones. The main drawback of 'what if' and "goal seeking' type of analysis is that one cannot impose restrictions on model variables and have the DSS generator solve for values of the 'what if' or 'goal seek' variables that satisfy those restrictions. Instead, one would have to muddle through many 'what ifs' or 'goal seeks' at random and check if the restrictiovs are satisfied. It is a manual process that is very time consuming. 'What's best' analysis with optimization allows one to impose such restrictions easily and do the analysis quickly. This type of analysis is playing an increasingly significant role within decision support systems. Practicing management scientists and managers should note this phenomenon. New tools and new ways of addressing problems are emerging.

Keywords: Optimization in DSS, Optimization in planning languages, Optimization application in business.

Asim Roy is the author of IFPS/OP- TIMUM, now marketed by Execucom Systems Corp., Austin, Texas. He has

!a Bachelor of Engineering degree from Calcutta University, India. He com- pleted his M.S. in Operations Re- search from Case Western Reserve University, Cleveland, and his Ph.D. in Operations Research from Univer- sity of Texas at Austin. He is currently an Assistant Professor of Management Science in the College of Business at Arizona State University. At Ex-

ecucom, he was manager of the operations research group. His main research interest is in the integration of optimization techniques with DSS languages and in making 'end-user" optimization a reality.

* The research was supported in part by the Faculty Research Development Award, Arizona ~tate University, and by the

• Department of Decision ana I.~ormation Systems Research Award, Arizona State University, 1986.

North-Holland Decision Support Systems 3 (1987) 27-35

1. Introduction

Optimization capabilities have recently been added to some decision support system (DSS) generators. The IFPS/OPTIMUM system was designed for the mainframe DSS generator called IFPS (see [6], [7], [9], [10]) and the VINO system optimizes models (only linear ones) built in mi- crocomputer DSS generators such as Visicalc, Lotus 1-2-3 and Multiplan [2]. The author of this paper is the developer of the pioneering optimization system, IFPS/OPTIMUM, and observed many of the applications of optimization within DSS. This paper reports on some of the DSS uses of optimization and presents simplified versions of the models actually used.

Many other individual applications of optimization within DSS have been reported. Russo and Sarker [11] reported one application of IFPS/OP- TIMUM at Exxon, and Roy, DeFalomir and Lasdon [4] reported an application at Ponderosa Industrial in Mexico. Roy and Huttner [8] reported two applications of IFPS/OPTIMUM at Prime Computers. Wagner [12] discusses the use of optimization within DSS for strategic planning. Carlsson [1] reports using 1FPS/OPTIMUM for solving a fuzzy muitiobjective programming problem. Geoffrion [3] has noted the importance of providing optimization capabilities to a variety of users in a new mode. Experience snows that providing these tools in the DSS toolkit is a very effective way to promote the use oi optimization.

The main purpose in providing optimization capabilities within DSS generators is to allow analysts to do more powerful model analysis than can be done with 'what if's' or 'goal seeking', t However, even a well-designed, user-friendly optimization system such as IFPS/OPTIMUM had

I In a model y •f(x), where x is a vector of variables, a typical 'what if' analysis changes one or more of the compo- nents in the x-vector and solves for y (where y could be a single variable or a vector of variables that is obtained as output). In 'goal seeking' (or backward solution), both x and y are single variables. One sets a target value for y, and solves for x. Goal seeking is therefore identical to 'what if' analysis (i.e., one sets a variable to some value and solves for another). Hence, any reference to 'what if' type of analysis in this paper would imply 'goal seeking' also.

0167-9236/87/$3.50 © 1987~ Elsevier Science Pubfishers B.V. (North-Holland)

28 A, Roy / From What l f to What's Best in DSS

difficulty in overcoming the fear of using a very complex tool. Much of that is changing as analysts learn to use optimization on the same models that were meant to be manually optimized using the 'what if' or 'goal seeking' mode of analysis. Two such models are presented next, ones that made the transition from 'what if' to 'what's best' mode of analysis. The main reason for the transition in these cases is that one cannot ask the 'what if' analyzer to obey restrictions on some model variables and solve for corresponding values of the "what if' variables. The analysts were frustrated

contains either data or an expression involving other model variables. For optimization, the IFPS model has some adjustable cells which contain the decision variables, and the model statements must compute the objective and constraints as functions of these decisions and of problem data. A directive set identifies which cells are decisions, constraints, and the objective, and specifies limits on the constraints and decisions.

To illustrate, the model and directive set for a simple product mix problem is shown. The model is self documenting.

100 COLUMNS PROD 1, PROD 2, TOTAL 110 PROFIT CONTRIBUTION = R E V E N U E - VARIABLE COST 120 REVENUE = QUANTITY* PRICE 130 VARIABLE COST = LABOR COST + PLASTIC COST 140 LABOR COST = LABOR HOURS* LABOR COST PER H O U R 150 PLASTIC COST = POUNDS OF PLASTIC* PLASTIC COST PER POUND 160 LABOR HOURS = QUANTITY* LABOR HOURS PER U N I T 170 POUNDS OF PLASTIC = QUANTITY* POUNDS PLASTIC PER UNIT 178 * * * 179 * * *QUANTITY WILL BE ADJUSTED BY OPTIMUM 180 QUANTITY = 5000, 6000 181 * * * 190 PRICE = 139, 189 200 LABOR HOURS PER UNIT = 0.5, 1 210 LABOR COST PER HOUR = 25 220 POUNDS PLASTIC PER UNIT- - 2,3 230 PLASTIC COST PER POUND = 7 240 COLUMN TOTAL FOR PROFIT CONTRIBUTION T H R U ' 250 POUNDS OF PLASTIC -- COLUMN PROD 1 + COLUMN PROD 2

having to try random solutions and then screening out the ones satisfactory to management.

Practicing managers and management scientists should note the availability of these new tools within the DSS tool kit. They can use these tools on many of their existing models and find better decisions for the organization more efficiently. This paper reports this new development through some applications. Following the overview of OP- TIMUM, the next two sections discuss two applications in detad.

The directive set identifies the spreadsheet cells of concern to OPTIMUM as the OBJECTIVE, the DECISIONS, and the CONSTRAINTS. This directive instructs OPTIMUM to maximize PROFIT CONTRIBUTION (in column TOTAL) by finding the optimal values of the decision variables, QUANTITY in columns PROD 1 and PROD 2, within the indicated bounds. The values of the decision variables are to be selected so that LABOR HOURS (in column TOTAL) and POUNDS OF PLASTIC (in column TOTAL) do not exceed the specified limits.

2. An Overview of OPTIMUM

OPTIMUM uses the IFPS planning language to define an optimization problem. IFPS is based on a matrix or spreadsheet, in which each cell

100 OBJECTIVE 110 MAXIMIZE PROFIT CONTRIBUTION (TOTAL) 120 DECISIONS 130 QUANTITY (PROD 1) BETWEEN 0 A N D 5000 140 QUANTITY (PROD 2) BETWEEN 0 A N D 6000

A. Roy / From What I f to What's Best in DSS 29

150 CONSTRAINTS 160 LABOR HOURS (TOTAL) .LE. 6000 170 POUNDS OF PLASTIC (TOTAL) .LE. 1800

Using this model, OPTIMUM diagnoses the linearity o~' the problem, generates the LP coefficient matrix, and transfers it to the linear programming solver. If the problem is nonlinear, it is solved by-a nonlinear optimization code. After solution of an LP, objective and right hand side ranges are available. In addition, IFPS offers ex- cellent reporting and graphics capabilities.

A major advantage of such a system is that problem generation, solution, and reporting are integrated. In the above example, the objective coefficients are computed in lines 110 through 150. The problem data in lines 190 through 230 is easily modified, and could be contained in a separate data file, or drawn from a data base using the IFPS DIMENSION database system.

A second advantage is ease of use. IFPS models are english-like, easy to understand, and utilize the spreadsheet format, which is widely accepted and understood. The ability to include many inter- mediate expressions in IFPS permits building a complex model in easy steps.

IFPS/OPTIMUM is mainly designed for small to medium size linear and nonlinear optimization problems. It recently added a mixed integer (linear) capability. However, many problems encoun- tered are nonlinear because of if-then-else logic, max, rain or round functions, etc. in the IFPS model. These lead to nondifferentiable, sometimes discontinuous optimization problems. Nonlinear programming algorithms designed for smooth problems often fail in these situations. It also cannot handle mixed integer nonlinear problems.

3. Collecting Accounts Receivable

Collection of accounts receivable has wide implications for the short term cash flow of a company. At one computer manufacturing firm, a separate group existed to follow up on accounts receivable. The receivables were classified by size (e.g., those under $10,000 and those over $10,000) and by age of the invoices (e.g., less than three months old, between three and six months old). The amount of time spent in collecting varied with size and age. This variation was also a function of

the difficulty of the collection process. The payoff from such efforts is often erratic. For instance, a vigorous follow up on a high value account did not always result in full payment.

Due to the rapid expansion of sales, it became more difficult for the staff to follow up adequately on the outstanding receivables. Before sanctioning additional staff, management called for an analysis of the collection operation to determine two things:

(1) the amount of additional staffing required, and

(2) opportunities for improvement in the collection operation.

That is, the management wanted to find if collection time was being used optimally, and if there was a need for additional staff when collection time is optimally allocated.

A simplified version of the model that was used for this analysis is shown in fig. 1. The collections staff provided information about the makeup of the receivable portfolio and the time requirements for each invoice type. Further refinement of the categories was done for the analysis. For example, value categories were increased from two to four. The amount of time required to follow up on each invoice type was based on actual experience. Estimates of the fraction that paid only part of their accounts (partial payment fractions) were also based on staff experience. Note that the partial payment fractions and the amount of partial payments may be considered as random variables. However, such uncertainties were not built into the model in order to keep the analysis simple. Also, it was assumed that the amount of collection in each category was linearly proportinal to the time spent. The model will now be explained in more detail.

The first three columns of the model (fig. 1) represent time periods. Column TOTAL is a special column used for summing variables over the first three columns. Lines 100 through 120 give the breakdown of existing invoices by age and value. This simplified model breaks down receivables into three ages categories [0 to 6 months old (HALF1), six to twelve months old (HALF2), and over one year old (OLDER)] and three value categories (under $10,000, between $10,000 and $100,000, and those over $100,000). Lines 150

30 A. Roy / From What l f to What's Best in DSS

I0 COLUMNS

20 * 30 * 40 * 50 * 60 * 70 * 80 *

90 *

HALF1 , HALF2 , OLDER , TOTAL

TO EXPLAIN THE COLUMNS: HALFI = INVOICE FROM THE CURRENT HALF YEAR HALF2 = INVOICE FROM THE SECOND HALF YEAR OLDER = INVOICE OLDER THAN ONE YEAR TOTAL = TOTAL OF ALL TIME PERIODS

NUMBER OF INVOICES IN EACH TIME CATEGORY BROKEN BY DOLLAR VALUE

I00 NUMBER LT I0 = 7000 , 2000 , 200 Ii0 NUMBER I0 TO I00 = 500 , 100, 20 120 NUMBER GT I00 = 150 , 50 , 20

130 * 140 * THE AMOUNT OF TIME EACH INVOICE TYPE REQUIRES FOR FOLLOW UP

150 TIME PER LT l0 = I , 0.5 , i 160 TIME PER I0 TO I00 ~ 2 , 3, 4.5 170 TIME PER GT I00 = 3, 4 , 6

180 *

190 * CALCULATE THE TIME REQUIREMENTS OF THE PORTFOLIO 200 TIME REQUIRED LTI0 = NUMBER LT I0 * TIME PER LT lO 210 TIME REQUIRED I0 TO lO0 = NUMBER I0 TO i00 * TIME PER I0 TO I00 220 TIME REQUIRED GT I00 = NUMBER GT I00 * TIME PER GT I00

230 * 240 * VALUE OF THE PORTFOLIO INCLUDING THE ISSUE OF PARTIAL PAYMENTS 250 PARTIAL I0 TO I00=30% * NUMBER I0 TO I00 , 50% * NUMBER I0 TO I00,0 260 PARTIAL GT I00 = 30% * NUMBER GT I00, 50% * NUMBER GT lO0 , 0 270 FULL VAL"E i0 TO 100 = NUMBER 10 TO I00 - PARTIAL I0 TO I00 280 FULL VALUE GT i00 = NUMBER GT I00 - PARTIAL GT I00

290 * 300 VALUE OF LT I0 = 8 * NUMBER LT i0 310 VALUE OF 10 TO I00 = 50 * FULL VALUE i0 TO I00 + 3 * PARTIAL I0 TO I00 320 VALUE OF GT I00 = 150 * FULL VALUE GT I00 + 6 * PARTIAL GT I00

330 TOTAL VALUE = SUM (L300 THRU L320) 340 * 350 * CALCULATE THE TIME SPENT ON COLLECTIONS AND AMOUNT COLLECTED 360 TIME ON LT I0 = I00 370 TIME ON I0 TO 100 = I00

380 TIME ON GT I00 = 100 390 TOTAL TIME SPENT = SUM (L360 THRU L380) 400 * 410 VALUE LT I0 COLLECTED = (TIME ON LT IO/TIME REQUIRED LT I0) * L300 420 VALUE i0 TO I00 COLLECTED = (L370/L210) * L310

4J0 VALUE GT 100 COLLECTED = (TIME ON GT 100/TIME REQUIRED GT I00) * L320 440 VALUE COLLECTED = SUM (L410 THRU L430) 450 *

460 DEFICIT ON LT I0 = TIME REQUIRED LT I0 - TIME ON LT i0

470 DEFICIT ON 10 TO i00 = TIME REQUIRED 10 TO I00 - TIME ON I0 TO I00 480 DEFICIT ON GT 100 = TIME REQUIRED GT I00 - TIME ON GT 100 490 *

500 COLUMN TOTAL FOR L390, L440 = SUM (COLUMN1THRU COLUMN3)

Fig. I. IFPS model.

through 170 specify the average hours required to follow up on an invoice of each type. Lines 200 through 220 calculate the total collection hours for all invoices. Lines 250 and 260 estimate the number of invoices that would make partial payments when followed up. It is assumed that full payments are made by accounts under $10,000. For the remaining categories, it is assumed that about 30% make partial payments in the first six months (HALF1) and it increases to 50% in the second six

months (HALF2). Lines 270 and 280 compute the number of invoices that are paid in full.

Lines 300 through 320 estimate the total col- lectible value of the accounts receivable portfolio, accounting for partial payments. The average partial payment estimates are based on past experience. In case of full payments, an average invoice value for the category is used. For instance, in line 300, the average invoice value for receivables under $10,000 is assumed to be $8,000. In line 310,

20 * 30 * OBJECTIVE - MAXIMIZE TOTAL COLLECTION BY OPTIMAL 40 * ALLOCATION OF COLLECTION TIME

50 * 60 MAXIMIZE VALUE COLLECTED (TOTAL) 70 * 80 * DECISIONS - HOW MUCH TIME TO ALLOCATE TO DIFFERENT 90 * CATEGORIES OF INVOICES IN DIFFERENT PERIODS

I00 * II0 DECISIONS 120 * 130 TIME ON LT I0, TIME ON 10 TO 100, TIME ON GT 100 (I-3) 140 ALL DECISION POSITIVE 150 * 160 CONSTRAINTS 170 * 180 * CONSTRAINT TYPE I - CANNOT ALLOCATE MORE TIME TO A 190 * CATEGORY THAN IS REQUIRED

200 * 210 DEFICIT ON LT 10 (I-3) .GE. 0 220 DEFICIT ON 10 TO 100 (I-3) .GE. 6 230 DEFICIT ON GT 100 (I-3) .GE. 0 240 * 250 * CONSTRAINT TYPE 2 - TOTAL TIME SPENT CANNOT EXCEED 260 * TIME AVAILABILITY

270 * 280 TOTAL TIME SPENT (TOTAL) .LE. 3900

Fig. 2. IFPS/OPTIMUM specifications.

? SOLVE DO YOU AGREE THAT THIS PROBLEM IS LINEAR (YES,NO OR STOP) ? YES PRINT INITIAL SOLUTION (YES OR NO) ? NO OPTIMAL SOLUTION ENTER SOLVE OPTIONS (OR END) ? ALL

HALFI HALF2 OLDER TOTAL

NUMBER LT 10 7000 2000 200 NUMBER 10 TO 100 500 100 20 NLq4BER GT 100 150 50 20 TIME PER LT 10 1 .5000 1 TIME PER I0 TO I00 2 3 4.500 TIME PER GT I00 3 4 6 TIME REQUIRED LT I0 7000 I000 200 TIME REQUIRED 10 TO I00 1000 300 90 TIME REQUIRED GT 100 450 200 120 PARTIAL I0 TO 100 150 50 0 PARTIAL GT 100 45 25 0 FULL VALUE 10 TO 100 350 50 20 FULL VALUE GT 100 105 25 20 VALUE OF LT 10 56000 16000 1600 VALUE OF 10 TO 100 17950 2650 I000 VALUE OF GT 100 16020 3900 3000 TOTAL VALUE 89970 22550 5600 TIME ON LT 10 740 I000 0 TIME ON I0 TO I00 I000 300 90 TIME ON GT I00 450 200 120 TOTAL TIME SPENT 2190 1500 210 VALUE LT I0 COLLECTED 5920 16000 0 VALUE I0 TO 100 COLLECTED 17950 2650 I000 VALUE GT i00 COLLECTED 16020 3900 3000 VALUE COLLECTED 39890 22550 4000 DEFICIT ON LT I0 6260 0 200 DEFICIT ON I0 TO I00 0 0 0 DEFICIT ON GT I00 0 0 0

ENTER SOLVE OPTIONS (OR END) ? END

Fig. 3. Selected IFPS/OPTIMUM solution output.

3900

66440

32 A. Roy / From What I f to What's Best in DSS

PRINT SENSITIVITY ANALYSIS (YES OR NO)

? YES

VARIABLE NAME OPTIMAL OBJECTIVE OPTIMAL RANGE (COLUMN NAME) SOLUTION COEFF (LOWER) (UPPER)

TIME ON LT i0 (HALF1) 740 8 TIME ON LT I0 (HALF2) I000 16 TIME ON LT I0 (OLDER) 0 8 TIME ON GT I00 (HALF1) 450 35.60 TIME ON GT I00 (KALF2) 200 19.50 TIME ON GT I00 (OLDER) 120 25 TIME ON 10 TO 100 (HALFI) I000 17.95 TIME ON i0 TO i00 (HALF2) 300 8.833 TIME ON 10 TO I00 (OLDER) 90 11.11

PRINT SHADOW PRICES (YES OR NO) ? YES

CONSTRAINT VARIABLE COLUMN

DEFICIT ON LT I0 NALFI DEFICIT ON LT I0 HALF2 DEFICIT ON LT I0 OLDER DEFICIT ON i0 TO I00 HALF1 DEFICIT ON i0 TO I00 HALF2 DEFICIT ON I0 TO 100 OLDER DEFICIT ON GT I00 HALFI DEFICIT ON GT I00 HALF2 DEFICIT ON GT I00 OLDER TOTAL TIME SPENT TOTAL

8 8.833

8

8

8

8

8

8

8

FINAL VALUE

6260 0

200 0 0 0 0 0 0

3900

SHADOW PRICES

0 - 8

0 - 9 . 9 5 0 - . 8 3 3 3 - 3 . 1 1 1 - 2 7 . 6 0 - 1 1 . 5 0

-17 8

VALUE TO ENTER

Fig, 4. Sensitivity analysis and shadow prices.

the average invoice value for receivables between $10,000 and $100,000 is estimated to be $50,000, and the average partial payment amount is assumed to be $3,000. Line 330 computes the total value for the three value categories. Lines 360 through 380 represent the decision variables for this model. The manager wants to find the optimal allocation of collection time among the different invoice types. Line 390 calculates the total allocated time. Lines 410 through 430 compute the expected collections for the different categories given the time allocations in lines 360 through 380. These relations reflect the critical assumption that the amount collected in each category is proportional to the time allocated. Line 440 computes the total collection. Lines 460 through 480 calculate the time short-fall from a given time

allocation. Line 500 defines the special column (TOTAL) calculation for the variables TOTAL T1ME SPENT and VALUE COLLECTED. The values for these variables are summed over all three columns.

In the actual case, attempts were made to find a satisfactory allocation using 'what if' analysis. However, out of frustration, the analysts finally resorted to IFPS/OPTIMUM to find the desired allocation more efficiently. Fig. 2 shows the IFPS/OPTIMUM specifications required to opti- mize the simplified model. Line 60 specifies the objective. Line 130 declares the decision variables of the problem - how much time to allocate to the different invoice types. Lines 210 through 230 specify constraints that ensure that the time allocated to a category does not exceed its require-

A. Roy / From What l f to What's Bes~ in DSS 33

ment. The constraint in line 280 ensures that the total time allocated is within the availability limit. Fig. 3 shows selected solution output. It shows that the mode' is diagnosed as a linear. In that case, a linear programming routine is used to solve the problem. Fig. 4 shows the corresponding sensitivity analysis output and shadow prices.

Sensitivity analysis answers the question, 'How sensitive is the optimal solution to changes in certain key variables?' The first column of the sensitivity analysis output in Fig. 4 shows the optimal values of the decision variables and the second column (Objective Coeff column) shows the unit contribution to the objective value of each decision variable. These unit contributions are computed by OPTIMUM from the model equa- tions. For the decision variable TIME ON LT 10(HALF1), the sensitivity analysis results in the columns of 'Optimal Range' state that the optimal solution would be unchanged for values of the objective coefficient between 8 and 8.833. For all the other decision variables (except TIME ON LT 10(OLDER)), the optimal solution is unchanged unless their objective coefficients (unit contributions) fall below 8. The sensitivity analysis essentially shows how much leeway you have in the objective coefficients, without affecting the optimal decision values.

The shadow price displayed in Fig. 4 answers the question 'How much will the objective change if I change a constraint limit?' For example, the shadow price of the constraint DEFICIT ON GT 100 (HALF1) is -$27.6, meaning that VALUE COLLECTED would drop by -$27.6 if the deficit is forced to be at least 1. The shadow prices essentially show which constraints affect the objective and how much, and therefore merit a closer examination.

The actual analysis verified the correctness of the existing policy of pursuing high value accounts first. It also showed that existing manpower is inadequate to follow up on the increasing volume of accot,nts receivable and that adding staff was justified. The management accepted and imple- mented the recommendations resulting from this analysis.

4. Dividend Policy Analysis

This application dealt with a controversial financial policy question - how much dividend

should a company pay. Financial experts usually have a variety of recommendations on this question. The chemical firm in this case did not have a formal corporate dividend policy. But certain fundamental changes in the company's financial and operating positions, such as increasing cash flow from a new acquisition, dictated such a need. After considering several factors, the company decided to adopt a policy of paying a stable and increasing dividend - an approach generally used by many companies. Within this policy frame- work, the company was interested in finding a stable dividend growth rate that could be main- tained in good and bad years, since the market would otherwise penalize the company's stock price if there is uncertainty in the dividends. In addition, it wanted the dividend policy to be in 'harmony' with other corporate goals such as growth in earnings per share, shareholder equity growth, net income growth, etc.

A financial model similar to the ove in fig. 5 was constructed to analyze the impact of van/ing the dividend growth rate (line 280). The input to the model came from the company's latest five-year plan. It essentially involved income statement and capitalization data (lines 20, 30, 40, 60, 100, 240, 290, 320, 370). The model allowed the growth rate in dividends to vary (line 290). It assumed that when dividends exceed plan estimates (line 300) they are financed by debt (line 70). If a dividend payout different from the plan is made, the model recomputes the different ratios and performance parameters pertaining to the company. They in- elude such items as the adjusted net income to common stock holders (line 90), adjusted common equity (line 150) and adjusted outstanding debt (line 170) and their percentages with respect to capitalization (lines 180, 200), earnings per share with increased dividends (line 230), net income growth rate (rate 350), shareholder equity growth rate (line 380), etc. The model essentially helps to accentuate the long-range implications of any dividend payout policy (line 280).

The model was initially supposed to take advantage of 'what if' and 'goal seeking' features of IFPS to examine alternative policies. The analyst soon realized that the processs was extremely time consuming and that he could not enforce restrictions on certain variable values within those techniques. Hence he resorted to IFPS/OPTIMUM to find the best dividend payout policy. A number of

I0 COLUMNS 1981-1985 20 OUTSTANDING DEBT = 3623,3390,3542,3290,2965 30 PREFERRED STOCK = 792,698,647,624,599 40 COMMON STOCK = 1561,1765,1876,1977,2139 50 CAPITALIZATION PER PLAN = OUTSTANDING DEBT + PREFERRED STOCK + COMMON STOCK

60 NET INCOME PER PLAN = 239,317,327,440,619 70 AFTER TAX INTEREST EXPENSE = 0.54 * 0.12 * ' 80 (INCREASED IN TOTAL DIVIDENDS OVER PLAN/2 + PREVIOUS CUMULATIVE INCREASE IN DIVIDENDS) 90 ADJUSTED NET INCOME TO COMMON = NET INCOME PER PLAN - AFTER TAX INTEREST EXPENSE 100 SHARES OUTSTANDING = 38.543,41.231,45.352,46.532,51.324 110 DEBT TO CAPITALIZATION PER PLAN = (OUTSTANDING DEBT/CAPITALIZATION PER PLAN) * 100 130 EQUITY TO CAPITALIZATION PER PLAN = (COMMON STOCK/CAPITALIZATION PER PLAN) * 100 150 ADJUSTED COMMON EQUITY ~ COMMON STOCKS -' 160 CUMULATIVE INCREASE IN DIVIDENDS - AFTER TAX INTEREST EXPENSE 170 ADJUSTED OUTSTANDING DEBT = OUTSTANDING DEBT + INCREASE IN TOTAL DIVIDENDS OVER PLAN 175 ADJUSTED CAPITALIZATION = CAPITALIZATION PER PLAN - AFTER TAX INTEREST EXPENSE 180 ADJUSTED DEBT TO CAPITALIZATION = 100 * (ADJUSTED OUTSTANDING DEBT/ADJUSTED CAPITALIZATION) 200 ADJUSTED EQUITY TO CAPITALIZATION = 100' (ADJUSTED COMMON EQUITY/ADJUSTED CAPITALIZATION) 220 EPS PER PLAN = NET INCOME PER PLAN/SHARES OUTSTANDING 230 EPS WITH INCREASED DIVIDENDS = ADJUSTED NET INCOME TO COMMON/SHARES OUTSTANDING 240 DIVIDENDS 1980 = 64.521 250 DIVIDENDS PAID = DIVIDENDS 1980 *' 260 (1+DIVIDEND GROWTH RATE/100), PREVIOUS * (1+DIVIDEND GROWTH RATE/100) 270 DIVIDENDS PER SHARE = DIVIDENDS PAID/SHARES OUTSTANDING 280 DIVIDEND GROWTH RATE = 10,PREVIOUS 290 DIVIDENDS PER PLAN = 58.546,63,457,69.654,74.321,85.324 300 INCREASE IN TOTAL DIVIDENDS OVER PLAN = DIVIDENDS PAID - DIVIDENDS PER PLAN ~10 CUMULATIVE INCREASE IN DIVIDENDS=PREVIOUS + INCREASE IN TOTAL DIVIDENDS OVER PLAN 320 NET INCOME 1980 = 79.986 330 NET INCOME AFTER TAX INTEREST EXPENSE = NET INCOME PER PLAN - AFTER TAX INTEREST EXPENSE 350 NET INCOME GROWTH RATE = ((L330~NET INCOME 1980)/NET INCOME 1980)'100,' 360 ((L330-PREVIOUS L330)/PREVIOUS L330)*100 365 CUMULATIVE NET INCOME GROWTH RATE = PREVIOUS + NET INCOME GROWTH RATE 370 COMMON EQUITY 1980 = 687.0 380 SHAREHOLDER EQUITY GROWTH RATE" ((ADJUSTED COMMON EQUITY - COMMON EQUITY 1980)/' 400 COMMON EQUITY 1980)*I00,((LIS0-PREVIOUS LI50)/PREVIOUS LIS0)*I00

Fig. 5. IFPS model.

10 *

20 MAXIMIZE CUMULATIVE NET INCOME GROWTH RATE (1984) 30 * 40 DECISIONS 50 DIVIDEND GROWTH RATE (1980) .GE. 0 60 * 70 CONSTRAINTS 80 EPS WITH INCREASED DIVIDENDS (1981-1985) .GE. 6.2 90 CUMULATIVE INCREASE IN DIVIDENDS (1984) .CE. 57 i00 SHAREHOLDER EQUITY GROWTH RATE (1981 THRU 1984) .GE. 4.7% ?SOLVE DO YOU AGREE THAT THIS PROBLEM IS NONLINEAR (YES OR STOP) ?YES PRINT INITIAL SOLUTION (YES OR NO) ?NO LOCAL OPTIMUM FOUND ENTER SOLVE OPTIONS (OR END)

?L280, L365, L380, L230, L310

1981 1982 1983 198~ 1985

DIVIDEND GROWTH RATE 8.022 8.022 8.022 8.022 8.022

CUMULATIVE NET INCOME GROWTH RATE 177.9 2!2.2 215.7 252.0 294.3

SHAREHOLDER EQUITY GROWTH RATE 125.5 12.35 5.662 4.71! 7.875

EPS WITH INCREASED DIVIDENDS 6.209 7.653 7.184 9.41 12

CUMULATIVE INCREASE IN DIVIDENDS 11.15 22.98 34.66 48.19 57.77

Fig. 6. IFPS/OPTIMUM specifications and the optimal solution.

A. Roy / From What I f to What's Best in DSS 35

different OPTIMUM formulations were actually used to arrive at the final dividend policy. Fig. 6 shows one such formulation that maximizes the cumulative net income growth over the 5-year planning horizon (line 20), keeps the earnings per share at an acceptable level (line 80), wants the shareholder equity growth rate to be more than 4.7% (line 100) and desires the cumulative increase in dividends to be at least 57. Given these restrictions, OPTIMUM finds the optimal dividend growth rate to be 8 percent, as shown in fig. 6. Management initially thought the dividend rate could grow at 10 percent.

If there was a multiple objective optimization capability in OPTIMUM, then this application would have been well suited for it. The analysts were actually trading-off between the different objectives in the different formulations. In each formulation, all the objectives except one were stated as constraints with lower bounds on them. An acceptable dividend policy emerged from the analysis.

5. Conclusions

Users of DSS tools are slowly realizing the value of optimization. They are gradually finding out the limit~,tions of 'what if' and 'goal seek' type of analysis where restrictions cannot be imposed on certain model variables and a backward solution done to find satisfactory values for the decisions. In addition, they are realizing that perfor- ming optimization manually by trying different 'what if's' on the decision variables is a waste time, and that the power of mathematics em- bedded in the optimization tools could do a better analysis more efficiently.

From the many applications seen so far, it can be concluded that optimization will play an increasingly useful role in the analysis of DSS models. From the author's contacts with many of the DSS vendors, it seems that a number of other leading languages are planning to include optimi-

zation capabilities in the f,:.ure. They would help to upgrade the 'what if' and 'goal seeking' mode of analysis to the 'what's best' mode with concur- rent increases in productivity of the analysts and an enrichment of the DSS toolkit.

Practicing managers and management scientists should be aware of these emerging tools. As shown in this paper, they can open up new application horizons.

References

[1] Carlsson, C., Fuzzy Multiobjeetive Programming With Composite Compromises, presented at the Task Force Meeting on Multiobjective and Stochastic Optimization at ilASA (Nov.-Dec., 1981).

[2] Cunningham, C., and L. Schrage, Optimization in Spreadsheets with VINO (Scientific Press, Palo Alto, CA, 1985).

[3] Geoffrion, A.M., Can MS/OR Evolve Fast Enough, In- ter/aces 13 (1983) 10-25.

I4] Roy, Asim, Emma De Falomir and Leon Lasdon, An Optimization-Based Decision Support System for a Prod- uct Mix Problem, Interfaces 12, No. 2 (April, 1982) 26-33.

[5] Roy, Asim, and Emma De Falomir, Selecting Products and Quantities: Making an Optimal Decision, Financial Case Report Series (Execucom Systems Corporation, Austin, TX 78766, 1981).

[6] Roy, Asim, Decision Support Systems - From Simulation to Optimization, Proceedings of the Fourteenth Hawaii International Conference on Systems Sciences 1 0981).

[7] Roy, Asim, Introducing a Nonlinear Optimizer into a Financial Planning Language, Tenth International Sym- posium on Mathematical Programming (Montreal, Aug., 1979).

[8] Roy, Asim, and Terrie Huttner, Optimization' with Deci- sion Support Systems at Prime Computers - Two Ap- plications, ORSA/TIMS Joint National Meeting (Hous- ton, Oct., 1981).

[9] Roy, Asim, Optimization and Planning Languages, Fourth European Congrezs on Operations Research (Cambridge, England, July, 1981).

[10] Roy, Asim, L. Lasdon, and J. Lordeman, Extending Plan- ning Languages to Include Optimization Capabilities, Management Science 32, No. 3 (March, 1986) 360-373.

[111 Russo, G., and S. Sarkar, A DSS for Plannig Petrochem- ical Production, Third International DSS '83 Transactions, G. Huher, ed. (June, 1983) 45-53.

[12] Wagner, G.R., Optimizing Decision Support Systems, Datamation (May, 1980)209-214,

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