Special Applications Issue || The Use of Models in Marketing Timing Decisions

The Use of Models in Marketing Timing DecisionsAuthor(s): Sidney W. HessSource: Operations Research, Vol. 15, No. 4, Special Applications Issue (Jul. - Aug., 1967), pp.720-737Published by: INFORMSStable URL: http://www.jstor.org/stable/168282 .

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THE USE OF MODELS IN MARKETING

TIMING DECISIONS*

Sidney W. Hesst

University of Pennsylvania, Philadelphia, Pennsylvania

(Received February 25, 1966)

Two cases illustrate the utility of simple models to aid pricing of obsolescent products. Both models yield surprisingly simple, easily implemented decision rules. They demonstrate that 'back of the envelope' modeling can still help management decision making.

THE FOLLOWING case studies illustrate the utility of simple mathematical models to

-make assumptions explicit -provide rational decision rules -specify necessary data collection

and thereby guide Atlas Chemical Industries, Inc., management faced with real marketing decisions.

Both cases concern pricing of obsolescent products-one a consumer product, the other an industrial commodity. Each was completed in less than 3 man-weeks.

CASE I-PROBLEM BACKGROUND

RECENTLY Atlas decided to introduce MODERN, slightly modified from our existing product, ANCIENT. Priced lower to be competitive, MODERN

would eventually replace ANCIENT in our product line. Because of the substantially lower price of the new product, we anticipated that retailers and wholesalers would find it difficult to sell the old product and would return substantial quantities of it for credit. On the other hand, current customers of ANCIENT would not change to MODERN immediately, so some sales of the higher priced product would continue.

The fundamental problem treated here is what strategy should Atlas use to minimize the impact of returned ANCIENT on profits?

Courses of Action

Three courses of action were proposed. All assumed a salvage value on returned ANCIENT.

* Presented at 12th Annual Marketing Conference, National Industrial Confer- ence Board, October 30, 1964.

t Formerly Manager of Operations Research, Atlas Chemical Industries, Inc., Wilmington, Delaware.

720



Models in Marketing Timing Decisions 721

Course of Action No. 1-Price Reduction

At some time in the near future the price of ANCIENT would be reduced to that of MODERN. At that point losses on inventory value sustained by wholesalers and retailers would be made up by Atlas in equivalent quantities of free MODERN. As the two products would now be equally priced, the major reason for returned goods should vanish and returns to us should disappear. This is a common practice in our industry.

Course of Action No. 2-Recall of Ancient

At some time in the near future, all ANCIENT remaining in wholesale and retail channels would be called back by us. Again the wholesaler and retailer would be reimbursed in free goods.

Course of Action No. 3-Do Nothing

We would continue to accept ANCIENT returns and credit at wholesale price but would not incur the expense of free goods.

Objective

Because of the costs incurred in providing free goods in courses of action No. 1 and No. 2, it should be clear that minimizing returns is not the real objective for ANCIENT strategy. The objective should be to maxi- mize total profit. This will be shown subsequently to be equivalent to minimizing the sum of the following costs:

1. Cost of returns. 2. Cost of free goods.

Model

The flow of ANCIENT and MODERN between Atlas, the wholesaler, the retailer and customer is illustrated in Fig. 1.

It is possible to construct a mathematical expression (i.e., model) showing Atlas profit measured from the time of the MODERN introduction and re- lated to the time of either

-A price reduction in ANCIENT, or -Recall of all ANCIENT

Price Reduction Model

While we will leave mathematical complexities to the appendices, it is still useful to describe the general structure of the models. First we can write an expression for the profit generation valid from the time of MODERN'S

introduction until just before the ANCIENT price cut. Profit per Month= (marginal profit per unit of MODERN) X (net unit



722 Sidney W. Hess

sales of MODERN per month)-(cost per unit of ANCIENT returned) X (net units of ANCIENT returned);

Unit Marginal Profit on MODERN = (selling price to wholesaler) - (variable cost of manufacturing) - (freight);

Unit Cost of Returned ANCIENT= (selling price to wholesaler) + (freight) - (salvage value).

At the time of the price change (to be determined) we inject a quantity of free MODERN into the distribution pipeline. This quantity is determined so as to compensate the wholesalers and retailers for their lost profits as a result of the price decrease. The quantity of free goods required will be proportional to the outstanding inventories of ANCIENT and will be introduced at a unit cost equal to the variable cost of manufacturing plus freight.

rl~~~~~ R1 C wholesale Sales >

0 E ~~~~~~~~~~~U A Net Returns L T Reta il Sales > S

T E ~~~~~~eurs A T

L s ~~~~~~~~~~~~~~~~~~0

S Net Mfgr. Sales L Wholesale Sales E Retail Sales E _

_ _ __ _ _ _ __ _ _ > E -----------------_ _ _ > R ------- _> _______i R

Fig. 1. Flow of ANCIENT and MODERN. (Solid lines are ANCIENT-

Dashed lines are MODERN)

Subsequent to the price decrease, Atlas' profit will be only (unit marginal profit on MODERN) X (unit sales of MODERN). The latter unit sales, however, will be influenced by the free goods placed in the pipeline at the time of the price change. At one extreme, we could argue that every free unit of MODERN given away will reduce future sales of MODERN by one unit. That is to say a retailer receiving a unit free will, in the future, have to order one less unit from his wholesaler. On the other hand, overloading the pipeline with free goods may yield some sales not otherwise attainable. The same retailer may make a sale because he had an extra unit in stock that he would not have had otherwise. Conceivably, this extra retail sale might lead to additional future business, or a full shelf of MODERN may encourage retailers to sell harder. Both of these reasons are implicit in decisions to promote by free samples. Nevertheless, one cannot give products away forever.

This model assumes that some fraction (between 0 and 1) of the free goods 'steals' from future Atlas sales of MODERN. If the fraction were 0.8, it would imply that the future Atlas sales of MODERN must be reduced



Models in Marketing Timing Decisions t723

by 80 per cent of the quantity of free goods.* This is not the sort of cost reported in a P. & L. statement, but it is nonetheless a real loss of sales otherwise attainable. It is relevant to the price reduction decision and is therefore considered in the model.

Having expressions for profit and costs before, during and after the ANCIENT price reduction, we can add these (by integration) and obtain an expression for the total profit as a function of the time of the price decrease [equation (A4)]. Maximizing this profit equation is shown to be equivalent to minimizing the sum [equation (A5)] of the following costs:

1. Cost of returned ANCIENT, 2. Cost of free goods, 3. Lost profits from free goods replacing potential Atlas sales.

ANCIENT Retail Sales Rate

i.J ~ k Times the Return Rate

0 Time

Optimal Time for Price Reduction

Fig. 2. Sales and k-times-Returns vs. Time.

In Appendix A we find that the optimal time for a price cut is either when:

1. The cost of returns equals the savings from field inventory reductions, or 2. Never.

The first condition occurs when the following two curves first intersect:

1. ANCIENT retail sales rate, 2. A constant, k, times the ANCIENT return rate [equation (A12)].

The constant k [equation (Al1)] depends on wholesale prices of the two products, manufacturing costs, freight, salvage value of ANCIENT, the free goods ratios (i.e., the units of free goods per unit of ANCIENT) and, to a lesser extent, the proportion of free goods not 'stealing' future ANCIENT sales.

* In the actual case we found the decision to be insensitive to values in the range 0.5 to 1.0. This model assumes that the low priced ANCIENT steals from competitive sales. Alternate assumptions are possible differing in detail but not in form.



724 Sidney W. Hess

Without knowing anything about the future retail sales and returns, one can argue that the retail sales will generally decrease. Returns (or returns times a constant) will start at 0, increase, and then drop off as inventories become exhausted. Figure 2 illustrates this, showing the two curves as a function of time.

Should these curves not intersect, in other words, if the retail sales are always at a higher rate than k times the return rate, it is cheaper to accept the returns and not decrease the price. In Fig. 3 total cost is plotted as a function of the time of price change. The characteristics of this curve are derived in Appendix A. If the curves in Fig. 2 intersect, the cost curve will show a minimum at the first intersection, a maximum at the second intersection, and a minimum at the third intersection. The third intersection will generally be the point at which both retail sales and returns are zero.

Second Intersection

Total Third Cost C| Intersection

-L First Intersection

0O 0 Time of Price Reduction

Fig. 3. Total cost vs. time of price reduction.

Note that as yet we have not said which of the minimums (the first intersection or the third intersection) is the lower.

Course of Action No. 2 Recall of Ancient

The model for ANCIENT recall is constructed similarly and has an anal- ogous solution to that of the previous model.

The optimal time for recalling ANCIENT is either when:

1. The cost of returns equals the savings from field inventory reductions, or 2. Never.

Again, the first condition occurs when the following two curves first intersect:

1. Its retail sales rate, 2. A different constant, h, times its return rate [equations (A18) and (A20)].




The constant, h, depends on costs, prices, etc., as does k. It will generally be less than k. When it is less, 'h times the returns' will always be below the 'k times the returns' curve. As shown in Fig. 4, the price reduction intersection (in our case) precedes the recall intersection. Aside from data collection, no action on a price cut or recall is required before this first intersection. The intersection of both curves can be forecast by extrapolating monthly data. Returns multiplied by the appropriate constants will generate the lower two curves in Fig. 4. An inventory audit will give data points on the retail sales curve. One has only to extrapolate to determine the decision points.

Just prior to the first intersection, sufficient data will have been collected to estimate whether the optimal action is to cut the price, wait and recall

ANCIENT Retail Sales Rate

o

k Times 4, . (Return I at

hTimes Return _ B?- Rate

0 T1 61

Time

Fig. 4. Determining decision times.

ANCIENT later, or do nothing. In particular, it is shown in Appendix A that doing nothing is cheaper than a price cut if the total retail sales beyond the first intersection (labeled T1) are more than, k times the total returns beyond T1. Also, doing nothing is cheaper than recalling ANCIENT if the total retail sales beyond 01 are more than h times the total returns beyond 01. Estimates of these future returns and retail sales are most easily ob- tained from graphs like those in Fig. 5.

In summary, if inventory audits are used to measure retail sales and if returns are recorded, a rational basis for deciding if and when the following courses of action should be taken would be available:

1. Reduce the price of ANCIENT,

2. Recall all ANCIENT from the field, or 3. Do nothing.



726 Sidney W. Hess

Numerical Example

For purposes of illustration and to get some feel for the magnitude of the costs involved, we made some simple assumptions concerning returns and retail sales patterns.

1. Retail ANCIENT sales will be proportional to outstanding inventory. 2. ANCIENT returns would increase asymptotically to become proportional to

outstanding inventory in approximately 12 months, after which returns would be about two-thirds as great as retail sales.

As described in Appendix B these assumptions permitted us to forecast

Units \ Cumulat ive

Retail Sales

Initial T otal Future Inventoryl\ I Retail Sales

Retail of ANCIENT - - - - - -

Inventory , Total Future

I I. ,, Returns -- --- - -- - -- -- - - - t

Cumulat ive Returns

v 1 0 Tl ~~~~~~~Time

Fig. 5. Extrapolating returns and sales.

the retail sales and returns shown in Fig. 6. Prior year ANCIENT sales are also shown. In Fig. 7, k times the returns are plotted with the retail sales forecast. Note that the k curve intersects in April, Year 2.

In the actual case h was negative. Therefore the h curve was negative and never intersects. Hence the recall of ANCIENT was not optimal.

The total cost of a price reduction as a function of the time of the reduction is plotted in Fig. 8. It can be observed that the minimum cost is incurred in April, Year 2. The maximum cost would result from a price reduction at the time of MODERN'S introduction in July, Year 1. An incorrect decision could thus entail a loss equivalent to one-month sales dollars. Of course, different retail sales and return patterns would have different cost functions, and the savings using this technique could be more or less.




Postscript

At the time the study began we felt a decision to reduce the ANCIENT

price was imminent. We think the results of our three-week study were partially instrumental in delaying such a decision.

To the surprise of all, promotion of MODERN stimulated sales of ANCIENT

sufficiently that returns never rnaierialized. The decision not to cut the ANCIENT price was a good one. Both products are still being produced and sold.

YeCar 1 Aver age Sales by Atlas

I Average Retail Sales Dec., Year 2 Through May, Year 3

Year 2 Average Sales w4 by Atlas\

Time of MODERN IntroductionJ

0 At las|

Year 1 Year 2 Year 3 Year 4 Year 5

Fig. 6. Estimated ANCIENT retail sales and return rates.

CASE II

WE SOLD two commodities for the same end use. The newer low-priced commodity, CHEAP, COUld generally be used in place of the traditional corn- fModity, DEAR. Gallon for gallon, the effectiveness of the two were nearly equal. To switch from DEAR to CHEAP, however, the customer had to make substantial one-time changes in equipment and procedures. Gradually, more and more customers were making the switch. They could not resist the price differential.

Unfortunately, we made less profit on a gallon of CHEAP, and the lower price was not increasing the total market. Would, however, a price cut for



728 Sidney W. Hess

DEAR slow down the penetration of CHEAP sufficiently to increase our total profits? Alternatively, would an increased price for DEAR-assuming other manufacturers follow-be more profitable in spite of an increase in the rate of market penetration of CHEAP?

Q3 \ r ANCIENT Retail Sales Rate

k Times | X Return

Year. Year 2 Year 3

Fig. 7. Optimum time for price cut.

Essentially, we needed to know how the price differential affected 'commodity switching.' Because the differential had always been constant, an analysis of historical data was not fruitful. An analogy from chemical kinetics was.

4J ~ ~~~~~~ mum Cost

t4

0 E-4

Deffbrence in the

Order of Minimum

One-Month's Cost Sales

Year l Year 2 Year 3

Time of Price Cut

Fig. 8. Total cost as function of price reduction.

In the simplest sort of chemical reaction, say the conversion of A to B, the rate at which A reacts to B is proportional to the amount of A present. As more and more A is reacted, the number of A molecules converting per



Models in Marketing Timing Decisions f729

second decreases. The mathematics of this chemical kinetics analogy predict that the proportion of A unreacted plots as a straight line against time on semilog paper. The slope of this line depends on the particular reaction and external conditions, for example: temperature. Generally,

100.. x

60-

Examplewith smaller price differential

4o -\ \ \0"

Extrapolation without price

a0 ~~~~~~~~~~~~~~~change

Exal \with greater price \ differential\

20-\

x Historical Data

0 After-the-Fact \

10 u i

0 1 2 3 4 5 6 7

Period

Fig. 9. DEAR Market share vs. time.

reactions of the type we are considering speed up when we increase the ten- perature, and the line we plot would be steeper.

In our analogy, A would be DEAR customers or market; B would be customers who have switched from DEAR to CHEAP. The rate at which they



730 Sidney W. Hess

switch-if the analogy followed-would be proportional to the number of DEAR customers remaining.

Historic data for the six prior periods showed the straight line character- istic of the kinetics model (Fig. 9). Extrapolations are also shown with

55

04~~~~~~~~~~~~~~~~~~~~

of i a

Avg. Price +to square a

co 45 'op *

c/Z

WS 2

6 7 8 9 10 11 12 13

DEAR Selling Price $t gallon

Fig. 10. Profit as a function o DEAR price.

and without changes in the price differential. The solid line has actually been used for forecasting.

We did not know how the slope of Fig. 9 changes with changing price diff erential. Again, we relied on our knowledge of the physical processes. Electrical current flow is proportional to the voltage driving force. Simi- larly, heat flow-proportional to temperature differences-and fluid flow-- proportional to pressure drops. In a few circumstances the change is proportional to the square of the driving force.

We assumed the slope in Fig. 9 to be proportional to some unknown




power of the price differential. We next added prices and variable costs of the two commodities and the overall market growth rate to our model of customer switching to yield a simple equation expressing cash flow as a function of time. Future cash flows were discounted to the present and summed (by integration).

This was repeated for different DEAR prices resulting in Fig. 10. The four lines relate to the assumed relation between Fig. 9 slope and price differential. In all four cases-powers higher than the third have no physical analogy-profits would increase with higher DEAR prices.

While the higher price would accelerate customer switching from DEAR

to CHEAP, larger short-run profits from DEAR would more than compensate. We should not cut the DEAR price, but try to increase it. Marketing personnel-not the model-would have to judge whether competition would let a price increase stick.

Management felt our conclusion was obvious-to them if not to us. Fortunately, the study only took a couple of weeks and had a useful by- product-the forecasting technique.

CONCLUSIONS

MATHEMATICAL marketing models are frequently useful in predicting the consequences of pricing and promotional strategies. They force us to make assumptions explicit and guard us from fuzzy thinking. Their solution will often result in rational decision riles and clearly specify the data necessary for correct application of the decision rules. Frequently they will only sub- stantiate management intuition, but even that is not without value.

APPENDIX A

DERIVATION OF THE DECISION RULES

AT TIME, t = 0, a lower priced product (MODERN) is introduced nationally to replace the existing, but similar, product (ANCIENT). Retail sales of ANCIENT continue but at a decreasing rate as customers switch to MODERN. Excess ANCIENT is gradually returned to the manufacturer for credit. It is assumed that the manufacturer, after I = 0, no longer sells ANCIENT. *

Two mutually exclusive courses of action are available to the manufacturer to prevent the returns:

1. At time T decrease the price of ANCIENT to the price of MODERN making up the wholesaler's and retailer's losses with equivalent quantities of free MODERN.

* If the manufacturer elects to continue the sale of ANCIENT, the model is still valid, but returns should be the net of actual returns from wholesalers less sales to wholesalers.



732 Sidney W. Hess

2. At time 0 call in all ANCIENT from the field making up wholesaler's and retailer's losses with equivalent quantities of free MODERN.

A third course of action is, of course, to do nothing and accept the returns. Let S = S(t) =unit sales (or returns) of ANCIENT per month,

S'=S'(t) =unit sales (or returns) of MODERN per month, I =I(t) = inventory of ANCIENT in the field.

All of these variables are functions of time, t. To indicate the direction and/or loca- tion of the movement or inventory we will use the following subscripts:

m = manufacturer, w = wholesaler, r = retailer, c = customer.

Thus Smw'(T) would be manufacturer's unit sales per month of MODERN to wholesalers at time T, Ir(T) would be retail inventories of ANCIENT at time T, and AN-

CIENT sales are illustrated in Fig. 11.

Al H wr E N L r F M S I

S 0

E

R

Fig. 11. ANCIENT Flow.

Unit prices and costs are given by p =ANCIENT price to wholesaler, p =MODERN price to wholesaler, cv variable manufacturing cost of ANCIENT,

C. =variable manufacturing cost of MODERN,

c =salvage value (before tax basis) for returned ANCIENT,

Cf =freight on ANCIENT,

Cf = freight on MODERN.

We will construct a mathematical model of the manufacturer's variable profit as a function of action times, T or 0, and then determine the values of T or 0 maximizing this profit.

Course of Action No. 1-Price Reduction

At any time before the ANCIENT price cut, the manufacturer's variable profit per month will be the difference between marginal profits from MODERN and the cost of returned ANCIENT. The total profit up to the price cut is the integral of this profit per month. Hence

T T

Profit from time 0 to Tf (P' C'- Cf')Smw'dt (P+Cf C8)Swmdt (Al)




At the time of the price cut, T, an amount of free MODERN proportional to the ANCIENT inventory in the field at that time will be introduced at a cost of c,' +cf' per unit. Letting

a =units of free MODERN per unit of ANCIENT in the field then

Cost of free goods = (cv' +c/)a[I (T) +I?(T)]. (A2)

The injection of free MODERN will have an important influence on later manufacturer sales in that much of the free product will substitute for what would have otherwise been purchased from the manufacturer. This, however, may not always be the case. The free goods may in some cases put MODERN on the shelf in stores that would not otherwise stock it. We will let

a=the fraction of free goods that 'steal' normal manufacturer sales*

MODERN total profits subsequent to the ANCIENT price cut will then be the unit profit times the net sales, i.e., those achievable without free goods, less those lost because of the injection of free goods. Hence

Profit from time T to o = (p -C V- CY){LSmw'dt -ia[Iw(T) +Ir(T)]} (A3)

Subtracting equation (A2) from the sum of (Al) and (A3), total profit as a function of T, is given by

Z(T) = J (pi ct Cf')Smw' dt- (p +ACf-C)Swm dt ?o ?o (A4)

-t[#pt +l (-O )(Cvt +c)][Iw(T) +Ir(T)]. The first term is total MODERN profits without free goods and is independent of

T. Hence it has no bearing on T and can be ignored. Instead of maximizing Z(T) we can minimize the last two cost terms, with signs reversed. Thus, our maximiza- tion problem reduces to minimizing the cost of returns plus the cost of free goods (both out-of-pocket and the cost of 'stolen' MODERN sales). Denoting this cost as L(T),

AC T

L(T) = (p +Cf Cs)f Swm dt +a[/p' + (1 -/) (cv' +cf')][Iw(T) +Ir(T)]. (A5)

Conditions for minimum loss are

dL(T)/dT = 0, (A6)

and d2L(T)/dT2 > 0. (A7) * In this formulation it is assumed the ANCIENT sales at the new low price would

'steal' from competitive sales, not from future MODERN sales. Other assumptions can be modeled and result in changes in (All) but not (A12) or (A13). Carefully de- signed long-term experiments might provide estimates of 3, but more likely-and in this particular case study-marketing personnel made subjective estimates of the numerical value.



734 Sidney W. Hess

But

dL(T) /dT = (p +Cf c8)Swm(T)

+a dp' + (1 -A) (cat icf')] { (d/dt) [Iw(T) + I,(T)] (A8)

and the rate of change of inventory is given by a material balance around the wholesaler and retailer, or

(d/dt)[Iw(T) +Ir(T)] = -Src(T) -Swm(T). (A9)

Substituting (A9) into (A8) and equating to zero

(p +cf -c)Swm(T) = ao[Op' + (I -f ) (Ci' +Cf )][Src(T) +Swm(T)] (A0)

This condition (A1l0) for an extrernum states that the cost of returns (left-hand side) must equal the savings in inventory reduction (right-hand side).

AVj7 Src

k S7

0 0 Time, t

Fig. 12. Sales and k-times-IReturns vs. Time.

If we combine constants by letting

k + 1 = (p +Cf-ce) a4/[#3p + (I -3) (c' +Cf')], (Al 1)

(Al0) will reduce to kSwm(T) =Src(T). (A12)

By (A7), (A12) will be a minimum if

k[dSwm(T)/dT] >dSrc(T)/dT. (A13)

For a numerical solution, one has only to plot the two sides of (A12). Their intersection will be at Ti. Values of Swm, the returns, can be tabulated by the manufacturer. Retail sales, Src can be determined from. periodic inventory audits. Only k must be estimated from (All).

Note that this derivation is entirely void of numerical assumptions. We can learn still more if we consider a graphical representation of (A12).

In the figures below kSwm and Src are each plotted schematically. It is reason- able to assume that the retail ANCIENT sales will decrease with time and that the returns will start at zero, increase to some maximum value and then decrease.

In Fig. 12, the curves never intersect until the inventory is exhausted. Sr, must approach zero with a slightly steeper slope than kSwm. Thus, the intersection at zero is a minimum., and the optimal time for an ANCIENT price cut is when the




inventory is exhausted, i.e., no price cut is optimal. The argument preceding (A16) also proves this. The conclusion is similar if k <0 and the kS& curve ap- proaches zero from below the time axis.

In Fig. 13 the curves intersect, in fact they intersect three times. The first

intersection, T1 is a minimum by (A13) because the slope of the return curve is posi-

Src

k Swm

Tj T2 TTime, T

Fig. 13. Sales and k-time-s-Return vs. Time.

tive and the slope of the retail sales curve is negative. Also by (A13) T2 is a maxi- mnum and the third intersection at T- oc is a minimum. This is illustrated in Fig. 14.

While at this point we do not know which of the two minimums is lowest, we can see that no price decrease is required before the first minimum at Ti. By plot-

maximum

Cost, < maximum L(T)

minimum -minimum

Tj T2

Time for Price Reduction, T

Fig. 14. Total cost vs. time of price reduction.

ting the returns and retail sales month by month, we can forecast this first intersection and, in addition, be able to estimate if L( 0) is less than, L(T1). If it is less, doing nothing is preferable to cutting the ANCIENT price.

Let a=p+cf -cs.

Then by (A5) and (All)

L x) =I Stm dt, (A14)



736 Sidney W. Hess

and T a

L(TM)N= a Swm dt+_a 4Iw(Tl)+Ir(Ti)I

=a~o Swmdt~k+1~t Swmdf~l Sre d0] (A15)

L( oo) <L(TY)>af Swm dt<k+lf Swumdt+k41f Sr& dt I I I~~~~T (AlO)

.xc4~kfSwm dt< S dt.

(A16) states that it is cheaper not to cut the price if k times the total returns from T1 on is less than the total retail sales from T1 on.

Courses of Action No. 2-Ancient Recall

This model differs from the previous one only to the extent that at some time 0 we call back all outstanding ANCIENT and have correspondingly higher free goods. The total cost to be minimized, G(8) is similar to (A5) and given by,

G(8) = (p +cf cs)f Swm dt + I [p'+(l (I -j3)(c' +cf')1 (A17)

+(cf -c,) } [1w(8) +1(8)1, where

1 =units of free MODERN per unit of ANCIENT in the field. Letting

h + = (p+cf -c,)/ t{el[p' +(1 -3)(cv +c/)1 +cf' -c, (A18) then

G() a 'Su d+7 a Iw(0)+Ir(8)]. (A19)

This is the same equation form as (A8) and, therefore, has a similar solution, namely that 0 such that

h&t(O) - e(O), (A20)

and h[dS,(8)/d8 >d&r(8)!d8. (A21)

For the particular numbers for this case (All) and (A18) show k >h. This means that the first minimum for policy No. I occurs before the first minimum for policy No. 2, i.e., T <81.

Hence, data can be collected up to T1, before a decision is required whether to

1t cut the price of ANCIENT at T1, 2. wait and recall all ANCIENT at 8l, or 3. do nothing.

Note, of course, that by an argument similar to the one preceding (AI6)

L(,0) =G( 00) <G(8i><z*h SW00 dt <fa 8ro dt. (A22) 1 1t




APPENDIX B

AN EXAMPLE

OUTSIDE OF the very general patterns sketched in Appendix A, no information was available to predict how ANCIENT retail sales and returns would proceed following the MODERN introduction. For purposes of illustration, and to get some feel for the magnitude of the costs of an incorrect decision, the following two rela- tions are assumed.

First, we assume that retail sales will decline in proportion to the decline in the wholesale and retail inventory. Thus if I(t) =IQ(t) ?Ir(t) and b is the constant of proportionality, then

$c- =bI(t). (Bl)

Second we assume that ultimately returns will also be proportional to inventory but since they must start at a rate of zero they reach this proportionality through an exponential transient or growth. Thus

Swm = gj (t) (1edit), (B2)

where g and o- are appropriate constants. Substituting (Bi) and (B2) into (A9).

dI(t)/dt = -bI(t) -gIQ(t) (1 -6et). (B3)

This is a separable differential equation whose solution is

In [I(t) = - (b ?g)t - (g/a)e7t +a constant. (B4)

Since I(t) =I(O) at t=0 the constant is evaluated to be g/x +ln [I(0)]. Hence

I(t) =I (O) exp E-(b +g)t +(g/ff)(1 -e7-7f)]. (B5)

From (B 1), at t = 0 and data available from recent retail sales audits

b =Src(O)/I(O) = 0.0892.

Now somewhat arbitrarily, for illustrative purposes, we assume that the transient of Swm is 95 per cent dissipated after 12 months. Therefore,

1 -e-20 - 0.95

or a =0.25.

Finally at steady state (i.e., for large t) it is assumed that the return rate will be hi as much as retail sales. Thus

g=3s b =0.0595.

Using these parameters 1(t), Swm and Src were computed and the latter two plotted in Fig. 6.



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