zi IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. EM-20, NO. 1, FEBRUARY 1973

and some of this work's concepts. A committee consisting of I. G. Durand, R. W. Mayo, J. A. Newburger, and S. M. Sutton were most instrumental in applying the cost relevance tree procedure towards planning underground plant programs.

REFERENCES [1] J. V. Sigford and R. H. Parvin, "Project PATTERN: A

methodology for determining relevance in complex decision-making," IEEE Trans. Eng. Manag., voL EM-12, pp. 9 - Î3, Mar. 1965.

[2] M. E. Esch, "Planning assistance through technical evaluation of

Abstrvct-Mmy models of research and development (R&D) proj­ects do not consider the intermediate outcomes and decisions that may be foreseen to arise during their evolution through the technical and commercial stages. Consideration of the sequential aspects of allocating scarce resources to a set of projects may make a great difference to the solution of the problem in terms of the optimum subset of projects to work on in the immediate future. The solution of this problem is important to R&D management

An approach is described based on the application of a stochastic linear-programming formulation to a portfolio of projects, each planned using a form of the decision i ee structure called a "project tree." A series of elementary examples are presented as a means of gaining insight into the method. The general formulation is then presented in detail and applied to a small problem.

INTRODUCTION

A NUMBER of authors have suggested the application of programming techniques to aid the selection of an

optimal portfolio of research and development (R&D) projects (e.g., see Asher [1], Beattie [2], Bell [3], Dean [4], and Lockett [5] ). Models of this type have been analyzed in some detail by Gear [6]. A shortcoming of nearly all these models is the implication that only the current decision is relevant to the evolution of each project.

In practice, a number of decisions may be involved during the technical and/or commercial stages of each project

Manuscript received November 11, 1971; revised July 5, 1972. This paper was presented at the National Conference of the Operational Research Society, Lancaster, England, 1971.

A. E. Gear is with The Open University, Bletchley, Bucking­hamshire, England.

A. G. Lockett is with the Manchester Business School, The Uni­versity of Manchester, Manchester, England.

relevance numbers,*' in Proc. 17th Nat. Aerospace Electronics Conf. (Dayton, Ohio, May 10 -12, 1965), pp. 346 - 351.

[3] A. L. Jestice, "Project PATTERN," paper presented to the Joint Nat Meet Operations Research Society of America and the Institute of Management Sciences, Minneapolis, Minn., Oct 7 - 9, 1964; also Pamphlet, Honeywell, Inc., Washington, D.C., 25 pp.

[4] N. Dalkey and O. Helmer, "An experimental application of the Delphi method to the use of experts," Manag. Sci., voL 9, no. 3, pp. 458-467, Apr. 1963.

[5 ] H. Southworth, Jr., "Sensitivity analysis for cost relevance trees," unpublished report available by request from author.

[6] J.E.Freund and F.J.Williams, Elementary Business Statistics - The Modern Approach. Englewood Qiffs, N.J.: Prentice-Hall, 1964, p. 206.

opportunity. A common example is the decision to undertake a feasibility study followed by a reappraisal to consider

[ continuation, termination, or postponement. Furthermore, I decisions to acquire additional resources at some cost, for , example, by recruitment of manpower, may be envisaged at s future times.

In recognition of this shortcoming, Hess [7] and Rosen [8] | have described models based on a dynamic programming

formulation in several time neriods to renresent the sequential nature of the process. These models do not allow resource

i constraints in time periods after the first to be included, and computational difficulties usually arise in large multiple resource problems.

Lockett [9] has presented an alternative approach based on f the use of a decision-tree structure to represent each oppor-i tunity in an R&D laboratory. This allows the multistage nature s of each project to be displayed and alternative portfolios of 1 decision trees to be studied, particularly with regard to 5 finalizing a selection of projects to work on in the immediate s future. Lockett [9] describes a method of analysis based on - simulation in what is essentially a heuristic approach. While

this may yield good solutions in practice, it does neglect g aspects of the sequential nature of the problem, particularly t the order in which uncertainties are resolved with respect to

decision options. :. Using the representational framework of decision trees, the à present paper is aimed at the development of a rigorous ;- method of analysis based on stochastic linear programming.

The field of stochastic programming has been comprehensively surveyed by McQuillan [10], and the philosophy of the

A Dynamic Model of Some Multistage Aspects of Research and Development

Portfolios ANTHONY E. GEAR AND A. GEOFF LOCKETT

GEAR AND LOCKETT: SOME MULTISTAGE R&D PORTFOLIOS 23

Key:

Decision Node

£ °< Probability

Probability

Estimated Resource Consumption

[ Amount of ' Amount of | Type 1 ; Type 2 |

I PERIOD!"""!

<H

I PERIOD 2~"l I PERIOD 3 ~ | I I I ! I i

D

-»[ 3 , 3

! +ΓΤ-

+ΠΣ

-CX

Fig. 1. Example project tree.

present approach has similarities to that applied by Wagner [11] to a production-scheduling problem. A series of illus­trative examples of increasing complexity are used to demon­strate the technique.

THE BASIC MODEL

The approach makes use of a decision-tree format (e.g., see Hespos [12] and Harvey [13]) to describe each project, known as a "project tree." This allows every project opportu­nity, each with its own pattern of choice and chance-branching points, to be diagrammatically planned on a common time scale divided into discrete subperiods. It is assumed that the sub periods are sufficiently fine to allow the placement of "milestones"' in the life of each project at the beginning of a subperiod. An example project tree is given in Fig. 1, which extends over three subperiods of time. This project has two feasible resource allocations (i.e., two versions) in time period 1 culminating in an uncertain intermediate technical outcome. If the project is started with the higher resource allocation and the most likely (probability = 0.7) chance outcome results, then a further decision point is foreseen.

The method allows each project to be flexibly planned, and a great variety of project types can be depicted. For example, the relative merits of a parallel or series strategy (e.g., see Abernathy [14]), alternative technical approaches, a delayed start, or alternative rates of resource usage can be explored for each and every project in one analysis. The method is able to represent uncertainties in the duration, resource inputs, technical outcomes, and commercial outcomes of each project whether current or new. Projects involving development, testing, modification, retesting, and so on, can also be included in the framework.

STOCHASTIC ANALYSIS

The resource-allocation problem with which this paper is concerned may be stated as: to what subset of activities should limited resources be applied in the short term (first time period) in order to be on an optimal path in terms of

e \HDÌ Θ Θ Lnry ΐτ»ΓΗ ΓΠΠ4

'ΠΡ-^ΠΠ--ΠΠ^

t = 2 (Β)

-GZK

; t = 3 ! (C)

Fig. 2. Problem 1.

maximizing the overall expected value of the sum of terminal values for the projects eventually completed. A full analysis must take account of the following.

1) The multistage nature of each project. 2) Overall resource availabilities and associated uncertainties. 3) Overrunning resource availabilities and the penalties

involved. 4) The attitude towards risk taking of the decision taker. The present paper develops a formulation to cover point 1)

of the above in detail. However, the structure of the formulation is such as to present no fundamental problems with regard to the addition nf points 2) and 3Ì. for the availability column is no more than a special sort of project with its own uncertainties and decisions regarding availability stretching over time analogously to the projects themselves. The introduction of an objective function other than the maximization of an expected portfolio value does, however, introduce theoretical problems [point 4)].

Before proceeding to consider the general multiproject, multitime period problem, a series of two project problems are presented as a means of gaining insight. As a large number of variables and constants are introduced by these problems, they are defined with the help of the figures. The x variables each refer to a decision path from a square node on a given project, and cither two or three subscripts, and sometimes a super­script, are used. The first subscript defines the project number, the second subscript defines the first-stage decision route, while the third subscript (if used) defines the second-stage decision routes following on from the first stage, if they exist in the problem. The superscripts p and q used in problem 3 are explained during the fornulation of that problem. The constants aXi bXi cY, a2, b2, sta, refer to the amount of resource required for a given project in a time period asdefined in the figures; probabilities of following various paths are shown as p, q, r, s; vl9 v2, etc., define the values of reaching a given end point. The five problems are shown in Figs. 2 - 6 and a formulation of each is given below, assuming each of the variables take a value of 0 or 1 only. Problem 1

This problem (shown in Fig. 2) concerns a project with an alternative starting decision and a second project with an alternative decision at the start of the second time period. No uncertainties are involved. A single resource is considered with availabilities A,B9 and C in time periods i = l , 2, and 3, respectively. The first three constraints ensure that the

ÏEEE TRANSACTIONS ON ENtiïNfcfcKÎNu MANAGEMENT, fEBRUARY 1973

π ^

PROJECT 2 Q-i2i- [_a2_}

EPH

PROJECT 2 Q *211 [ a 7

Fig. 3.

resource availabilities are not exceeded in any of the time periods; the fourth constraint ensures that only one decision path may be chosen initially for project 1 ; the last constraint ensures that ή project time 2 is chosen in the first period then it is continued in the other periods.

Maximize:

[»1*11 +^2*12 + ^ 3 ^ 2 1 1 + i ; 4 ^ 2 1 2 ]

Subject to:

01*11 + 02*12 + 03*21

bxxxx + £2*12

Fig. 4. Problem 3.

CxXXi + c3X2i <C

£1*11 + ^4*2 1 < C

c 2 * n + c3x2X <C

C2X\ 1+^4*2 1 < £ *

period 3.

^1*1 1 + C2*12

X l l + *12

Problem 2

* 2 1

<A : period 1

+ b3x2 11 + £4*212 <B : period 2

+ c3x2 11 + C4X212 < C : period 3

< 1

= 0. *2 1 1 *2 12

This problem (shown in Fig. 3) involves a probabilistic hrp.nrhino rsninf fnr pnr-h nrniprt hni in different t ime periods.

The probabilities are given as p, q, r, and s, and the pair from

#1*1 1 +#2*2 1 u5-*2\ * w ' l - * i n r > t/2-M i 2f

V 2 1 C 3*2 1 + C 1 * 1 1 I P + C

2 * 1 1 2 P

A second-stage decision is involved in project 1 and a second-stage chance-branching point in project 2. It is assumed that the outcome of the chance node will be known when the second-stage decision is made on project 1. Thus it is necessary to introduce a variable for each second-stage route and for each chance outcome (with probabilities p or q), as shown in Fig. 4. The formulation in this case becomes:

Maximize:

[(pv3 + qvA > 2 1 + POi * I 11P + νιχ\12*9 + Φι*ιιι(Ι + ν2χι12<Ι)]

Subject to:

<A

+ bxxXiX<i+b2xll2q<B

<c

+ cixi 1 1 q + c i x \ 1 2 ^ < ^

period 1

period 2

period 3

* n

* n

- * n i P

- * m 6

= 0

■ * H 2 ? = 0 .

each node are exhaustive. The problem is one of finding a set The values of xx x XP and xx x2P and of xxxx'l ana * i i 2 ^ are of starting decisions that are feasible for all the uncertain the optimal strategies if paths with probabilities p or q, future states of the world (hereafter called "futures"). Only a respectively, occur, provided that the solution values for xx x

single variable is required for each project. A resource and x2l are adopted. This is the first problem for which it is constraint is necessary for each possible future, and therefore necessary to define a route variable for each possible sequence two and four are required for the second and third time of chance events prior to the decision point concerned, for periods, respectively. That is, each route leaving the decision point. The last two constraints

Maximize: (called "sequencing" constraints) ensure that if project 1 is , chosen in period 1, a complete decision path is continued for

[(pvx + qv2)Xl !+ (rv3+sv4)x2l J ^ a l t e r n a t i y e f u t u r e s t a t e s t h a t c a n GCCUU

Subject to: SI. Y- - + Λ- Y - - <? A ■r\c%fir\A 1

xxXi + b3x2l<B J v period

2*!!+ b3X2l<B \

bxxx

b2

Problem 4 in this case, the decision-branch point in project 1 occurs

before the chance-branching point of project 2 (see Fig. 5). This simplifies the formulation t o :

GEAR. AND LOCKETT: SOME MULTISTAGE R&D PORTFOLIOS 25

PROJECT 1

PROJECT2

Fig. 5. Problem 4.

Π-Μ

PR0JECT2

t = 1 (A)

Fig. 6. Problem 5.

Maximize:

[vxlx + vxl2 + (pv3+ qvA)x2l]

Subject to:

01*11 +02*i2 +tf3*2i:<^l '- periodi

*>i*ii + b2X\2 + b3x2\<B :

bxXn +b2xX2 + b4X2i<B :

2 1 < C : )

period 2

άχΧιι · c2Xi2 i c3x2

ΟχΧχι +C2Xi2+C4X21<C : )

* n + *i2 <1.

\ period 3

Problem 5 This problem (shown in Fig. 6) involves chance followed by

decision branching on project 1, and the reverse on project 2. As the opportunity to select between the alternative routes at period 2 in project 1 only occurs for one future (probability p), the variables Χχ i x and xli2 are all that are necessary to formulate the problem, apart from first-period variables. That is,

Maximize:

[pvxXm +pv2x1 1 2 +qv3xxl + v4x2i + (rv5 + sv6)x22]

Subject to:

Û I * I I + Û2*2i + ö3*22 <r^ : period 1

*4*ai + M 2 2 + M l l + M i l 2 < ^ · \ b3xlx + b4x2l + bsx22 <B

\ period 2

C 4*21 + C5*22 + C1-*111 + C2**112 < ^

C*X2\ +C6X22 + c l*l l l + C2*X'112 < ^

<C

Xf) i + X^ 9 = 1

C 3*l 1 + C 4*2 1 + C5*22 period 3

v21 v22

- X 111 - X 112 < 0 .

GENERAL FORMULATION

The problems considered in the previous section lead to some generai conclusions.

1) Where there are a number of chance states at which a decision point is reached, then each decision path from this point should be defined in terms of a list of variables, one corresponding to each chance state.

2) The number of constraint rows for a particular resource type at each period is given by the number of chance states that can occur at that period.

3) Only project variables that lead to an end-point value without involving further project decisions carry a non­zero-value coefficient in the objective function. Other variables um appear in the solution due to the sequencing constraints (which appear as equalities) for a given project which connect the variables from one time period to the next.

A formulation in terms of a stochastic integer program designed to find the optimal set of first-period decisions in the light of subsequent uncertainties and flexibilities to adapt is developed with the aid of Fig. 7, which shows à generalized project tree. The notation is different to that used earlier for the explanatory examples, but is completely general in character. All the variables introduced in thé formulation should strictly be taken as zero - one variables. The integer nature öf this condition may be relaxed in order to obtain solutions in reasonable computer running times.1

For time period Î, a Variable set xfji is introduced to represent the selection' (x«^ = 1) or rejection (x(fi = 0) of decision route // of project ί in the first time period (*= 1, "%N, where N is the number of projects under consideration, and /Ί = 1, •·%/|ΐ, where / / i is the number of alternative first-period decision routes for project /). The set of constraints

Jil

7i=l (1)

is introduced to ensure that no more than one route is selected for each project at the outset.

The inequalities

Σ Σ %rxV\<^r. r=l,.",Ä (2) i=l / Î = 1

1 There is no guarantee that the use of a linear rather than integer/ programming code will provide a useful solution in practice.

26 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, FEBRUARY 1973

*· etc. i I etc. ! I ·> TIMEPERIODI

t = 2

1 - *

F

VALUE OF END POINT

Key Q Chance Node I I Decision Node

Fig. 7. Generalized "project tree.'1

ensure that the overall availability Ar of resource type ris not exceeded in time period 1, and R is the number of resource types; a\j γ is defined as the amount of resource type r used in the first period by decision route; of project /.

It is necessary to introduce constraints to ensure that overall resource availabilities are not exceeded in time periods after the first:

N ^ift Σ Σ W/xiitft<Aftr- ' = 2, - , Γ /=! jt=l

r = l ,

,Ft. (3)

In the above, xtftft is a variable to represent selection of route / of project / in time period t given that future ft has occurred. The formulation assumes that all the states of the world that may have occurred up to time period t are defined by the set / / = 1, "*,Ft. A particular future thus defines a complete set of chance outcomes in terms of routes from each chance node up to the given time stage. The future may also include the definition of a set of availabilities/if/, if these are also to be treated as uncertain. The quantity Gijyf is defined as the amount of resource type r required by project / if route it is selected in time period r, if future ft has occurred. Other quantities are: Γ, the number of planning periods; Ft, the number of states of the world up to time period t;Ffft, the number of decision routes open at time period t for project /, if future/y has occurred.

For each future, it is necessary to introduce sequencing relationships between the variables of each project to ensure that a complete route through a project is selected or rejected. These take the form

JK *<>'i - Σ */)2/2 > i=\,'",N

f = 1 "· F (4)

xVsft~ Σ *i/m/f+i' j = l , ·

/f+i = 1,

t = 2,

',Ν •Jit ',Ft+\ ',T-l (5)

+ iry-ii» r*air\r\A c T., * 'IJt+l

'. ST..-Î Ϊ . - Ι Λ + ÄO +V»/3

subset of routes Jif*+l that can be reached if jc y = 1 in (4) or Xjjtft = 1 in (5), respectively.

As mentioned earlier, in this formulation all variables should strictly be zero - one only, but useful solutions to the linear-programming problem may sometimes be obtained by relaxing this condition.

Objective Function Taking the stage-by-stage decisions on each project into

account and the order in which uncertainties are resolved as the projects evolve, an objective function to maximize the overall terminal value of the portfolio of routes selected can be vvi l l t v i .

N Σ Σ

=i A=i

N Jift χύ\+Σ Σ

Ft Σ W i xiftft· (6)

for time oeriod t - 1. and

In this expression the terms v^ and v\jtft a r e e a c n special forms of average values for project /which need to be defined with care. The term VÌJX is the expected value of the return for project / if xijx - 1 and if no further decision stages are involved in project / after the first on route/. The term vijtft *s

the expected value of the return for project / if Xijtft ~ 1 and no further decision variables which are the last in the chain on the project have a nonzero value attached in the objective function.

Problem Size The number of constraints and variables required in the

mathematical formulation of a particular problem depends critically on the structure of the project trees. Upper bounds for the cenerai problem are <*iven below.

GEAR AND LOCKETT: SOME MULTISTAGE R&D PORTFOLIOS 27

(OS

1

VtriabUt

Nan \ ,

PROJ 1

2 iPROJ 2 3IPRCJ 3 4 IPROJ""* 5.PR0J 5 6jPR0J 6 7 8 y

lui il

PERIOO 1 PE 2 FUT 1 PE 2 FUI 2 PE 2 FUT 3 PE 2 FUT 4

12 FE Z FUI l 13

Si g PE 3 FUI Z

IFTTFUT 3 ΓΤΤΠΓΓΤ PE 3 FUT 5

171 FÉ 3 FUT 6 13ÌFE 3 FUT 7 19 ?Ó 21 n ιί

?5 Ì6

C "5 ri!T R • -. r. : ri...: FR0J4 FUT 1 PROJ* FUT 2 PR0J6 FUT l PR0J6 FUT 2 PR0J6 FUT 3 PR0J6 FUT 4 PR0J6 FUT 5 J

Ì7ÌPROJ6 FUT 6 Î8PR0J6FUT 7 inir.nr>i£ CUT 0

OBJECTIVE

1

2

. . . )..

5

C M |

1

0

•5

ΓΤ

SI

1

0 Ü |

0

"ΊΠ 0

«JL· 0 n

[

10

* 1

ì

i

4

<**

1

2

2 ! 2

2 2 0 0 0

" F 1

0

LJL 0

c

1

"ΠΓ

1

ô 2 2 2 2 2 '2 2 2 2 2 2 2

1

TV

TABLE I

LINEAR PROGRAMMING TABLEAU UNDER UNCERTAINTY

1

3 ! 1 1

1 "* -1

Πχ

1 j

1 3 6 3 6 3 3 6 6 3

■J. 6 6

TT"

s""

-1 -l"

1 -i i-i

-1 1-1

3

1 2 ! 2 2 2 2 Ü

2 0 2 0

1 2 0 2

Li I

2 2

1

2 2

1

1

l . i i 1 i

1 0 j 10 |2.4512.45 J 1

«M J

u Û

1

«M I

u 0

1

1 t 1

1 °

~JPf\

o 1

Ί

f

<M 1 so 1

i

1

I i

S O

2

I 1

1

| 0 jo.06 jo.54 b.06

1 0 I

2

i i

Jo.54

trs

2

1

0.1

i·^ I

SO

1 2

o j cs. \ao I

so k—· ! M ko I

1

! 2

I i

1

H 1.26

l~i-

1 2

j T ~

!θ.14 jl.26

«*> 1 CM I so 1

4

1

«.H SO

4

J

fsT^ 1 1

f s T

1

4

1 1

■■4»

1 1

Jj.135ll.215l0.135il.215

CSI S O

—r

1

Csf SO

4

1

c\ 1 CM J

j 1 3E ti

<sH so I

<

1 l· i 1<

~r

1

| 0.315 2.83^0.315

]! 1 13: col loc I

1

j «- j l j

1 < j 10. |

9

UI 9Ί j i

t

ΓΓΓ1 8

-1 "8

1 6

: r 8 ~ 8 \ U 1 8 1

1 "* ! ""

1 1 1

- | T ~ =. 1 0 1 - J j j •ΤΤΊ ■I °-J -1 òj "T ° 1 - 1 Ü 1

■I ° - 0

-4-üA m] 0 j

J2.835! J 1

The maximum number of variables is given by

TV T Ft

Σ Σ Σ /=1 t=\ ft=\

Jift (7) PROJECT 1 P ^ l 2 \

a ö ö u n i u i g , u variable is needed for every path and every time period for each possible future.

The maximum number of constraints is given by

T N T-\ N + R^Ff+Y^ Ì2Jift'

ί=1 ι=1 ί=1 (8)

The terms correspond to the project constraints [as repre­sented by (1)], resource constraints [as represented by (2) and (3)], and sequencing constraints [as represented by (4) and (5)], respectively. Reference to the above relationships shows that a critical factor is the number of possible futures at the various stages. This is given by the product of the numbers of branches at each preceding chance point, and may become very large.

The above bounds are of little value because there is usually a large number of redundant variables, and hence sequencing constraints. This is illustrated in the next section, where a small example is presented.

AN EXAMPLE

A portfolio of six potential projects is shown in Fig. 7, each with a different pattern of choice and chance-branching characteristics, resource inputs, and eventual payoffs.

Using the formulation, the linear-programming tableau for

PROJECT4, ËIHH 3

Fig. 8. S ix project examples.

the six project problems of Fig, 8 is shown in Table 1. Rows 1 - 6 corresDond to (1), ensuring that each project is not

I ß IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, FEBRUARY 1973

TABLE II INTEGER AND LINEAR PROGRAMMING SOLUTIONS

TO THE TABLEAU OF TABLE I

Variable

ph x21l x22l x3ll xZ2l

χ33χ x41l x5Ii x61l x62l

x412l2 x41222 x422l2 x42222 X6I3I3

xöl323 x61333 x61343 x61353 | x61373 x61383

■ x623l3

x62323 x62333 x62343 x62353 x62363 x62,7, I 0 j x 6 23 83 1 ÎWÎWLIO VALU!· I

Integer Solution

1 1 X

1

1

1.0 1.0

1.0 1.0 1,0 1.0 1.0 1.0 1.0 ! 1.0 3X3 Γ

! Linear Solution

0.44

1.0

1.0

1.0 0.11 1.0

0.5 1.0 0.5

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

— 3 X U

First stage decision variables

Future

stage

decision

variable:»

j selected in more than one starting version; row 7 corresponds to (2), ensuring that the first-period resource availability of 10 units is not exceeded; rows 8-19 correspond to (3), ensuring that the resource availabilities of 9 and 8 units are not exceeded in time period r = 2 and 3 for all possible futures; rows 20-29 correspond to (4), ensuring the correct se­quencing of variables for projects i = 4, and 6; row 30 is the objective function.

The integer and linear solutions for the six project formulations of Table I are given in Table II. Note that the solutions provide the optimal set of first-period decisions in the light of the flexibility to modify subsequent actions according to chance outcomes. The values of the variables xijtft in the solutions indicate an optimal strategy if future ft occurs for time period t. The integer solution is now discussed in detail below.

Discussion of the Integer Solution The solution gives a clear indication of the optimal

first-period resource-allocation decision. Projects 1 and 5 are rejected; projects 2 (second decision path), 3 (first decision path), 4, and 6 (first decision path) are accepted.

For the subsequent decision-node project 4 (if it occurs by chance), the solution is to always choose the first path. Similarly for project 6, the solution is to choose the second decision path (at the decision node at the beginning of period 3) for all possible futures.

It is interesting to compare the integer and linear solutions. The main difference is that a different initial decision path of project 3 is chosen by the linear solution. Also, there are fractional values chosen for projects 1 and 5 which do not appear in the integer solution.

The first-period solution to this problem may be compared with the solutions obtained by Lockett [9]. It may be important to note that one of his "efficient" solutions, labeled portfolio I, is substantially similar to the optimal solution derived in the present paper. Thus in a large problem, which for practical reasons cannot be formulated and solved exactly, the simulation approach proposed by Lockett may prove useful.

SUMMARY AND CONCLUSIONS

The paper has proposed a method of handling uncertainties in multistage processes as currently foreseen for each of a set of projects under consideration. It is assumed that each project is an independent entity and is related to the ether projects only through the need to draw on a common pool of scarce resources, usually of several types.

The method of representing each project by means of a decision tree on a time scale appears very powerful. Not only the R&D stages but also the subsequent commercial stages may be depicted on a single planning diagram. It is not clear at present in what detail it is necessary to represent projects and resources or to what planning horizon the problem should be formulated. These problems are common to most strategic models at this time.

A method ç>f analysis is described which makes use of certain concepts of stochastic linear programming. This allows a porti Olio öl ueeisiön trees ΐο be analyzed under conditions of resource scarcities over time. A problem here is to decide on an appropriate number of time periods into which to divide the overall planning period.

Alternative methods of analysis may be based on simula­tion [9] or on chance-constrained programming [15], [16]. These methods have the advantage of allowing solutions to large problems to be found. However, a large set of solutions will usually be generated leaving the difficult problem of final selection to the decision taker. It is unlikely that any one of this set will correspond to the optimal solution as defined in the present paper. Nevertheless, many solutions in the set may meet conditions of "sufficiency" in view of the inherent uncertainties.

In the authors' view, the greatest advances will stem from attempts to model in as much detail as possible the complex­ities and uncertainties of actual situations. Simplifying assumptions to meet practical data demands and computa­tional difficulties can then be tested on a more rigorous basis. Hopefully, this paper provides a contribution to this end by describing a method of handling dynamic aspects of portfolio-selection problems.

Note added in proof: The zero - one problem solved for the

IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, FEBRUARY 197 3 29

6-project example described in the paper was done on an ICL (1906 A) computer using 12 s of central processing time. A modified form of the Balas Algorithm was used (E. Balas, "An additive algorithm for solving linear programs with zero - one variables," Oper. Res., vol. 13,pp. 517 -546, 1965).

REFERENCES [ 1 ] D. T. Asher, "A linear programming model for the allocation of

R and D efforts," IRE Trans. Eng Manag., voL EM-9, pp. 154 -157, Dec. 1962.

{2] G J. Beattie, "Allocating resources to research in practice," in Proa NATO Conf. Applications of Mathematical Programming Techniques. English Universities Press, June 1968.

[3] D. C. Bell, J. W. Chilcott, A.W.Read, and R. A. Salway, "Applications of a research project selection method in the North Eastern Region Scientific Services Department," R&D Dep., United Kingdom C.E.G.B., Rep. RD/H/R2.

[4] B. V. Dean and L. E. Hauser, "Advanced material systems plan­ning," IEEE Trans. Eng. Manag., vol. EM-14, pp. 21 - 4 3 , Mar. 1967.

[5 ] A. G. Lockett and P. Freeman, "Probabilistic networks and R&D portfolio selection," Oper. Res. Quart, voL 21, no. 3, Sept. 1970.

[6] A. E. Gear, A. B. Lockett, and A. W. Pearson, "An analysis of some portfolio selection models for R&D," IEEE Trans, Eng Manag., vol. EM-18, pp. 66 - 77, May 1971.

A n Approach Toward Improving the Creative Output of Scientific Task Teams

ROBERT D. DOERING

Abstract-Administration of an innovative task team today requires a capability to integrate not only the interrelated technical disciplines but also the relationships of the individual members as well. The team leader needs a tool to measure and characterize the members so that he can predict their interactions and structure his task teams accordingly.

The Myers - Briggs Type Indicator Test, which places individuals in a scheme of personality types as established by the Swiss psychologist Carl Jung, was used effectively in a scientific task-team management situation. Jung theorized that much apparently random variation in human behavior is actually orderly and consistent and reflects certain basic differences in the way people prefer to use perception and judgment in a given situation. Perception in the task-team context involves the process of becoming aware of the problem; judgment includes the process of making a decision about the problem. Clearly, if the task-team members do "riot "see" the same problem, they will evaluate courses of action differently and the result will be poor communications, strained personal relations, and disorganized progress toward solution of the problem.

In Jung's concept of intellectual differences, each person's mode of functioning can be described in terms of four pairs of behavioral patterns.

Extrovertism - Introvertism (E/I). The extrovert's primary interests lie in the outer world of people and things, while the introvert's lie in the inner world of concepts and ideas.

Sensation - Intuition (S/N). A sensing individual trusts only his five

Manuscript received January 18,1972; revised June 22, 1972. The author is with the Department of industrial Engineering and

Management Systems, Florida Technological University, Orlando, Fla.

[7] S.W. Hess, "A dynamic programming approach to R and D budgeting and project selection," IRE Trans. Eng. Manag, voL EM-9, pp. 170 - 179, Dec. 1962.

[8] E. M. Rosen and W. E. Souder, "A method for allocating R&D expenditures," IRE Trans. Eng Manag, vol. EM-12, pp. 8 7 - 9 3 , Sept. 1965.

[9] A. G. Lockett and A. E. Gear, "An approach to dynamic model­ling in R&D," presented at the 38th Meeting Operations Research Soc America, Detroit, Mich., Oct 1970.

[10] C. E. McQuillan, "Stochastic programming," Water Resources Cen., Univ. of Wisconsin, Madison, Rep. C-1228, 1969.

[11] H. M. Wagner, Principles of Management Science with Appli­cation to Executive Decisions. Englewood Cliffs, N.J.: Prentice-Hail, 1970.

[12] R. F. Hespos and P. A. Strassman, "Stochastic decision trees for the analysis of investment decisions," Management Sci, voL 11, no. 10, pp. 244 - 259, 1965.

[13] R. A. Harvey, "A probabilistic approach to aeronautical research and development," Aeronaut J., voL 74, pp. 373 - 380, 1970.

[14] W. J. Abernathy and R. S. Rosenbioom, "Parallel strategies in development projects," Management Sci, voL 15, no. 10, pp. B486 - 505, 1969.

[15] A. Charnes and W.W.Cooper, "Deterministic equivalents for optimising and satisfying under chance constraints," Oper. Res., voL 11, no. 1, pp. 18-39, 1963.

[ 16 ] A. Charnes and A. C. Stedry, "A chance-constrained model for real-time control in research and development management," Management Sci, voL 12, no. 8, pp. B353 - 362, 1966.

physical senses to correctly define the situation. Intuitive perception, on the other hand, is an indirect perception of the deeper meaning and possibilities within a situation.

Thinking - Feeling (T/F). These are the two forms of judgment used to evaluate a situation and make a decision. Thinking judgment is a logical, functional process aimed at an impersonal evaluation of fact; feeling judgment is a process of evaluation in which a situation is given a personal or subjective value.

Judgment - Perception (J/P). A preference for a judging attitude by an individual leads him to live a life that is controlled, caîëfuuy planned, and orderly. A person preferring a perception mode will be more open and receptive to all experiences with activities characterized by spontaneity and flexibility.

When profiles are exactly the same, maximum communication and cooperation are possible because the problem is being perceived in the same dimensions and the judgment logic is identical. On the other hand, if there are no preferences in common, each individual will misunder­stand the other and tend to undervalue what the other is explaining or proposing. Members with like profiles do not make the most effective innovative task teams, however, because they are prone to make the same errors and overlook the same problem areas.

A balanced team of selected profiles is more effective. To achieve this balance, all profile preferences should be included to augment and supplement each individual view and approach to the problem. The administrative task of the team leader will be greater because of the differing views, but if all members have at least two or three common preferences in their profile, communication is ensured and the leader can coordinate and control the team without endless effort.

The concept is illustrated by an industrial case study of interaction in a Research and Development Engineering Group.

INTRODUCTION

Typically, problem solution today requires a methodology to integrate the interrelated parts of the problem, and this, in turn, involves a complex relationship of a number of disciplines. Thus new

Technical and Management Notes