Engineering Costs and Production Economics, i (1982) 69- 80 Elwier Scientific Publishing Company, Xmster4am - Printed in The Netherlands

eller

University of Bradford, Bradford, W. Yorkshire BD7 1 DP (U.K.)

Memorial University of Newfoundland, St. John’s, Newfoundland A 1B 3X5 (Canada)

and

V. Supriyasilp

University of Bradford, Bradford, W. Yorkshire 807 1DP (U.K.)

69

ABSTRACT

Generally, in the contracting industry, the contractor works in an environment of risk and uncertainty caused by the economic factors such as fluctuations in the costs of labour and materials. This paper examines the overall effect of’ these factors on the policy for claiming payments under the type of con- tracts with clauses for price escalation, Particular reference has been made to the

formula ‘F’ of the Water Tube Boiler Maker’s Association ( W, T. B.A.), England. It has been shown that under certain circumstances the contractors c t2 optimise the benefit, or minimize the loss, arising from changes in tJle costs of labour and materials due to inflation. An appropriate algorithm based on dynamic programming is given.

In recent years, most businesses have been uncertainty for Aements in the

economic environment. in economic factors such as inflation an in rates in t risk commercial projects. strated that there is

prediction of net costs by contractors. One in three of the estimates are at variance from the actual total net cost by more than six percent. The probability of the estimated cost of each

being in error by more than forty percent is approximately one in three [ 11. This situation has compounded the risk ele-

for the profit a contractor can expect

0167-l 88X/82/OOOQ-OOOO/$O2,75 0 1982 Elsevier Scientific Publishing Company

to make on projects with relatively long time scales and large investments. To reduce this degree of risk, it is normally necessary for the contractor to include large contingencies in initial estimates of the contract when he is tendering or negotiating the contract. If the contingencies are overestimated, the probabil- ity of the contract going to another con- tractor is increased. On the other hand, if the contractor does not allow for inflation and interest rates correctly, his initial tender would bt: too low and he would suffer signif- icant losses. Accordingly, it is desirable to negotiate a. contract equable to both parties which not only reduces the risk of loss by the contractor but also will not require the client to pay exce sslively .

It is possible for the client and the con- tractor to draw up a special type of contract or to use one of the standard forms, such as firm fixed price, fixed price with incentive and cost reimbursable f2,3,4,5]. Fixed price contracts with an escalation clause are predominantly used in heavy engineering industries. In this type of contract, the escala- tion in cost of lahour, material and, in many cases, overheads arising during the course of its execution is adjusted with the contract price. This adjustment is referred as the Con- tract Price Adjustment (CPA) or Prime Cost Adjustment in some industries [ 61.

The objectives of this paper are to demon- strate a method for analysing the inherent risk associated with CPA type of contracts and propose an approach to devise an optimal policy of placing interim CPA claims.

CONTRACT PRICE ADJUSTMENT - BACKGROUND

CPA originated in the U.K. during World. War II. The original premise was that with the Government controlling wage rates and

aterial costs, escalation of costs on these ts was outside the contractor’s con%1 ould be reimbursed on a net cost basis.

This practice continued in tre U.K. after the war and became progressively more so:phisti- cated.

In January 1968, the Economic Develop- ment Committee for Building approved a steering group on Price Fluctuation, The terms of reference of this committee were to assess the cumbersome practice of claims for CPA. This practice required the contractor to submit complete details relating to the work for which !re was claiming CPA. These details included labour, materials, overheads and a considerable effort was required to check and substantiate them. The committee was accordingly asked to devise simple formulae which only took account of labour and material indices and the period of time over which the contract was carried out [ 71. In 1969 the steering croup issued a report and proposed a set of formulae for Contract Price Adjustment. Other bodies such as the Water Tube Boiler Maker’s Association (W.T.B.A.) [8], and the Civil Engineering Economic Development Committee then issued similar formulae [ 9, lo].

The fixed variable elements of a contract

The “fixed” element of the contract price, which is not subject to CPA, is normally deemed to include profit and certain fixed overheads; for a number of formulae in general use, the fixed element is agreed as a fixed percentage. These percentages normally vary between 15% to 20% of the contract price. The balance is the “variable” element on which CPA is allowable and is calculated using the agreed formulae. This variable ele- ment then has to be segregated into labour and material components; every expenditure is deelr,ed to fall into one of these two cate- gories. The proportion of these two generally vary with the class of work. For boilers, heat exchangers, and other similar components, proportions of 60% and 40% respectively are recommended.

71

Labour indices

The labour index expresses labour costs and includes earnings, overtime, holiday pay, national insurance, and redundancy pay with reference to a particular time datum. Altera- tions in working hours are also taken into account. The labour indices vary from industry to industry. W.T.B.A.‘s labour index is based on the information published by the Department of Employment, U.K.

Material indices

Most engineering contracts require the use of a wide variety of materials, some of which are of a relatively high value. The index for a particular class of work has to be designed to take account of the mixture of materials used. Appropriate material indices are published by the Department of Trade and Industry, U.K. For W.T.B.A. formulae, the relevant indices are published in the ‘Trade and Industry’ price index 3 1 l-3 12, Table 2: Steel Industries Home Sales.

Terms of payment

Several methods are used for payments to the contractors. Although payment in full on the successful completion of a contract is a useful incentive, it is generally unacceptable to the contractor, except on relatively small projects of time scales less than one year.

In the 1966 edition of the General Condi- tions of Contracts Model Form A for home contracts with erection, there is provision for a 90% payment of the sum of each interim certificate covering expenditure for the variable element by the contractor. These stages of interim payments are agreed by the contracting parties and are usually taken at the 25% 50% and 75% stages of work completion. A 10% retention is released in two stages, when the commissioning phase is comnleted and the plant is taken over by the

client, and when period is completed

It is seen that

the agreed maintenance WI.

the contractor, in his management of the contract, must allow for servicing costs (i.e. interest) incurred .mtil interim costs are recovered, in addition 13 the servicing costs of the 10% retention.

3 the case of very large contracts running over . everal years, it is usual to make pay- ments against an incidence of expenditure curve (Fig. l), which is of the S-type and is discussed in the next section. The curve is based on either an agreed physical programme of work, or the contractor’s anticipated expenditure, or the value of work completed; any payments are made to major subcontrac- tors. When the contractor chooses to work (or to pay to subcontractors) in advance of an agreed programme, no early progress payments are usually allowable. However, should the contractor fall behind the contract programme, the client would then have the right to redraw the curve and retard (or reduce) the progress payments.

According to the latest General Conditions of Government Contracts for Building and Civil Engineering Works, the contractor is entitled to be paid during the progress of the work for 97% of the value of the work completed and 90% of the value of materials

! EXPENDITURE CURVES

8

iii

i?

-8 8 6”:

00 “*

0

MONTHS

Fig. 1. Comparison of S curves for various values of param- eter n.

72

required and deIivered to the site [6]. The contractor may submit lelaims for payment of such expenditure at intc:rvals& of not less than one month.

Planned payment curve (S-curve)

A suggested mathenlatical expression for cumulative expenditure as a function of time, is as follows [K?]:

6 C,t m exp(r/r)*

” ---

6 + P exp(t/r)” (1) and

here: C is cumulative expenditure or effort at time t; C= is the cost of the project if activ- ity was continued for an infinite time; 7 is a parameter with dimensions of time and related to fihe total duration T of the project; 0 is a pararnefer related to the time, when the maximum rate of expenditure occurs; n is a parameter which normally lies between 1 .O and 3.0 and is character&d by the complex- ity of the project, at the finishing end, and called the lzaming parame*a.er; m is a param- eter whose value in most case is approximate- ly equal to ,yt [ 51. This is characterised by the complexity of t;“le project at the commence- ment.

In the present study the value of m is taken equal to n. This gives:

Cwtn exp(r/r)n c=

0 + t” exp(t/r)” (2)

A set of parameters of the S-curve which generate curves typical for thf: boiler jmanu-

ring industry and which are used in the present study is as follows:

CT_= 100.0

n = 1.617

7 = 13.41

6 = 152.8

ese parameters were obtained from ata obtained from contractors.

Figure 1 shows various expenditure curves obtained by varying the value of parameter n.

CPA formulae

Though not specifically given in the official literature, the general formulae for CPA can be expressed in algebraic terms as follows:

CPA(f) = CPA,j&) + CPAL(?)

= CPAF(t) . C(f) (3)

Where: CPA(t) is the amount of the price variation to be paid to the contractor at time t; CPA&t) and C&(t) are the amount of CPA attributed to variations in materials and labour indices respectively at time t; C&¶F(t) is the price variation factor which is multi- plied by the contract value at time f to obtain the adjustment; C(r) is the planned. value to be adjusted at time t; A0 is the co- efficient representing the proportion of the planned cost allowed for adjustment (e.g. in WTBA formula ‘F’, A 0 = 0.85); VM, IQ are the co-efficients representing the propor- tionate values of the contract relating to materials and labour respectively (e.g. in (WTBA) formula ‘F’, VM = 0.40 and IQ = 0.60, i.e. material and labour are assumed to constitute 40% and 60% respectively of total cost); Mt , Lt are the index figures for materials and labour, respectively, at time t; MO, Lo are the base index figures for materials and labour respectively .

The CPA formulae vary with the class of work. For the Water Tube Boiler Maker’s Association’s (WTBA) formulae ‘F’, the methods of calculating the values of variations in materials and labour indices at time t are different from the general formulae above. Clause (1.1) in formulae ‘F’ states:

73

. . . the contract price should be adjusted by the percentaTe variation between the WTBA labour cost .‘izdex ‘B ’ ruling for the month in which the tender price basis date falls and the average of monthly indices for the labour period /8/.

Clause (1.2) states the same method as clause (1.1) except for materials, i.e. to take the average of the relevant monthly indices instead of taking only the current price index into account. Hence

CY’AF(t) = A, t Lj/t -Lo - +VLC-

j=l Lo 3

(5)

In Fig. 2 the CPAF factor for a contract of thirty months (July 1971 -December 1973) is shown. Though this factor is given specif- ically for the WTBA formulae ‘F’, one would expect other formulae for this period of time to have a similar shape.

CPA - W.T.6 A. FORMUA “F”

Fig. 2. CPA factors at each month (July 19‘71-December 19’73).

OPTIMISATION OF CPA CLAIMING POLICIES

Since the total value of CPA claims is the If the value of r per month is very small.,

same regardless of the dates of interim claims, terms in r with a higher degree than 1 can be

benefits can be gained only by maximising the total. accrued interest from all these claims. This can be achieved when there is some flexibility in the date of CPA claiming.

From initial considerations of discounted cash flow, one can expect that the more frequently and the earlier the interim CPA claims are made, the greater the total value of these claims is to the contractor. Haw- ever, this is not always so, since the optimal policy for interim CPA claims depends upon the relative sizes of the certified contract price amounts and on the shape of the S-curve and the trends of both material and labour indices.

Let the CPA factors at time periods (say months) K(K = 1,2,. . . , n) be CPAF(K), and CPA value at time perioa K be CPA(K). Then at any period, K, with planned cumula- tive expenditure C(K), the CPA(K) can be calculated as :

CP4K) = CPAF(K) X(K) (6)

For convenience let

F(K) = CPA(K) (7)

Suppose there are m interim CPA claims at months Xi, i = 1,2, . . . , nz, then if we con- sider only the total value of CPA claLns at the end of the contract, the value with the rate of interest are r% per month will be as foul0 ws :

V = F&)(1 +@ -xl + (F(x,) --F(x,)) (1 + r)Xm -‘2 +

1*. + (F(x& -F(X,&) (1 + dxm -xm

m = c F(xi.& [_(I + v)‘m -xi + (1 + r)xm -xi-1 1 + F(xm)

i=2 (8)

m-l 1 1 = (1 +rjrn C F(xi) - -

(1 +r)xi () +,)%+I + F(xm) i=l

(9)

74

neglected. Then

m-2 Y= g F(xj) ((1 +r(x*-xi))-(1 +‘(xm-xi+l))l

Pl

c F(xm_*) I(1 + mm --m-19) -1 I + Fern)

m V= t fC F(xj9 (xi+1 -xj) J + F(xm) (10)

i=l Start

i=i%i) = C -Xy eXp(Xj/T)"

- CPAF(Xj) 8 + Xf eXp(Xj/T)”

(11)

Clearly V is a function Of Xj, i = 1,2, l . . , m. The function V wil! be maximised, subject to tirx cosxtraints on these variables Xi, namely that Xi+1 - xi 2 6 (the time between two claims should be at least six months) for i=l,2 ,..., m-l,andx,>6.

In order to maximise the CPA claims, one has to examine the effect of these constraints.

The first CPA claim could be at any month after 5 months, i.e. at the 6th, 7th, . . . or nth month. If one chooses to claim for CPA at month 7, say, then the second claim can be chosen at any month from the 13th month to the end of the contract. Subsequent claims wi!l be constrained in a similar manner. This

as been shown diagrammatically in Fig. 3. It is not necessary to consider which month

is the best for each particular claim in lation, but to consider which path, or set

of months, should be taken as the best deci- sion with a virtw to maximising the total value

CPA claims subject to the principle of cash ws. The problem of finding values of the

lesx,,x,, . . . , xm such that a function Y (of these variables) zssumes a maximum

s an optimisaton problem that can using dyjlamic programming tech-

es. Details of tJ-.ese are given in Appendix 1.

6

l-l

and so on

Fwst claim Seconfl Claim

n

Fig. 3. Dynamrc programming algorithm to devise claiming policy.

EFFECTS ON INTERIM CPA-CLAIMING POLICY OF THE CHANGING SHAPE OF THE S-CURVE

The value of accrued CPA claims, taking interest into account , depends very much on the policy of allocation of the expenditure or, in other words, on the shape of the S-curve.

For illustration purposes, the effects on CPA claims of changing or varying the value of the parameters n and T of Eqn. 2 are con- sidered in the following worked example:

For a contract with:

Contract period dates: July 197 1 -December 1973 (30 months period) Basic date (month): June 197 1 CPA formulae: WTBA formula ‘F’.

The optimal policy computed with Eqn. 9 for the varying value of pt and T are given in Tables 1 and ;! respectively. In Tables 1 and 2, one can notice that in each cast: the best policy for interim CPA claims is when the number of interim CPA claims is 3, which is the maximum number of interim claims that can be made within the terms of the contract.

75

TABLE 1

Effects of changes in parameter n of S curve on the CPA claiming policy. Contract period (July 1911-December 1913)

Parameter NPAY * Best policy to Max. value Worst policy to Worst value n claim at months claim at months

1.211 1 2 3

1.611 1 2 3

2.02 1 1 2 3

2.411 1 2 3

2.811 1 2 3

16,30 11.9513 11,30 13,21,30 12.2563 11,24,30 11, 11,24,30 12.3484 12,18,24,30 i9,30 11 .I896 11,30 15,23,30 12.0553 11,24,30 12,18,24,30 12.1505 11,18,24,30 22,30 ‘11.5984 11,30 18,24,33 11.8051 11, 11,30 12,18,24,30 11.8411 11, 11,23,30 24,30 11.4312 11,30 18,24,30 11.4898 11, 11,30 12,18,24,30 11.4938 11, 11,23,30 24,30 11.1622 11,30 18,24,30 11.1610 11, II,30 12,18,24,30 11.1672 11,17,23, 30

11.6151 12.0698 12.3304 11.3182 11.8419 12.1409 11.0511 11.4165 11.8113 10.9858 11.0648 11.4311 10.9181 11.0648 11.0979

*Number of interim CPA claims.

TABLE 2

Effects of changes in parameter T of S curve on the CPA claiming policy. Contract period (July 1911-December 1913)

Parameter NPAY* Best policy to Maximum Worst policy to Minimum 7 claim at months value claim at months value

5.41 1 2 3

9.41 1 2 3

13.41 1 2 3

11.41 1 2 3

24,30 il.1541 11,30 10.9177 18,24,30 11.1565 11, 11,30 10.9796 12, 18,24,30 11.1566 11, 11,23,30 10.0806 23,30 11.5811 11,30 11.0316 18,24,30 11 .I168 II, 11,30 11.3511 12,18,24,30 11.8021 11, 11,23,30 11 .I695 19,30 11 .I896 11,30 11.3182 l&23, 30 12.0553 11,24,30 11.8479 12, 18,24,30 12.1505 11,18,24,30 12.1409 16,30 11.9010 11,30 11.5321 13,21,30 12.2012 11.24,30 12.0076 11, 11,24,30 12.2953 11, 18,24,30 12.2842

*Number of interim CPA claims.

‘THE SENSITIVITY

1 II JOKES

‘lt’he trends of materials and labour indices

play a very important role in the interim CPA claims policy as well as the accrued amounts of the expenditure and it is of interest to illustrate their effect on the interim CPA claiming policy with the help of the above example.

76

Variations in labour and material indices were fitted using the following polynomials: L hear.

YM 106.0409 + 0.08079X

YL = 134.0320 + 1.7556X

where YM and YL are material and labour price indices, respectively, and X is the time.

For this fit the best CPA claim policy is to claim at months 11, 17, 23 and 30. Quadra tic :

YM = 112.4391 -0.3918X + 0.M37X2

YL = 137.9449 + 1.021X + 0.0237x2

‘The best CPA claim policy is now to claim at inonths 12, 18, 24 and 30. In the present caSe this is the minimal sufficient fit (i.e. the polynomial of lowest degree from which there is no significant deviation).

For another thirty month (contract with:

Contract period: January 3 972-June 1974 Basic date (month): December 197 1

the minimal suffic;:ent fit for material indices of this period is the third degree polynomial trend :

Y,v = 107.0471 + 2.691X-0.2494.1:2 + 0.008PX3

and the minimal sufficient fit of labour indices for the same period is a first degree polynomial trend i.e.

YL = 140.3722 + 2.0764X

e best policy for interim CPA claims wiBl e to claim at months 12, 18, 24 and 30,

iich again is the same policy as in the

Optimal number of interim CPA claims and optimal timing

En the examples considered so far, the al policy has been to make as many

s as possible, which is what one

would intuitively expect; however this is not necessarily always so, as is illustrated in the example given below:

Contract period dates: July 196%June 1974 Basic date (month): June 1969

The results are given in Table 3. It shows that the policy of making only five interim CPA claims, at months 21, 27, 34,41 and 48 is the best.

In certain cases, such as shown in Fig. 4, the cumulative total of CPA’s at a certain time may fall (i.e. the CPA’s for some months may be negative). Then, if the maximum number of CPA claims are made, it is neces- sary to make a claim at a time when the amount will be negative, which will detract from the total accruecl value of the claims. Negative claims are likely to occu;: after the end of material procurement periods, or when the labour or material indices themselves decrease.

Fig. 4. CPA value (%I: July 1971-December 1973.

CONCLUSIONS

A mathematical model to describe CPA claiming by a contractor is developed to- gether with inethods for obtaining optimal claiming policies.

The study shows th6.t both the optimum

TABLE 3

Results of CPA claims for various numbers of interim claims. Contract period ~July 1969-June 1974)

No. of interim claims

Months claimed Value of CPA claims

Max. at h onths: 36,60 26.36% Min. at h .onths: 54,60 23.03% Max. at h onths: 28,44,60 27.82% Min. at h Months: 21,54,60 24.27% Max. at h ionths: 25,37,48,60 28.53% Min. at h Months: 21,46,54,60 25.68% Max. at h lonths: 25,32,40,48,60 28.85% Min. at n ionths: 21,42,48,54,60 26.84% Max. at 1 lonths: 21,27,34,41,48,60 28.98% Min. at h [onths: 24,31, 37,43,49,60 27.54% Max. at 1 ionths: 21,27,34,40,47,54,60 28.48% Min. at h months: 21,27,33,39,45, bl, 60 28.14%

timing and the value of interim CPA claims are sensitive to changes in the rate of expendi- ture and to the trends of material and labour indices. This indicates that the policy of cost planning (or, in other words, policy of allo- cating the expenditure) is important when devising optimal policies for CPA claiming.

It is also shown that it is not always neces- sary to make the maximum allowable number of interim CPA claims in order to maximisc the value of CPA payments. This is due to possible negative values of certain claims, caused by the downward trend of the labour and materials indices.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the con- structuve comments made by Dr. A. Redlack and Mr. C. Vaughan, School of Business Administration and Commerce, Memorial University of Newfoundland.

REFERENCES

1 Barnes, N.M.L. and Thompson, P.A., Engineering Bills of Quantities, CIRIA England.

1971. Civil Report 34,

(Received October 5, 1981; accepted November 17, 1981)

6

7

8

9

10

11

12

13

Volpe, S.P., 1972. Construction Management Practice, John Wiley, London. Scott, P., 1974. The Commercial Management of Engineering Contracts, Gower Press, England. Turner, J.H.W., 1963. Construction Management for Civil Engineers, C.R. Books, London. Gupta, Y .P., X976. Probabilistic approaches in the analysis of financial risk and uncertainty associated with capital project contracts, Ph.D. thesis, University of Bradford, England. General Conditions of Government Contracts for Building and Civil Engineering Works (edn. l), 1973. H.M.S.O., London. Economic Development Committee for Building, 1974. Price Adjustment Formulae for Building Contracts, H.M.S.O., London. Supriyasilp, V., 1975. Major Contract Financial Risk Analysis, Ph.D. thesis, University of Bradford, England. Economic Development Committee for Civil Engineer- ing, 1973. Price Adjustment Formulae for Civil Engineering Contracts, (edn. l), Civil Engineering Works, H.M.S.O., London. Economic Development Committee for Civil Engineer- ing, 1974. Price Adjustment Formulae for Civil Engineering Contracts, (edn. 2), Civil Engineering Works, H.M.S.O., London. Standard Conditions of Contract Model Form A, Home Contracts with Erection, 1966. The Institute of Electrical Engineers, London. Keller, A.Z. and Singh, M.P., 1975. Uncertainty in project expenditure prediction in: Keller, AZ. (Ed.) Uncertainty in Risk and Reliability: Appraisal in Management, Adam Hilger, London. Roberts, S.M., 1964. Dynamic Programming in Chemical Engineering and Process Control, Academic Press, New York.

APPENDIX 1

DYNAMIC PROGRAMMING APPROACti

General definitions

The basic terminologies of dynamic programming are presented as follows:

(11 smge The problem can be broken down (decom-

posed into smalter subproblems), and each subproblem is referred to a:: a ‘stage’. A stage here signifies a portion of the decision problem for which a separate decision can be made. The resulting decision must also be meaningful in the sense that if it is optimal for the stage it represents, then it can be used directly as part of the optimal solution to the entire problem.

In this probIem, the stages are the succes- sive claims for the CPA.

(2) State Each stage has a number of ‘states’ ass&

ciated with it; the ‘state’ of the system is a variable, or set of variables, which can be used to describe the system at any stage.

In the problem, the state can be OX of the set of months available to be selected for a particular CPA claim.

i.r) stcrge decisiclln The decision-making at each stage involves

the selection of one of the states of the stage, this is usually referred to as a ‘stage decision’,

(4 Object&e furwtion III the type of problem being considered,

it will be required to optimise a certain func- tion (in our problem the accrued value of CPA claims) by a suitable set of decisions in passing from stage to stage. This function will be referred to as the ‘objective function’.

IS) Return function Associated with each stage decision is a

‘return function’ which evaluates the con- tribution of the choice of a state (i.e. a deci- sion) to the objective function. In the CPA

claiming problem, the return function is R (xi, xi,1 ) as explained later.

(6) Policy A policy is a set of decisions, represented

by the states chosen at each stage. In partic- ular, an optimal policy is represented by the set of states chosen to maximise the objec- tive function.

In this problem, by selecting an optimal feasible month (state) for each CPA claim (stage), the selected set of states is then said to comprise an optimal ‘policy’ for the entire decision problem.

(7) PrincipIe of Uptimality The ‘Principle of Optimality’ is the corner-

stone of dynamjc programming. It states ‘An optimal policy has the property that whatever the initial state and initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision’ [ 131. Use of the Principle of Optimality guarantees that the decision made at each stage is the best decision in light df the entire problem.

Applicatian of the dynamic programming approach to optimise dates of CPA claims

The CPA claiming problem may be formu- lated as follows:

Consider an exampie of optimising interim CPA claims for a contract of thirty-months duration, subject to six-months elapsed time for consecutive interim claims. The process commences at the beginning of the contract period (state 0, stage 0) with the accrued value of CPA claims (PO) = 0. If there are m claims (i.e. m -1 interim claims), consider the system at each successive claim (i.e. at ‘stage’ 1,2,. . . ,m), and the number of months elapsed (i.e. the ‘state’ of the system) when each claim is made. Suppose the ith claim is made at month xi. Then the contribu- tion of the first i claims, P(xl,x2,. . . ,xi),

to the total accrued value (&) is

where R (x~,x~+~ ), which is the return in passing from claim j at time xi to claim U+l)

l

at time xj+l, will be equal to r*i;(Q (xi+, -xi) for j<m-1 and F(x,) for j = m.

Since PO = 0, this can be expressed as:

i-l V&,X*,.. . ,Xi) = r c Wj) (Xj,l -Xj) for i < m-l

j=l

and

Let Gj(xj) be the optimum value of Vi for all possible policies x1 ,x2,. . . ,xi, and using the dynamic programming algorithm :

c, (X,) = F(O)(x, -0) = 0

Gi(q) = Max Gi_, (XI-~) + I F(x~_~) (q-xi_, ) Qr i Q m-l

xi-r

G&m) = Max Cm-1 (+-,I +Wm) Xm-8

The optimal policy is obtkned,

XOlYl tY2 ,-9**.,Ym I,%#,

together with the optimal value the total accrued value of the claims.

For simplicity, one specifies the

of vm, number

of interim CPA claims equal to 3, i.e. m=4. Since the first interim CPA claim cannot. be placed before month 6 and the last interim CPA claim (or in &her words, the third interim CPA claim) cannot be placed after month 24, according to the constraints, the number of months (states) available to be considered for these 3 interim CPA claims is 24 - 6 + 1 = 19.

The first interim CPA claim can be made at any month between 6 and 12, the second interim claim (stage 2) can be made at any month (state) from 12 to 18, and similarly the third claim can be made at any month from 18 to 24.

Considering the second claim (stage 2)

79

first, month 12 can be used provided the first claim (stage 1) has been placed at month 6 i.e. if the second claim is to be made at month 12 then the first one must have been made at month 6, For each xi, the optimal stage xi-1 is recorded and is called Opt (xi).

Therefore

W&2) = 6; G,(12) = G, (6) +R(6,12) = 6F(6)

Similarly, month 13 can be used for the second claim provided the first claim has been made at months 6 or 7. One selects which of months 6 and 7 gives the maximum return, i.e. compare

P, (6) f R (6,13) = 7F(6)

P, (7) + R (7,13) = dF(7)

Suppose month 6 gives Gz(13), the maxi- mum. Then O;& (13) = 6; and so on. At least a set of Optz(xz) at the second claim is obtained as follows:

Q75I1.2) = 6, Opt,(13) = 6, Opt,(l4) = 8,

opt, (15) = 6, O&(16) = 7, Opf,(17) = 6, and

Opr,(18) = 10

Table 4 shows all maximum values G2(x2) for each value of states in stage 2 including a set of Opt,(x2) as shown in the last column

Now consider at the third claim (stage 3). The set of Opt&,), which can be obtained by the same procedure as in the second claim, is as follows:

Q&(18) = 12, OptJ19) = 13, Opt,(20) = 13,

Opt,(21) = 13, Opr, (22) = 14, Opcb (23) = 12,

Opt{241 = 14

The resulting calculations of stage 3 are shown in Table 5.

At the final claim (i.e. i = m = 4), one selects which of the months (states) in stage 3 gives maximum return, i.e. compare

P&8) +R(18,30) = 9.0 (see Table 6)

P,(19) +R(19,30) = 11.5

P, (20) + R(20,30) = 10.5

P&21) +R(21,30) = 10.0

80

P,(22)+ R(22,?0) = 12.5

$(23)+ R(23,30) = 12.0

P,(24)4 R(24,30) = 11.0

Hence, month 22 gives the maximum. Then @t&30) = 22, with G,(30) = 12.5. The resulting calculations of this stage are shown in Table 6.

The optimal policy can be obtained by considering from the last stage (Table 6)

TABLE 4

towards the first stage (Table 4), as follows:

*, = Opt.&&; Xm = 30

*, = Opt, CYs)

*I = OPqY,)

Hence

*II = 22, since 22 = Opt,(30) (from Table 6)

*, = 14, since 14 = Opr,(22) (from Table 5) and

*, 6 8, since 8 = O&(14) (from Table 4)

The optimal policy is to claim at months 8, 14,22 and 30.

Calculation of stage 2

Stage 1 xi, feasible states P*(xJ =~,(~,)+~(~,J,) G,W,) O&(x,) O~YI for stage 2

6 7 8 9 10 11 12

Stage 2 12 6 4.5 - - - - - - 4.5 6 13 697 4.6 3.7 - - - - - 4.6 6 14 6,798 4.8 4.5 5.0 - - - - 5.0 8 I: . . 6,7,&g 5.1 4.9 3.6 3.5 5.1 - - - 6 iI 6,7,8,9,10 4.2 4.8 4.0 4.5 4.0 - - 4.8 7 17 6,7,8,9,10,11 4.9 4.7 3.6 4.2 3.7 4.0 - 4.9 6 3d 6,7,8,9,10,11,12 3.9 3.8 4.0 4.4 4.5 4.2 4.0 4.5 10 .C)

TABLE 5

Calculation of stage 3

Stage 2 Xi, ieasible states for siagt: 3

Stage 3 18 12 19 12,13 20 12,13,14 21 12,13,14,15 22 12,13,14,15,16 23 12,13,14,15,16,17 24 12,13,14,15,16,17,18 I

TABLE 6

P&l) =Jyq +R (X,J,) GJ(x3) W&J or y1

12 13 14 15 16 17 18

6.5 - - - - - - 6.5 12 7.0 7.2 -. - - - - 7.2 13 7.3 7.5 6.5 - - - - 7.5 13 6.5 7.0 5.8 5.2 - - - 7.0 13 7.0 7.2 7.4 7.2 6.0 - - 7.4 14 7.5 7.4 6.3 7.0 7.2 7.3 - 7.5 12 7.3 7.3 8.0 7.5 7.0 6.5 6.0 8.0 14

.I.

Calculation of the final stage

Stage 3 Xi, feasible states k(x,) = P&s) + R (X&J &(x4) W,(q) or ys for 3tage 4

18 19 20 21 22 23 24

Stage 4 3 18,19,:29,21,22,23,24 9.0 11.5 10.5 10.0 12.5 12.0 11.0 12.5 22

_m