LINEAR PROGRAMNING

We have now reached the stage of having a target output, "import and

export for each industry.

,

^.

Our targets are con-sistent, and seem to be·

reasonable. However, .it would be post;;ib+e to pre'duce many ,e'ther combina-tioa's which are both consistent and apparently reasona1:ile~ . Whàt the planne ne~_ds is the best of all pos~>ible· c~mbinations, in the light ·of ctrrent kn~wledge and ~xpectations. Can this best be found? In theory it ·can be, thoughi~in practice our-'techniques are still rudim.entary.

The best solution would be that which maximized net eutput ~t sorne constant set of priees. The appropriat~.priees are for~ig~ trade pribes.

.. .. ^'

The final demand for each eommodity is given; therefore.the only question is whether to produee it at home, o,r import it. The resourees used _,. for producing i t •~«ld pr.oduçe something ·e·lse for export; if. the _. incom~ _{., .!>.} this w•uld earn· cou'ld in foreign markets buy more of the desired commodity than cotild be ma.de at home wi th the same resour.ces, dornes tic output s'-lould be

.... ,,.. .."1

eut and imports iner~ased. If·we know the quantity of resourees required to make a uriit of each· eommodity, the . . .priees which. expor,

.

t^' s would fetch^{; .} , and the priees at whieh imports cou.ld be bought, we ean calculate the most profitable output of each industry whose produet can be imported or

exporte~.,.·

. . ^:

.!

The mathematical technique for making these calculations :is· krtown as 'programming' , whether 'linea~' . or 'n<:m-linear' depending on the na turE of·the problem. Programmin~ is_a technique with many uses. For example, in development planning it can be use to choose between different

, techniques .for mP.king the sp_me commedi ty; this is ·indeed ·the purp.o_se for

whieh i t is -most o'ften u.sed. Our example, in which i t is to be' use.d for

•n . . :

choosing between home production and foreign trade is som~what e~oteric.

REPRODUCTION/038/80

Page 58. ^.,

The choice between linear and non-linear programming depends on the

.. - .

problem. If cast per unit and priee do not vary with the size of output, the problem is linear~ and is easily solved. But if cost or priee changes as output changes,. the problem is non-linear; and is more difficult. It is di ffi cul te, first ,. bE?,.cause i t is ·not et;?.sy to fi ;nd by established means the -.on-lin-ear. equations which express exactly how priees and. costs, vary with ·output; an·d second,ly,.becaus,e, even i.f one had the

equatio~s ; t~e/

would be difficult to -solve unless they were unrealisti6ally simple.: Mathematicians have:given us~easy methods for solving l~near pr,grammes;

but are still trying to find easy ways to solve non-linear programmes~·

Though there.ehas been sorne progress, we are in practice usually confi'~ed to the use of linear programming.

Now thip is a great ~andicap; since our problems are definitely not

li~ear. Whether the priees ruling in foreign trade will change with .the volume of. purc.hase.3 or sales will depend on the coun trf't sl size; a small èountry, may be able to treat all foreign trade priees· as give·n. How.,.,er·, even -in a small country the quanti ty of re sources per unit of output .. wilL ,depend upon the sfilale of production. Hence t'a use only l'iriea·r programming

to determine the allocation of resources would give the wrohg answer.·~

We can escape this difficulty by programming not the use of all the country's resour~es, but only the use of the last 1 per cent èr so. One thus assumes that an increase or reduction in the output of any commodity by 1 per cent will not affe~t its foreign trade priee o~ its domestic cost. The priees and costs used for the. calculations' will not be average costs or priees, but the marginal costs and priees expected when the level of output of the final year has b\en attained. Hav_ing· thus by programming found which are 'the niore prqfi table commodi ties' ~~ can increase what' was proposed for them, and reduce the ethers, relying once more on kno'i!ledge of output and market conditions.

:···

TABLE XXIII. Proposed Outputs

REPRODUCTION/038/80 Page 60.

-

-.

•

^. .

^. Ex-Service ^;

r!:!~~-E~;:_1:!~i~

____

^'-:'

_________

~~~g~~~~::::::

~gm~~g~~~= Value

======

A 485 48.5 10

B 160 160.0 1

c

171 24.43 7

D· 113 4_5.2 2.5

E 145 _.~9.0 5

F 96 48.0 2

G 173 55.8 3o ^Î

-·--- - ^.

-:--Next we deal with the interm~diate inputs, sho~n in columns and r~ws

. f

A through G in Table XXII. Di vi ding each row horizontaJ.ly by i ts appro-pria te priee turns values into quantities. Then dividing each celuma vertically by the appropriate quantity of the commodity "(taken from the final column of Table XXIII) yields the input per unit of each commodity.

Th~se intermediate inputs appear in the firs~ seven rcws o~ the matrix in Table XXVI, on the le ft of the equ,al sign.

Next row H, Value Added, in Table XXII, must now be broken down into the primary ~nputs Ciactors ~f production), here assumed to be Land, Water, Stone, Unskilled Labou·r, Skilled Labour and Ca pi t'al. The number of factors of production depends on the time horizon. In the short period labour and capital are highly specialized, so one has to work with a great many

different kinds of labour and cap{tal. But over ten years the ·amounts of new labour and new capital are very large, and these can be allocated to many different purposes before becoming specialized, so at the margin the number of primary units is small • . At this stage capital is not the amount invested, but the annual capital cost of the required capital, using the standard formula given on page 56. The unit of capital is the amount which can be hired for one unit of the national currency. Water could be treated

REPRODUCTION/038/80

-:-REPROUCTION/038/80 Page

62. ·

is therefor Final Demand. This is given in columns K through R o~ Table

XXII,· e~tept that the figures in that Table will have to be adjusted downwards becaus~ Final Demand ·th~re.~xce~d~ what is

production, since there is an impprt: surpl~~- of 40.

. • .. . . _,r:

cr,. ·.·.

adjustment in a ·~ornent. ~

.l'

yielded by domestic

W~ shall come to this

.,...

In the mat rix in Table XXVI the symbois; A,!; •

7

G are used-.to

indicate output, while the symbols a, ••••••• ₁

g

indicate ·the: F"inal Demand for each commodity. The difference is used up in intermediate production.

The first seven :equc9:tions express this differen·c€!;. Outputs ·have a plus sign; inputs ·a minus sign. Output is 1.0 if a ·commodi ty .does not use up any of itself in production; if it doee use up s6me of itself,. this is

subtr~cted•

It will be obse~v,ed that instead of syil~ls d and g we have figures

· This is b.ecause D and G. are not traded in s.ignifica.nt quanti t i es. Value is to be maximized at. foreign trade priees, on t:tie . assumption that i t makes no difference to the c0untry whether Final Demand is met from home

o~tput or from imports. We are assuming that Q, cbâstruction~ can be neither import~d .nor exported; and that though D, Local Manufactures, does enter.t into foreign trade, in practice hEfàvy '·t ransport .costs keep imports to zero and exports to a small, alreadj' known., amount. He nee Final Demands d and g are not unknowns. They are found by adding Row D,

columns K through R of Table XXII, which", when divided by a priee of 45.2 yields 1 .• 504 un~:ts; and by adding the 'same columns in Row G, which, divided by-a priee o!: 55.8,. yields 2.921 units.· The'ot·het Final Demands are the five unknowns., which are shawn also 'i~ the maximand. at foreign trade priees.

t 't

.. ^; ..

.. = ...

····· ··--

·-·-:, .

"·

REPRODUCTION/038/80 ^·t

assumptions, each 'yielding different resulta. Since ·the amount involved

i~1 ^small, the a,djus.tment can be made simply by reducing eaeh item

. ' . ^'\ -^~

....

^{' : :}

REPRODUCT'ION/03B/80

Page

65. ·

^.1

. ^; .. . · ..

We now have all the data required for programming. The total quantity of each factor :::of production ,is known. What it takes_to

ma~e

each commodi ty, bath by wày d_f pMmary factors of production, and

( . \ . ^. .

by way of intermediate ·use of cÏther commodi ti:es,. is known. The value of ,each çomm?dity is also

kno~n. ~ow· ~~ hav~ ~o

^find

~he ~a~i ~urn

~al 'Ife wnich can be produced in thi~ _ ~i tuatiori.: This vàlu.e is formally

. . .

termed Final Demand, but the use of the 'word Dernand should not mislead the reader. What we shall really get is net output (~rosé out~ut of each commodity minus intermediate use), since Finalj.Demand includes

.·. . ' .

net experts.

1 : ' .

The forrnal mathematical stateinent of the' p·roblem i~;; given in Table XXVI •

• J.: ; ! ••• . ... 'l''

1 '

This rnodel cannat give

~ ' realist{c

éoltitfbn, because of

i~~

non~.negativity constraints

. ·' ,.

These state that Final Demand cartnot be negative. Now l"i:nal Demand equal.e Domeetic Uee

pltie

^Net

Ex~orts·.

If Final Demand cannot be negative, imports cannat exceed Domestic Use; therefore intermediate 'û'ses cannot be supplied by importing. Such a solution cannat be

., ..

·· accepted· when prograînmil1g for an underd.evelope:d country, for in . practic. .. ,., there will 'be many commoditi-es (e .. g. crude steel) imported rnainly for

in~ermed{ate tises •

. 1

'( ,. ·.·t .

.. · .j

...

REPRODUCTION/038/80 Page 66.

We g~t:··out of this ·difficul ty by changing the non:..riegati vi ty

cons-traints ·to ^l'

AL :

^o,.

^~

^{••• ·.,}

^··G~o

^·.

^..

^':_.··,

indi·catiiig 'that :it is në>w Out which cannat ·be negative~ The. ~riables

a, •• ~~, f are . now redundant;· We get · rid of the~ : by· àubsfi tu :ting in the objective · function·, ti:sing :.thé ·definition§ ·give~ in· equ~tions (1'), (2), (3)

(5) and (6)." This prôduces the· riew objèctive function .which appears in

·Table XXVII. :''

Now we are in a different kind of trouble. So lon·g ·as we had an equation for each commodity, with Final Demand non-negative, the solution wouid h~ve to yiel~ ·some Outpcit fo~ ~~ery co~modity ~hi~h has intermediate uses. With these equations gone, the number of constraints shrinks

drastically. Linear programming cannet yield. outputs for more commodities 'than there ~re t~~~trairit~~ In .. this e~a~ple ther~ aré~~~ven commodites

and eight constraints, but this is only:becaude w~ ~re ~orking with so few commodi ti es. In reë;t}istiç, exerçt~e.~ti.t· ·the number of commodi ti es will

.... ^. .

greatly exceed the number of factors of production, if one is working with

~s long a horizon a~ ten yea~~. The ~~luticiri ~ill th~n indicate that the econofuy should produce only a~ uftf~ali~fiê~llY s~all 'riufub~r of' commodities

. ^'

This difficultl aris~~ becacise we are ~ppl~{ng linear programming to a situati~n which is definit~ly not linear. Îf costs were constant and

. . . . . .

priees infinitely elastic it would indeed pay an economy to specialize in very few commodities. One cannat hope to get' the right 'ans~er if one uses inappropriate models.

One way round this difficulty is to apply additional artificial constraints. Jssume that the planned output for each commodity is not far out from what it should be, and therefore do not let the programmed output vary from the planned by more than say

5

per cent either way. Putting a limit to the expansion of the more profitable commodities reduces the number whose output can fall to zero. This method can be used to produce sharp distinctions by reiteration at different levels; by imposing limits of say 10 per cent, then 5 per cent, then 3 per cent and then 1 per cent,

(1)

T .i' BLE XXVIlo Programming Model₂ Stage II REPRODUCTION/038/80 Maximize Z

=

^{42.57 À} + 67.11 B + 18.14

c -

12.8"1 D + 21.95 E + 21.98 F- 6.30 G Page 68.

Subject to:

lA B

c

^{D E F G ~}

(1) -0.042 +1.000

.. .

-0.186 = 1.504

•• . .

(2) •• -0.029

..

^{... .:. ... o54 +1.000 = 2.921}

(3) 11.0 +14.0 +4.0 +2.0 +1.0 +1.0 +2.0 170.200

(4) 1.0 + 4.0 +1.0 +1.0 +1.0 +2.0 +1 .o 35.6

(5) 1,o + 2.0 +2.0 +2.0 +l.o.: +1.0 +4.0 50.4

(6) 23.0 +44.0 +9.0 +11.0 +6.0 ·+7.0 +19é0 467.4

(7) 2.0 +10.0 +2.0 + 4.0 +4.0 +5.0 + 6.0 102.6

(8) 2.0 6.0 +2.0 +10.0 +8.0 +9.0 + 8.0 147.8

A _{0 ' • • • '} G ⁰

'

' ; ,

. .

; . 1

REPRODUCTION/038/80 Page 69.

one can arrive at a rough arder of priority for the commodities. This is only a subterfuge; a somewhat doubtful method of evnding the fact that our madel does not fit the situation we are analysing.

First Trial

We are now ready to proseed. The problem to be solved is given in Table XXVI in its revi~ed for~. Final Demand is to be ~aximised. subject to eight constraints. The fi~st two indicate that final demands d and

g are already known. The ether six indièàte the maximum availabilities of the six factors of production.

···

~

^{··The' solbtibn1 of"this}

~roblem ·

^{is given·iQ'}Table XXVIII.

Pr~gram med

Final Demand cornes to 730~0 whereas ·Planned Final Demand (Table XXVI)

' . . .

came only to· 721.8 ...

1. As stated in the Preface, one of the objectives in writing of this chapter has been to present the.material in such a ~~y that the average persan likely to read it may check the estimates for himself. This

cannat be achieved iri this section on linear progtamm.ing. To do all these calculations on

a

desk calculator "'ould take too .IT)uch time, and would almost certainly result in many a;rithmetical errors.. This calculation was done ~n the Princeton electronic computer;

i

a~.grateful to my colleague Mr. E.S. Pearsall, who :pt.ogrammed it for me, and to Mr. John Hill, who programmed an earlier version. The work made use of computer

facilities supported in part by National Science Foundation Grant NSF-GP 579o

1 ; •

a

360 . 4

b

'169.9

c

66.3

i

. 257 . 9

1lt6 . 0 38. 8

~EFRO:OUe'f:t:Ot:/0,8/SO

Page

70 •

e

212 . 2 125. 7

111 0 1 6. 0 5. 6 13 . 0 -135 . ?

-24o.L~

f

= ^{;;:: ~ --~} __ ^{i--~~ - ~:~: : - ~~--=}

^--~-r--~---0 ·-'"""'"

L

----.·- - -⁰ - -··- ._. •--•• ^{! 1 ' i}

Table·XXVIII nlso translates the results ;!n~o a programmes .fo~ for eigp t~

Final Demand equals Domestic Use (Table XXll) p:)_us net exports. ~ Domestic Use is first adjusted upvJards to equal Final Demand in total. · Net. experts then ïollm.r

b~ subtraction.

This, at firat sight, is a niost impressive result. The Agenc-;t does not need to gues.s what ~he .outputp_.and foreign trade should be; giveh assumptions about· pri~es and càsts' i t can calculate 1tlhat outputs \vill maxirri:l.ze the national income, given available ::r:es.ourçe:S. .T.he assumptions are heroic, but i t is

striki~g to f'ind a tool \·ihich forces them tp :r.ield an answer.

Second Trial

Let us look more closely nt the answer. What progl~amming tells us is that given the resources which could produce the planned output, it would be better to prod'..lCe the programmed output. But are the resources given?

Take nnskilled labour. The sixth constraint in Table XXVII states that the amount of unskilled labour should not exceed

467 . 4

units. This constraint is needed if full employment is expected by Year

10,

but in an over-populated country, \'li th open or disguised unemployment, this constraint is not necessary.

REPRO,DUCTION/038/80 ... ,, Page

71

Similarly the third constraint limits the use of land ta

170.2

units. This constraint is not required in sparsely populated countries. ·Sven' in over-populated countries more land can be brought into' cultivation by reclamation, terracing and irrigation, in which ca5e, instead of a land constraint one should simply add the cast of preparing the land ta the cast of the commodi tièiç; ·.ùich which will produce. It is not necessary to introduce ~11 the factors of produc-tion into the equations.

If unskilled labour is as.sumed to be abundant, we elimina. te the sixt.h con-straint in Table XXV:J:I and solve again. The result is shawn in the third column of Table XXIX. Then we replace the unskilled labour constraint and assume that land instead is abundant, removing the third constraint. The result is shawn in the fourth column.

TABLE XXIX.

Original Plan

First ^V!i thout . Without

Trial Labour Land

\

· -- --- - - ---- ^-· ---- _ ____ _

^.....

^-

^..

- ·- - - -

^-----~---.-..

--- ^-

^...

---

^..

_

^....

a

382. 0 360 . 4 3,72.4 536 . 6

b

145.0 169.9 291.2 ••

c

42. 3 66 . 3 -85 . 2 66 . 7

e

145 . 0 212 . 2

^232.~·

:21 :9. 4

f

7.5 -76 . 8 -75 . lt -74.7

., ______________ ....

----

..

---

^·--

- --- -

^{... ..---~--}---·- ·-·

721 . 8 732.0 735 . 4

- -

^~·----·

- - ---.- - "'- ----oJ·--- -- - ---·-· - --- -

^{..: ....}

_ --.- --- --- - ·----· -- - ·--- ----·

REPRODUCTION/03a/8p Page 72.

The choice of constraints obviously makes a creat difference to the oJlswer; in this example not so much difference to the total, but an enormous difference to the allocation among different commoàities.

: . ^~ Third Trial

In the second trial \ve assumeà that one or ether of the facto~·s was abundant. Now sorne other factors are not abundant, but can be made available at a priee. The supply of skilled labour ean be increased by training - wl,ich

· takes a long time - or by importation .from abroad. 'rhe f?Upply of: capital can

also be increased by importation from abroad. If more· of a factor can be obtain at a priee, we should ·riot prbceed as if its quantity were fixed. The correct procedure is to subtract the cost of the factor from the priees of the commoditü into which it enters, and maximize the surplus.

We can tell \vhether to imp.ort more or less of any importable factor lJ:r comparing i ts priee wi th i ts marginal product. V!hen a linear programme is solvec it yields two sets of solutions. One set,.which He have been using, is the best combinat ion of outputs.- The oi;her set, knovm as the 'dual' , is :the priees of th factors of production,_·i.e. the marginal productivity, corresponding to that

• co:~bination of outputs. If the dual priee of an importable factor .exceeds the cost of importing it, the national incarne can be inereased by import:i.ng more of i t. i:Je have be en assuming that the priee of each factor is uni ty. Table XXX shows the dual priees derived from the second trial when the unskilled labour constraint \"/as moved.

TABLE XXX. Dual Factor Priees

Land 3.2

\-Jater 0

Stone 0

Skilled 1.2 Capital 1.7

He cannat import more land, and the cast of importing skilled labour is likely to exeeed the domestic priee by more than 20 per cent, so we can ignore these two. But if capital can be hired for 1.0 when it yields 1.7, this constraint should disappear.

REPRODUCTION/038/80 Page 73.

This is achieved by subtracting the element of capital cost in each commodity from the priee of the commodity. One must subtract not only the direct capital input, but also the capital in the intermediate inputs. To gct at the total capital input, one has to solve the input equations yielJed by each column of Table XXVI, the typical equation being

0.961A-0.078C-ü.042D-0.058F=2

the figure on the right side being the number of units of capital. Solving these seven equations yields the following capital inputs, intermediate plus primary.

A 3.6 E 9o2

B 13.6 F 11.9

c

^{2.8 G 11.8}

D 12.1

The priee of capital being unity, we subtract these figures from the priees to get a new objective function, comparable with that of Stage I , XXVI.

Maximize S•44.9a+131.4b+19.2c+19.8e+28.1!

This transformations in Stage II, Table XXVII, into Maximize. S:40.0A+61.2B+16.1C-11.1D+14.0E+12.4F-4.7G

The solution is given in Table XXXI, except that, for the sake of comparability with the earlier solutions, the commodities are valued at priees which includc the cast of capital. Both columns assume that unskilled labour is abundant. In the first column no additional capital is imported; in the second column unlimited capital is available at a priee of 1 •. 0 per unit.

TABLE XXXI. Programmed Final Demand I I I

a b c e f

Limited Capital 372.4

291.2 -85.2 232.4 -75.&4 735.4

Capital Priced 602.4

REPRODUCTION/038/80

-Conclusion

REPRODUCTION/038/80 Page

75 .

He obsc:rved at the outset that whi le linea.r programming is a useful tool

He obsc:rved at the outset that whi le linea.r programming is a useful tool

Dans le document Development planning the arithmetic of planning (Page 58-78)