- Email: [email protected]
Disjunctive programming and the generalized Leontief input–output model
Available online at www.sciencedirect.com
Applied Mathematics and Computation 198 (2008) 551–558 www.elsevier.com/locate/amc
Disjunctive programming and the generalized Leontief input–output model Aniekan Ebiefung a
a,*
, Michael Kostreva b, Ishita Majumdar
b
University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States b Clemson University, Clemson, SC 29634, United States
Abstract This paper considers a generalization of the Leontief input–output model that is useful in modeling the concept of choice of technology. It is shown that a disjunctive programming problem, together with its dual problem, may be used to effectively solve the new input–output model. Various theoretical results are presented together with an illustrative example. Three production sectors, each having two possible choices of technology, are provided in the example and only one linear program is required for finding the solution. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Leontief models; Input–output; Disjunctive programming; Linear programming; Choice of technology
1. Introduction Input–output analysis is the name given to an analytical framework developed by Professor Wassily Leontief in the late 1930s, work for which he received the Nobel Prize in Economic Science in 1973. One often speaks of a Leontief model when referring to input–output models. The term inter-industry analysis is also used, since the fundamental purpose of the input–output framework is to analyze the interdependence of industries in an economy. An input–output model, in its most basic form, consists of a system of linear equations, each one of which describes the distribution of an industry’s product throughout the economy. The input–output model has been generalized and applied to solve many and varied problem instances. For example, Leontief [1], Ebiefung and Kostreva [2], applied the model to the problem of choosing the right combination of technologies for a given economy. For other application instances, see [3–6]. The purpose of this paper is to generalize the model in the context of disjunctive programming. The generalized model will be applied to select an optimal combination of technologies for the economy. Such technologies ensure that the economy produces at minimum production cost.
*
Corresponding author. E-mail address: [email protected] (A. Ebiefung).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.08.055
552
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
2. Notation and fundamental relationships An input–output model is constructed from observed data for a particular economic location: a nation, a region, a state, etc. The economic activity in the area must be divided into a number of segments or producing sectors. The necessary data are the flows of products from each of the sectors (as a supplier) to each of the sectors (as a consumer). External demands and inventories are also considered as inputs. These inter-industry flows (or inter-sectoral flows) are measured for a particular time period (usually a year) and in monetary terms. Assume that the economy is divided into n sectors. If we denote the total output of sector i by Xi, the total final demand for product of sector i by Yi, and the unit of flow of output from sector i to sector j by aij, borrowing the notation in [4], we obtain the equation n X aij X j þ Y i ; i ¼ 1; . . . ; n: ð1Þ Xi ¼ j¼1
In matrix notation, after re-arrangement, this is equivalent to ðI AÞX ¼ Y : The existence of solutions depends on whether or not the matrix (I A) is singular; that is, it depends on whether or not (I A)1 exists. The matrix A is known as the matrix of technical coefficients or technology matrix. If (I A)1 exists, then the unique solution of the system is given by X = (I A)1Y. The inverse matrix (I A)1 is often referred to as the Leontief inverse. 3. Formulation as a disjunctive program The purpose of this section is to investigate a novel generalization of the basic Leontief input–output model, namely, one from disjunctive programming that is useful in modeling choice of technology. We start with some notation. Notation n number of sectors in the economy mj number of technologies available to sector j Tj the technology selected by sector j Sp {T1, . . ., Tn} = the technology Q state for the economy S {S1, . . ., SQ}, where Q ¼ nj¼1 mj Ap the technology matrix representing a given state of the economy, where 1 6 p 6 Q bp the demand for goods corresponding to a given state of the economy xj production quantity of sector j corresponding to a given state of the economy aij the monetary value of the output of sector i needed by sector j to produce one unit of monetary value of its own output using the given state of the economy
3.1. Formulation We will assume that change of technology does not involve a change in product, but could lead to an increase in the quantity of products produced. We also assume one period of activities. The conditions that the economic system has a feasible technology state x for some Sp 2 S is given by GLM: ðI Ap Þx P bp ;
ð2Þ
x P 0; p
p
where 1 6 p 6 Q. The matrix A P 0. Consequently, (I A ) is a Z-matrix; a matrix in which the off-diagonal elements are non-positive [7]. If the GLM is feasible, then the LP:
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
553
zo ¼ min et x s:t: ðI Ap Þx P bp
ð3Þ
x P 0; has a solution, which is the least element of the feasible region of (2) by [8,9]. Thus, a solution of (3) satisfies the GLM. Consider the following programming problem: PP: wo ¼ max fw : ^S p 2S ðw up bp 6 0; up 2 CÞg ð4Þ where ^ denotes the AND operator and C ¼ fup : up ðI Ap Þ 6 et ;
up P 0g
Assume that (3) and (4) are both feasible and have finite optimal objective values. We consider the following theorem. Theorem 1.If the economic system, GLM, has a feasible technology state with production vector x, then there is a w vector which solves the PP. u Proof. Suppose that the economic system has a feasible technology state Sp2S for some p, 1 6 p 6 Q. Let x be the production vector corresponding to Sp. Then there exists a technology matrix Ap such that ðI Ap Þx P bp ;
x P 0;
p
where b is the corresponding demand vector. Since (I Ap) is a Z-matrix, we can take x to be the least element of the polyhedral set [8,9] U ¼ fx : ðI Ap Þx P bp ;
x P 0g:
By [6,8,9], x is the optimal solution of the LP: zo ¼ min et x s:t: ðI Ap Þx P bp x P 0; where 1 6 p 6 Q. By duality theory, the dual program (DP): wo ¼ max up bp s:t: up ðI Ap Þ 6 et up P 0 has a solution upo P 0 such that zo ¼ et xo ¼ up0 bp ¼ wo : We can now rewrite the above dual program as zo ¼ maxfup bp : up 2 Cg: But, then zo ¼ maxfup bp : up 2 Cg ¼ maxfw : w up bp 6 0; up 2 Cg P maxfw : ^S p 2S ðw up bp 6 0; up 2 CÞg: This implies that zo P w. Thus, the maximum value of w is w0 = z0. This completes the proof. Observe that problem (4) is equivalent to wo ¼ max w s:t:
^S p 2S fw up bp 6 0;
up ðI AP Þ 6 et ;
up P 0g
h
ð5Þ
554
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
Alternatively, (5) can be written as (DDJP): wo ¼ max up bp s:t:
^S p 2S fup ðI AP Þ 6 et ;
ð6Þ
up P 0g:
The dual of problem (6) is (DJP): zo ¼ min et x _ fðI Ap Þx P bp ; s:t:
ð7Þ
x P 0g;
S p 2S
where ¤ is the OR operator. Problem (7) is called a disjunctive program [10]. The following theorem shows the relationship between the generalized Leontief model and the disjunctive program. As before, we assume that both problems are feasible with finite optimal objective values. Theorem 2. The generalized Leontief model (GLM) is equivalent to the disjunctive programming problem (DJP). Proof. We shall show that a vector x solves the GLM if and only if it solves the DJP. If x solves the GLM, then by Theorem 1, it solves the DJP by the relationship between Eqs. (4)–(7). Conversely, suppose that x solves the DJP. Then there exists at least one l, 1 6 l 6 Q, such that, ðI Al Þx P bl ;
x P 0;
Consequently, given Al, there is a technology state Sl for the economy with production vector x. Hence, x solves the GLM. This completes the proof. h We now examine the behavior of the DJP when the economy has no feasible technology state. Theorem 3. Suppose that the economy has no feasible technology state, then 1. The disjunctive program (DJP) is infeasible. 2. The dual of the disjunctive program (DDJP) is infeasible or unbounded.
Proof. Case 1: Consider proof by contradiction. Suppose that the DJP is feasible. Then there exists h, 1 6 h 6 Q, such that U 5 /. In particular, there is an ^x 2 U such that ðI Ah Þ^x P bh : This implies that given the technology matrix Ah there exists a technology state Sh 2 S, and a production vector ^x, such that ðI Ah Þ P bh ;
^x P 0;
where 1 6 h 6 Q, Sh 2 S. This contradicts the assumption that the economy has no feasible technology state. Hence, if the economy has no feasible technology state, the DJP is infeasible. Case 2: If the economy has no feasible technology state, then the LP min
z ¼ et x
s:t:
x 2 U;
is infeasible for each p, 1 6 p 6 Q. This is because (I Ap) is a Z-matrix and no technology state means that the feasible region is empty [6,8,9] and therefore, the LP is infeasible. Hence, the dual program max w ¼ up bp s:t: up 2 C; is infeasible or unbounded for each p, 1 6 p 6 Q. Therefore, the DDJP is infeasible or unbounded.
h
Theorem 4. The economy has feasible technology state if and only if the disjunctive programs, DJP and DDJP, are feasible and bounded.
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
555
Proof. The first part of the theorem follows from Theorem 3. For the second part, suppose the dual is feasible and bounded. Then there exists at least one LP, indexed by h*, with optimal objective value z* = w*. Since row j of the LP is from sector j of the economy, the economy has a feasible technology state. h 4. Example Consider an economy with three sectors, say, 1, 2, and 3. Suppose that the sectors’ outputs are in pairs of shoes, pounds of food, and four packs of light bulbs. Suppose that for each sector, there are two technology alternatives, Technology 1 and Technology 2. The sectors’ operations are based on a single period of activities. Let e = (1, 1, 1) be the cost vector, that is, one unit of each of the output costs 1 dollar. We define the states of the economic system as 0 1 i B C S p ði; j; kÞ ¼ @ j A; k where i, j, k are technologies used by industries 1, 2, and 3, respectively. Suppose that under the current operations, the technology matrices and demand vectors are as follows: 0 1 0 1 0:6 0:1 0:3 150 B C B C S 1 ð1; 1; 1Þ : A1 ¼ @ 0:3 0:6 0:1 A; b1 ¼ @ 500 A; 0
0:1 0:3 0:6 0:1
B S 2 ð1; 1; 2Þ : A2 ¼ @ 0:3 0:6 0
0:2 0:1 0:6 0:1
B S 3 ð1; 2; 1Þ : A3 ¼ @ 0:3 0:1 0
0:1 0:2 0:6 0:2
B S 4 ð1; 2; 2Þ : A4 ¼ @ 0:1 0:1 0
0:2 0:3 0:1 0:1
B S 5 ð2; 1; 1Þ : A5 ¼ @ 0:1 0:6 0
0:3 0:3 0:1 0:1
B S 6 ð2; 1; 2Þ : A6 ¼ @ 0:2 0:6 0
0:2 0:1 0:1 0:3
B S 7 ð2; 2; 1Þ : A7 ¼ @ 0:6 0:1 0
0:0 0:2 0:1 0:3
B S 8 ð2; 2; 2Þ : A8 ¼ @ 0:6 0:1 0:2 0:3
0:6 1 0:1 C 0:3 A; 0:1 1 0:3 C 0:1 A;
20 1 100 B C b2 ¼ @ 150 A; 200 0 1 150 B C b3 ¼ @ 200 A;
0:6 1 0:1 C 0:4 A; 0:1 1 0:2 C 0:3 A;
80 1 100 B C b4 ¼ @ 100 A; 200 0 1 150 B C b5 ¼ @ 400 A;
0:6 1 0:5 C 0:3 A; 0:1 1 0:4 C 0:5 A;
20 1 150 B C b6 ¼ @ 400 A; 20 0 1 100 B C b7 ¼ @ 150 A;
0:6 1 0:5 C 0:4 A; 0:1
200 1 100 B C b8 ¼ @ 150 A: 200:
0
0
0
0
We assume that bp P 0 if bp represents the requirement at the end of the period, and bp < 0 if bp represents the net number of units available at the start of the period.
556
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
Table 1 Solutions to the disjunctive program Economic state
Objective value
Pairs of shoes: x1
Food: x2
Bulbs: x3
S1(1, 1, 1) S2(1, 1, 2) S3(1, 2, 1) S4(1, 2, 2) S5(2, 1, 1) S6(2, 1, 2) S7(2, 2, 1) S8(2, 2, 2)
469.23 818.18 375.00 550.72 276.66 1033.63 6341.27 1662.75
415.38 363.63 375.00 0.00 186.66 98.65 1682.54 423.53
0.00 136.36 0.00 246.37 0.00 571.74 2492.06 692.16
53.84 318.18 0.00 304.34 90.00 363.22 2166.67 547.06
Let x1 = number of pairs of shoes to be produced by sector 1. x2 = number of pounds of food to be produced by sector 2. x3 = number of four packs of light bulbs to be produced by sector 3. As before, we assume that technology change does not involve change of product but could lead to change in the quantity produced. 4.1. Solution by disjunctive programming The primal disjunctive program, Eq. (7), is: min x1 þ x2 þ x3 8 < 0:4x1 0:1x2 0:3x3 P 150 0:3x1 þ 0:4x2 0:1x3 P 500 : 0:1x1 0:3x2 þ 0:4x3 P 20 8 _ < 0:4x1 0:1x2 0:1x3 P 100 0:3x1 þ 0:4x2 0:1x3 P 150 : 0:2x1 0:1x2 þ 0:9x3 P 200 _ 8 _ < 0:9x1 0:3x2 0:5x3 P 100 0:6x1 þ 0:9x2 0:4x3 P 150 : 0:2x2 0:3x2 þ 0:9x3 P 200: The solutions to the eight distinct linear programs corresponding to the DJP are summarized below: Observe that the minimum cost is attained when the economy chooses state S5(2, 1, 1). Consequently, given the current economic conditions, the economy will produce at minimum cost if sector 1 chooses Technology 2, sector 2 chooses Technology 1, and sector 3 chooses Technology 1 (see Table 1). Remark. The economy can produce, depending on other circumstances, at any of the 8 states since they all have solutions. For example, if the country is at war and more shoes are needed, then the economy could decide to produce at state S1 or S2. For a different set of data, the economy may have a feasible technology state, while some states are infeasible. 4.2. Solution by dual disjunctive program In this section, we solve Example 4 using the dual disjunctive program (DDJP). For each technology state Sp of the economy, we define the Lagrange multipliers up1, up2, up3, where p = 1, . . ., 8. The corresponding DDJP is:
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
max s:t:
557
w w 150u11 þ 500u12 þ 20u13 6 0; 0:4u11 0:3u12 0:1u13 6 1;
ð8Þ ð9Þ
0:1u11 þ 0:4u12 0:3u13 6 1; 0:3u11 0:1u12 þ 0:4u13 6 1;
ð10Þ ð11Þ
w 100u21 þ 150u22 200u13 6 0;
ð12Þ
0:4u21 0:3u22 0:2u13 6 1; 0:1u21 þ 0:4u22 0:1u13 6 1;
ð13Þ ð14Þ
0:1u21 0:3u22 þ 0:9u13 6 1; w 150u31 þ 200u32 þ 80u33 6 0;
ð15Þ ð16Þ
0:4u31 0:3u32 0:1u33 6 1;
ð17Þ
0:1u31 þ 0:9u32 0:2u33 6 1; 0:3u31 0:1u32 þ 0:4u33 6 1;
ð18Þ ð19Þ
w þ 100u41 100u42 200u43 6 0; 0:4u41 0:1u42 0:2u43 6 1;
ð20Þ ð21Þ
0:2u41 þ 0:9u42 0:3u43 6 1;
ð22Þ
0:1u41 0:4u42 þ 0:9u43 6 1; w 150u51 þ 400u52 þ 20u53 6 0;
ð23Þ ð24Þ
0:9u51 0:1u52 0:3u53 6 1; 0:1u51 þ 0:4u52 0:3u53 6 1;
ð25Þ ð26Þ
0:2u51 0:3u52 þ 0:4u53 6 1;
ð27Þ
w þ 150u61 þ 400u62 þ 20u63 6 0; 0:9u61 0:2u62 0:2u63 6 1;
ð28Þ ð29Þ
0:1u61 þ 0:4u62 0:1u63 6 1; 0:5u61 0:3u62 þ 0:9u63 6 1;
ð30Þ ð31Þ
w þ 100u71 150u72 200u73 6 0;
ð32Þ
0:9u71 0:6u72 0:0u73 6 1; 0:3u71 þ 0:9u72 0:2u73 6 1;
ð33Þ ð34Þ
0:4u71 0:5u72 þ 0:4u73 6 1; w þ 100u81 150u82 200u83 6 0;
ð35Þ ð36Þ
0:9u81 0:6u82 0:2u83 6 1;
ð37Þ
0:3u81 þ 0:9u82 0:3u83 6 1; 0:5u81 0:4u82 þ 0:9u83 6 1:
ð38Þ ð39Þ
A solution of this LP is $276.66, which agrees with the opimal value of the DJP. To find the corresponding production quantities, we look for a constraint of the form w upbp that is tight and has a positive dual price. The 24th constraint satisfies this condition. Thus, w ¼ 150u51 400u52 20u53
ð40Þ
gives the objective function value. Consequently, the 25th–27th constraints’ dual prices give the values of the variables. For example, the 25th constraint is tight 0:9u51 0:1u52 0:3u53 ¼ 1
ð41Þ
558
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
and has the dual prices of 186.667, which is equal to x1, the number of pair of shoes to be produced by sector 1. The 26th constraint is not tight 0:1u51 þ 0:4u52 0:3u53 6 1
ð42Þ
and has dual price of 0.00, which is equal to x2, the number of pounds of food to the produced by sector 2. The 27th constraint is tight 0:2u51 0:3u52 þ 0:4u53 ¼ 1
ð43Þ
and has dual price of 90.00, which is equal to x3, the number of four packs of light bulbs to be produced by sector 3. 5. Conclusion The mathematical model considered in this paper is a novel generalization of the Leontief input–output model. Starting from the classical foundations, it allows for a fixed number of variations or states. It may be compared with another model, considered earlier by the authors [2], and seen to be distinct from the earlier work. The new model has a feasible region that is the union of linear polyhedral sets, rather than the intersection of such sets. This feature adds realism, especially in applications such as choice of technology. Indeed, not all possibilities of the choice of technology may be feasible in realistic applications. The earlier models could not comprehend such a scenario, yet the new model can handle it with ease. It is worth a remark here that the computational aspect of the new model is positive. That is, even though it deals with a union of polyhedral sets, a non-convex set, it can be turned into a linear programming problem, by means of duality. This dual linear programming problem, once generated, can be solved by standard packages such as LINDO or MATLAB, which are widely available. Thus, specialized computer codes are not required, only standard, widely available packages. This fact is an advantage that should not be overlooked. In addition, another computational feature should be noticed. The dual linear programming problem, above, has a large number of variables and a large number of constraints. That these factors limit the applicability of the new model can be conjectured, but, so far, it is a subject for further research. Finally, since it is widely recognized that input–output models describe some very basic, but essential, workings of economies, it seems beneficial to study them in new forms and to unite them with themes such as choice of technology [1,2]. At this stage, input–output models may be considered as part of technology since they are so widely accepted. Yet, we see that there is room for more research to merge input–output models with results from mathematical programming [2,8–10] and to make the input–output paradigm even more comprehensive and influential in the 21st century. References [1] W.W. Leontief, The choice of new technology, Scientific American (1985) 37–45. [2] A.A. Ebiefung, M.M. Kostreva, The generalized Leontief input–output model and its application to the choice of new technology, Annals of Operations Research 44 (1993) 161–172. [3] W.W. Leontief, F. Daniel, Air pollution and the economic structure: empirical results of input–output computations, in: A. Brody, A.P. Carter (Eds.), Input–Output Techniques, North-Holland, New York, 1972. [4] R.E. Miller, P.D. Blair, Input–output Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1985. [5] T.T. Raa, The Economics of Input–Output Analysis, Cambridge University Press, Cambridge, 2006. [6] A.A. Ebiefung, M.M. Kostreva, The generalized linear complementary problem: least element theory and Z-matrices, Journal of Global Optimization 11 (1997) 151–161. [7] M. Fiedler, V. Ptak, On matrices with nonpositive off-diagonal elements and positive principal minors, Czechhoslovak Mathematics Journal 12 (1960) 382–400. [8] A. Tamir, Minimality and complementarity properties associated with Z- and M-functions, Management Science 7 (1974) 17–31. [9] R.W. Cottle, A. Veinott, Polyhedral sets having a least element, Mathematical Programming 3 (1972) 238–249. [10] E. Balas, Disjunctive programming, Annals of Discrete Mathematics 5 (1979) 3–51.
Applied Mathematics and Computation 198 (2008) 551–558 www.elsevier.com/locate/amc
Disjunctive programming and the generalized Leontief input–output model Aniekan Ebiefung a
a,*
, Michael Kostreva b, Ishita Majumdar
b
University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States b Clemson University, Clemson, SC 29634, United States
Abstract This paper considers a generalization of the Leontief input–output model that is useful in modeling the concept of choice of technology. It is shown that a disjunctive programming problem, together with its dual problem, may be used to effectively solve the new input–output model. Various theoretical results are presented together with an illustrative example. Three production sectors, each having two possible choices of technology, are provided in the example and only one linear program is required for finding the solution. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Leontief models; Input–output; Disjunctive programming; Linear programming; Choice of technology
1. Introduction Input–output analysis is the name given to an analytical framework developed by Professor Wassily Leontief in the late 1930s, work for which he received the Nobel Prize in Economic Science in 1973. One often speaks of a Leontief model when referring to input–output models. The term inter-industry analysis is also used, since the fundamental purpose of the input–output framework is to analyze the interdependence of industries in an economy. An input–output model, in its most basic form, consists of a system of linear equations, each one of which describes the distribution of an industry’s product throughout the economy. The input–output model has been generalized and applied to solve many and varied problem instances. For example, Leontief [1], Ebiefung and Kostreva [2], applied the model to the problem of choosing the right combination of technologies for a given economy. For other application instances, see [3–6]. The purpose of this paper is to generalize the model in the context of disjunctive programming. The generalized model will be applied to select an optimal combination of technologies for the economy. Such technologies ensure that the economy produces at minimum production cost.
*
Corresponding author. E-mail address: [email protected] (A. Ebiefung).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.08.055
552
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
2. Notation and fundamental relationships An input–output model is constructed from observed data for a particular economic location: a nation, a region, a state, etc. The economic activity in the area must be divided into a number of segments or producing sectors. The necessary data are the flows of products from each of the sectors (as a supplier) to each of the sectors (as a consumer). External demands and inventories are also considered as inputs. These inter-industry flows (or inter-sectoral flows) are measured for a particular time period (usually a year) and in monetary terms. Assume that the economy is divided into n sectors. If we denote the total output of sector i by Xi, the total final demand for product of sector i by Yi, and the unit of flow of output from sector i to sector j by aij, borrowing the notation in [4], we obtain the equation n X aij X j þ Y i ; i ¼ 1; . . . ; n: ð1Þ Xi ¼ j¼1
In matrix notation, after re-arrangement, this is equivalent to ðI AÞX ¼ Y : The existence of solutions depends on whether or not the matrix (I A) is singular; that is, it depends on whether or not (I A)1 exists. The matrix A is known as the matrix of technical coefficients or technology matrix. If (I A)1 exists, then the unique solution of the system is given by X = (I A)1Y. The inverse matrix (I A)1 is often referred to as the Leontief inverse. 3. Formulation as a disjunctive program The purpose of this section is to investigate a novel generalization of the basic Leontief input–output model, namely, one from disjunctive programming that is useful in modeling choice of technology. We start with some notation. Notation n number of sectors in the economy mj number of technologies available to sector j Tj the technology selected by sector j Sp {T1, . . ., Tn} = the technology Q state for the economy S {S1, . . ., SQ}, where Q ¼ nj¼1 mj Ap the technology matrix representing a given state of the economy, where 1 6 p 6 Q bp the demand for goods corresponding to a given state of the economy xj production quantity of sector j corresponding to a given state of the economy aij the monetary value of the output of sector i needed by sector j to produce one unit of monetary value of its own output using the given state of the economy
3.1. Formulation We will assume that change of technology does not involve a change in product, but could lead to an increase in the quantity of products produced. We also assume one period of activities. The conditions that the economic system has a feasible technology state x for some Sp 2 S is given by GLM: ðI Ap Þx P bp ;
ð2Þ
x P 0; p
p
where 1 6 p 6 Q. The matrix A P 0. Consequently, (I A ) is a Z-matrix; a matrix in which the off-diagonal elements are non-positive [7]. If the GLM is feasible, then the LP:
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
553
zo ¼ min et x s:t: ðI Ap Þx P bp
ð3Þ
x P 0; has a solution, which is the least element of the feasible region of (2) by [8,9]. Thus, a solution of (3) satisfies the GLM. Consider the following programming problem: PP: wo ¼ max fw : ^S p 2S ðw up bp 6 0; up 2 CÞg ð4Þ where ^ denotes the AND operator and C ¼ fup : up ðI Ap Þ 6 et ;
up P 0g
Assume that (3) and (4) are both feasible and have finite optimal objective values. We consider the following theorem. Theorem 1.If the economic system, GLM, has a feasible technology state with production vector x, then there is a w vector which solves the PP. u Proof. Suppose that the economic system has a feasible technology state Sp2S for some p, 1 6 p 6 Q. Let x be the production vector corresponding to Sp. Then there exists a technology matrix Ap such that ðI Ap Þx P bp ;
x P 0;
p
where b is the corresponding demand vector. Since (I Ap) is a Z-matrix, we can take x to be the least element of the polyhedral set [8,9] U ¼ fx : ðI Ap Þx P bp ;
x P 0g:
By [6,8,9], x is the optimal solution of the LP: zo ¼ min et x s:t: ðI Ap Þx P bp x P 0; where 1 6 p 6 Q. By duality theory, the dual program (DP): wo ¼ max up bp s:t: up ðI Ap Þ 6 et up P 0 has a solution upo P 0 such that zo ¼ et xo ¼ up0 bp ¼ wo : We can now rewrite the above dual program as zo ¼ maxfup bp : up 2 Cg: But, then zo ¼ maxfup bp : up 2 Cg ¼ maxfw : w up bp 6 0; up 2 Cg P maxfw : ^S p 2S ðw up bp 6 0; up 2 CÞg: This implies that zo P w. Thus, the maximum value of w is w0 = z0. This completes the proof. Observe that problem (4) is equivalent to wo ¼ max w s:t:
^S p 2S fw up bp 6 0;
up ðI AP Þ 6 et ;
up P 0g
h
ð5Þ
554
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
Alternatively, (5) can be written as (DDJP): wo ¼ max up bp s:t:
^S p 2S fup ðI AP Þ 6 et ;
ð6Þ
up P 0g:
The dual of problem (6) is (DJP): zo ¼ min et x _ fðI Ap Þx P bp ; s:t:
ð7Þ
x P 0g;
S p 2S
where ¤ is the OR operator. Problem (7) is called a disjunctive program [10]. The following theorem shows the relationship between the generalized Leontief model and the disjunctive program. As before, we assume that both problems are feasible with finite optimal objective values. Theorem 2. The generalized Leontief model (GLM) is equivalent to the disjunctive programming problem (DJP). Proof. We shall show that a vector x solves the GLM if and only if it solves the DJP. If x solves the GLM, then by Theorem 1, it solves the DJP by the relationship between Eqs. (4)–(7). Conversely, suppose that x solves the DJP. Then there exists at least one l, 1 6 l 6 Q, such that, ðI Al Þx P bl ;
x P 0;
Consequently, given Al, there is a technology state Sl for the economy with production vector x. Hence, x solves the GLM. This completes the proof. h We now examine the behavior of the DJP when the economy has no feasible technology state. Theorem 3. Suppose that the economy has no feasible technology state, then 1. The disjunctive program (DJP) is infeasible. 2. The dual of the disjunctive program (DDJP) is infeasible or unbounded.
Proof. Case 1: Consider proof by contradiction. Suppose that the DJP is feasible. Then there exists h, 1 6 h 6 Q, such that U 5 /. In particular, there is an ^x 2 U such that ðI Ah Þ^x P bh : This implies that given the technology matrix Ah there exists a technology state Sh 2 S, and a production vector ^x, such that ðI Ah Þ P bh ;
^x P 0;
where 1 6 h 6 Q, Sh 2 S. This contradicts the assumption that the economy has no feasible technology state. Hence, if the economy has no feasible technology state, the DJP is infeasible. Case 2: If the economy has no feasible technology state, then the LP min
z ¼ et x
s:t:
x 2 U;
is infeasible for each p, 1 6 p 6 Q. This is because (I Ap) is a Z-matrix and no technology state means that the feasible region is empty [6,8,9] and therefore, the LP is infeasible. Hence, the dual program max w ¼ up bp s:t: up 2 C; is infeasible or unbounded for each p, 1 6 p 6 Q. Therefore, the DDJP is infeasible or unbounded.
h
Theorem 4. The economy has feasible technology state if and only if the disjunctive programs, DJP and DDJP, are feasible and bounded.
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
555
Proof. The first part of the theorem follows from Theorem 3. For the second part, suppose the dual is feasible and bounded. Then there exists at least one LP, indexed by h*, with optimal objective value z* = w*. Since row j of the LP is from sector j of the economy, the economy has a feasible technology state. h 4. Example Consider an economy with three sectors, say, 1, 2, and 3. Suppose that the sectors’ outputs are in pairs of shoes, pounds of food, and four packs of light bulbs. Suppose that for each sector, there are two technology alternatives, Technology 1 and Technology 2. The sectors’ operations are based on a single period of activities. Let e = (1, 1, 1) be the cost vector, that is, one unit of each of the output costs 1 dollar. We define the states of the economic system as 0 1 i B C S p ði; j; kÞ ¼ @ j A; k where i, j, k are technologies used by industries 1, 2, and 3, respectively. Suppose that under the current operations, the technology matrices and demand vectors are as follows: 0 1 0 1 0:6 0:1 0:3 150 B C B C S 1 ð1; 1; 1Þ : A1 ¼ @ 0:3 0:6 0:1 A; b1 ¼ @ 500 A; 0
0:1 0:3 0:6 0:1
B S 2 ð1; 1; 2Þ : A2 ¼ @ 0:3 0:6 0
0:2 0:1 0:6 0:1
B S 3 ð1; 2; 1Þ : A3 ¼ @ 0:3 0:1 0
0:1 0:2 0:6 0:2
B S 4 ð1; 2; 2Þ : A4 ¼ @ 0:1 0:1 0
0:2 0:3 0:1 0:1
B S 5 ð2; 1; 1Þ : A5 ¼ @ 0:1 0:6 0
0:3 0:3 0:1 0:1
B S 6 ð2; 1; 2Þ : A6 ¼ @ 0:2 0:6 0
0:2 0:1 0:1 0:3
B S 7 ð2; 2; 1Þ : A7 ¼ @ 0:6 0:1 0
0:0 0:2 0:1 0:3
B S 8 ð2; 2; 2Þ : A8 ¼ @ 0:6 0:1 0:2 0:3
0:6 1 0:1 C 0:3 A; 0:1 1 0:3 C 0:1 A;
20 1 100 B C b2 ¼ @ 150 A; 200 0 1 150 B C b3 ¼ @ 200 A;
0:6 1 0:1 C 0:4 A; 0:1 1 0:2 C 0:3 A;
80 1 100 B C b4 ¼ @ 100 A; 200 0 1 150 B C b5 ¼ @ 400 A;
0:6 1 0:5 C 0:3 A; 0:1 1 0:4 C 0:5 A;
20 1 150 B C b6 ¼ @ 400 A; 20 0 1 100 B C b7 ¼ @ 150 A;
0:6 1 0:5 C 0:4 A; 0:1
200 1 100 B C b8 ¼ @ 150 A: 200:
0
0
0
0
We assume that bp P 0 if bp represents the requirement at the end of the period, and bp < 0 if bp represents the net number of units available at the start of the period.
556
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
Table 1 Solutions to the disjunctive program Economic state
Objective value
Pairs of shoes: x1
Food: x2
Bulbs: x3
S1(1, 1, 1) S2(1, 1, 2) S3(1, 2, 1) S4(1, 2, 2) S5(2, 1, 1) S6(2, 1, 2) S7(2, 2, 1) S8(2, 2, 2)
469.23 818.18 375.00 550.72 276.66 1033.63 6341.27 1662.75
415.38 363.63 375.00 0.00 186.66 98.65 1682.54 423.53
0.00 136.36 0.00 246.37 0.00 571.74 2492.06 692.16
53.84 318.18 0.00 304.34 90.00 363.22 2166.67 547.06
Let x1 = number of pairs of shoes to be produced by sector 1. x2 = number of pounds of food to be produced by sector 2. x3 = number of four packs of light bulbs to be produced by sector 3. As before, we assume that technology change does not involve change of product but could lead to change in the quantity produced. 4.1. Solution by disjunctive programming The primal disjunctive program, Eq. (7), is: min x1 þ x2 þ x3 8 < 0:4x1 0:1x2 0:3x3 P 150 0:3x1 þ 0:4x2 0:1x3 P 500 : 0:1x1 0:3x2 þ 0:4x3 P 20 8 _ < 0:4x1 0:1x2 0:1x3 P 100 0:3x1 þ 0:4x2 0:1x3 P 150 : 0:2x1 0:1x2 þ 0:9x3 P 200 _ 8 _ < 0:9x1 0:3x2 0:5x3 P 100 0:6x1 þ 0:9x2 0:4x3 P 150 : 0:2x2 0:3x2 þ 0:9x3 P 200: The solutions to the eight distinct linear programs corresponding to the DJP are summarized below: Observe that the minimum cost is attained when the economy chooses state S5(2, 1, 1). Consequently, given the current economic conditions, the economy will produce at minimum cost if sector 1 chooses Technology 2, sector 2 chooses Technology 1, and sector 3 chooses Technology 1 (see Table 1). Remark. The economy can produce, depending on other circumstances, at any of the 8 states since they all have solutions. For example, if the country is at war and more shoes are needed, then the economy could decide to produce at state S1 or S2. For a different set of data, the economy may have a feasible technology state, while some states are infeasible. 4.2. Solution by dual disjunctive program In this section, we solve Example 4 using the dual disjunctive program (DDJP). For each technology state Sp of the economy, we define the Lagrange multipliers up1, up2, up3, where p = 1, . . ., 8. The corresponding DDJP is:
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
max s:t:
557
w w 150u11 þ 500u12 þ 20u13 6 0; 0:4u11 0:3u12 0:1u13 6 1;
ð8Þ ð9Þ
0:1u11 þ 0:4u12 0:3u13 6 1; 0:3u11 0:1u12 þ 0:4u13 6 1;
ð10Þ ð11Þ
w 100u21 þ 150u22 200u13 6 0;
ð12Þ
0:4u21 0:3u22 0:2u13 6 1; 0:1u21 þ 0:4u22 0:1u13 6 1;
ð13Þ ð14Þ
0:1u21 0:3u22 þ 0:9u13 6 1; w 150u31 þ 200u32 þ 80u33 6 0;
ð15Þ ð16Þ
0:4u31 0:3u32 0:1u33 6 1;
ð17Þ
0:1u31 þ 0:9u32 0:2u33 6 1; 0:3u31 0:1u32 þ 0:4u33 6 1;
ð18Þ ð19Þ
w þ 100u41 100u42 200u43 6 0; 0:4u41 0:1u42 0:2u43 6 1;
ð20Þ ð21Þ
0:2u41 þ 0:9u42 0:3u43 6 1;
ð22Þ
0:1u41 0:4u42 þ 0:9u43 6 1; w 150u51 þ 400u52 þ 20u53 6 0;
ð23Þ ð24Þ
0:9u51 0:1u52 0:3u53 6 1; 0:1u51 þ 0:4u52 0:3u53 6 1;
ð25Þ ð26Þ
0:2u51 0:3u52 þ 0:4u53 6 1;
ð27Þ
w þ 150u61 þ 400u62 þ 20u63 6 0; 0:9u61 0:2u62 0:2u63 6 1;
ð28Þ ð29Þ
0:1u61 þ 0:4u62 0:1u63 6 1; 0:5u61 0:3u62 þ 0:9u63 6 1;
ð30Þ ð31Þ
w þ 100u71 150u72 200u73 6 0;
ð32Þ
0:9u71 0:6u72 0:0u73 6 1; 0:3u71 þ 0:9u72 0:2u73 6 1;
ð33Þ ð34Þ
0:4u71 0:5u72 þ 0:4u73 6 1; w þ 100u81 150u82 200u83 6 0;
ð35Þ ð36Þ
0:9u81 0:6u82 0:2u83 6 1;
ð37Þ
0:3u81 þ 0:9u82 0:3u83 6 1; 0:5u81 0:4u82 þ 0:9u83 6 1:
ð38Þ ð39Þ
A solution of this LP is $276.66, which agrees with the opimal value of the DJP. To find the corresponding production quantities, we look for a constraint of the form w upbp that is tight and has a positive dual price. The 24th constraint satisfies this condition. Thus, w ¼ 150u51 400u52 20u53
ð40Þ
gives the objective function value. Consequently, the 25th–27th constraints’ dual prices give the values of the variables. For example, the 25th constraint is tight 0:9u51 0:1u52 0:3u53 ¼ 1
ð41Þ
558
A. Ebiefung et al. / Applied Mathematics and Computation 198 (2008) 551–558
and has the dual prices of 186.667, which is equal to x1, the number of pair of shoes to be produced by sector 1. The 26th constraint is not tight 0:1u51 þ 0:4u52 0:3u53 6 1
ð42Þ
and has dual price of 0.00, which is equal to x2, the number of pounds of food to the produced by sector 2. The 27th constraint is tight 0:2u51 0:3u52 þ 0:4u53 ¼ 1
ð43Þ
and has dual price of 90.00, which is equal to x3, the number of four packs of light bulbs to be produced by sector 3. 5. Conclusion The mathematical model considered in this paper is a novel generalization of the Leontief input–output model. Starting from the classical foundations, it allows for a fixed number of variations or states. It may be compared with another model, considered earlier by the authors [2], and seen to be distinct from the earlier work. The new model has a feasible region that is the union of linear polyhedral sets, rather than the intersection of such sets. This feature adds realism, especially in applications such as choice of technology. Indeed, not all possibilities of the choice of technology may be feasible in realistic applications. The earlier models could not comprehend such a scenario, yet the new model can handle it with ease. It is worth a remark here that the computational aspect of the new model is positive. That is, even though it deals with a union of polyhedral sets, a non-convex set, it can be turned into a linear programming problem, by means of duality. This dual linear programming problem, once generated, can be solved by standard packages such as LINDO or MATLAB, which are widely available. Thus, specialized computer codes are not required, only standard, widely available packages. This fact is an advantage that should not be overlooked. In addition, another computational feature should be noticed. The dual linear programming problem, above, has a large number of variables and a large number of constraints. That these factors limit the applicability of the new model can be conjectured, but, so far, it is a subject for further research. Finally, since it is widely recognized that input–output models describe some very basic, but essential, workings of economies, it seems beneficial to study them in new forms and to unite them with themes such as choice of technology [1,2]. At this stage, input–output models may be considered as part of technology since they are so widely accepted. Yet, we see that there is room for more research to merge input–output models with results from mathematical programming [2,8–10] and to make the input–output paradigm even more comprehensive and influential in the 21st century. References [1] W.W. Leontief, The choice of new technology, Scientific American (1985) 37–45. [2] A.A. Ebiefung, M.M. Kostreva, The generalized Leontief input–output model and its application to the choice of new technology, Annals of Operations Research 44 (1993) 161–172. [3] W.W. Leontief, F. Daniel, Air pollution and the economic structure: empirical results of input–output computations, in: A. Brody, A.P. Carter (Eds.), Input–Output Techniques, North-Holland, New York, 1972. [4] R.E. Miller, P.D. Blair, Input–output Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1985. [5] T.T. Raa, The Economics of Input–Output Analysis, Cambridge University Press, Cambridge, 2006. [6] A.A. Ebiefung, M.M. Kostreva, The generalized linear complementary problem: least element theory and Z-matrices, Journal of Global Optimization 11 (1997) 151–161. [7] M. Fiedler, V. Ptak, On matrices with nonpositive off-diagonal elements and positive principal minors, Czechhoslovak Mathematics Journal 12 (1960) 382–400. [8] A. Tamir, Minimality and complementarity properties associated with Z- and M-functions, Management Science 7 (1974) 17–31. [9] R.W. Cottle, A. Veinott, Polyhedral sets having a least element, Mathematical Programming 3 (1972) 238–249. [10] E. Balas, Disjunctive programming, Annals of Discrete Mathematics 5 (1979) 3–51.