Question

Example Suppose that the economy of the city of Everett depends on three industries Logging-timber operation, electricity and aircraft-manufacturing plant. Monitoring the operations of these three industries over a period of one year, we were able to come up with the following observations: 1. To produce 1 unit worth of timber, the Logging-timber industry must consume 0.3 units of its own production, 0.3 units of electricisy and 0.3 units of aircraft to run its operations 2. To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of Logging-timber, 0.1 units of its own production, and 0.5 units of aircraft. 3. Finally, the aircraft-manufacturing plant requires 0.3 units of Logging-timber 0.6 units of electricity and 0.2 units of its own production to produce I unit of aircraft. Find the production level of and the internal demands assuming that the above model is closed, that is, no goods leave or enter the system each of these industries in order to satisfy the external Solution

Answer 1

Production level of each of these industries in order to satisfy the external and internal demands that the above model is closed Let x1, x2, x3 be the total output of industry S1 (Logging timber operation), S2(electricity), S3(aircraft manufacturing plant), respectively. Then, To produce 1 aircraft manufacturing unit of Logging timber operation,ele ant timber Logging timber operation 0.3 0.3 0.3 electricity electricity 0.4 0.1 0.5 aircraft manufacturing plant 0.3 0.6 0.2 aircraft x10.3x1 + 0.3x2 + 0.3x3 X2 = 0.4X1 + 0.1X2 + 0.5X3 X1 = 0.3x1 + 0.6x2 + 0.2x3 To solve the system, we take all variables on left side and make the right side 0, then we get -0.7x1 + 0.3x2 + 0.3x3-0 0.4x1- 0.9x2 + 0.5x3- 0 0.3x1 +0.6x2 - 0.8x3 0 which is a homogeneous equation and must has a solution

Multiplying every equation by 10 and then we get the augmented matrix 7 3 3 0 4 9 4 -9 5 0 3 6 -8 0 51 47 7R2+4R1 0 -47 7R3+3R1 51 which can be transformed by elementary row operations to the matrix 47/51 R2/-51 0 -14 R+R2 0 0 The system has infinitely many solutions, taking parameter the production level can be described as 17 -47 X1 (14/17) t For example if we take t-51 then x1-42 x2=47, and x=51. 0 1 0 x2(47/51) t 51 Any multiple of 51 will work as solution. 0