Линейное программирование. Практикум. Учебное пособие для бакалавриата

7. Tranportation problem

z(X) =1750 <2170.

The plan

is optimal. We have proved that testing the initial basic plan obtained by the Least Cost Rule.

Questions for self-control

1)Formulateamathematicalmodelofthetransportation problem.

2)What is a closed transportation problem?

3)Formulate the stages solving the transportation problem?

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7. Tranportation problem

Exercises for independent work

1) Find the initial plan for the transportation problem

using a) North West Corner Rule; b) the Least Cost Rule.

2) Find the initial plan for the transportation problem

using a) North West Corner Rule; b) the Least Cost Rule.

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7. Tranportation problem

3) Solve the transportation problem a)

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7. Tranportation problem

c)

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CONCLUSION

Linear programming as a science appeared in the first half of the twentieth century to solve applied problems of economics. Linear programming is still actively used to solve both theoretical and applied problems in various branches of science. Therefore, linear programming takes an important place in the knowledge system of a modern economist.

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ANSWERS FOR QUESTIONS

AND EXERCISES

Section 2

Questions

1) Linear programming is a branch of applied mathematics that deals with solving problems of maximization or minimization of a linear function subject to linear constraints (it means that constraints are linear equations or linear inequalities).

2) A linear programming problem is the following optimizing problem

f = c1x1 + +cmxm +c0 → max (min)

Subject to

a11x1 + +a1mxm ≤ b1,

,

an1x1 + +anmxm ≤ bn,

d11x1 + +d1mxm = q1,

,

dk1x1 + +dkmxm = qk, x1 ≥ 0, ,xm ≥ 0 .

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Answers for questions and EXERCISES

f= CT X +C0 → max (min)

AX ≤ B,

3)DX = Q,X ≥ 0.

4)The linear function f to be maximized or minimized

is called the objective function.

(x1*,...,xn* )T of the linear

programming problem is called the optimal solution.

AX ≤ B,

6) DX = Q.

7) x1 ≥ 0, ,xm ≥ 0 .

8) A vector X = (x1,...,xn )T satisfying the constraints of the linear programming problem is called a feasible solution.

9) The set of all feasible solutions is called the feasible

set.

10) A linear programming problem is said to be feasible if the feasible set is not empty.

11) A linear programming problem can have a unique solution, infinitely many solutions, or no solutions.

12) A set M is said to be convex if for any x,y M and any t [0;1] it follows that tx +(1−t)y M .

13) The feasible set of a linear programming problem

AX = B,

X ≥ 0.

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Answers for questions and EXERCISES

15) Any linear programming problem can be transformed to the canonicals form.

16)

f = CT X +C0 → max

AX ≤ B,

X ≥ 0.

17) If a linear programming problem written in the canonicals form system of basic constraints reduced to the unit basis, then the linear programming problem can be transformed to the standard form.

Exercises

1)

f= x1 −5x2 +2x3 → max

x1 −x2 +4x3 =7,

a)x1 −4x2 +y1 =2,

x1 +2x2 +3x3 −y2 = 6,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0,y1 ≥ 0,y2 ≥ 0.

f= x1 +2x2 −x4 → min

3= 3,

b)2x1 −2x2 +6x4 +y1 =1,x1 −x2 +2x3 −x4 +y2 =2,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0,x4 ≥ 0,y1 ≥ 0,y2 ≥ 0.x1 +2x2 −5x

2)

f = −x1 +4x2 −4 → max

a)x1 −x2 ≤1,3x1 −2x2 ≤5,x1 ≥ 0, x2 ≥ 0.

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Answers for questions and EXERCISES

f = x1 −3x2 +1 → min

b)x1 +2x2 ≤1,2x1 −x2 ≤5,x1 ≥ 0, x2 ≥ 0.

3)

x1

f = (1 −1 2) x2 → minx3

x1

b) f = (1 −3 1 4) x2 → minx3

x4

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