Линейное программирование. Ч.1. Учебное пособие для бакалавриата
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Answers for questions and EXERCISES
5) The level line is perpendicular to the gradient.
6)
a)Build the feasible region.
b)If the feasible region is not empty then set aside the gradient vector from the origin.
c)Construct a straight line (the level line) perpendicular to the gradient and move this line in the positive direction of the gradient vector.
d)The last intersection of the level line with the feasible region is the point of maximum.
7)
a)Build the feasible region.
b)If the feasible region is not empty then set aside the gradient vector from the origin.
c)Construct a straight line (the level line) perpendicular to the gradient and move this line in the positive direction of the gradient vector.
d)The first intersection of the level line with the feasible region is the point of maximum.
8) The last intersection of the level line with the feasible region is a side.
9) The first intersection of the level line with the feasible region is a side.
10) There is no last intersection of the level line with the feasible region when moving in the positive direction of the gradient.
Exercises
1)xmin =(3;2), fmin =12 .
2)xmax =(4;6), fmax =16 .
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Answers for questions and EXERCISES
3)xmin =(1;2), fmin =7 .
4)xmax =(3;5), fmax =21.
5)fmin = −∞.
6)xmin =(2+4t;3+t); t [0;1] , fmin = −20 .
7)xmax =(2+2t;3−2t); t [0;1] , fmax = −4 .
8)The problem is infeasible.
9)fmax = +∞, xmin =(2;3;0;0;2), fmin =13 .
10)xmin =(4 −4t;3+2t;0;5−2t;10t); t [0;1] , fmin = −9 , xmax =(2;0;8;0;1), fmax = −1.
Section 4
Question
1)The simplex method is used if the following conditions fulfilled.
a) A linear programming problem is written in the canonicals form.
b) The system of main constrains is solved by JordanGauss method so we know basic and free variables.
c) The free terms of main constrains are non-negative. d) The objective function is expressed in terms of the
free variables.
2)The evaluation of the free variable xj in the simplex table is the coefficient γj in the row corresponding objective function (the index row) at the intersection with the column corresponding to the free variable xj .
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Answers for questions and EXERCISES
3) The basic solution X1 is obtained from the constraints under the assumption that all free variables are equal to zero.
X1 =(b1,b2, ,br ,0, ,0).
4)The basic solution is a corner point (a vertex) of the feasible region.
5)From a geometric point of view the main idea of the simplex method is moving from one corner point (a basic solution) of the feasible region to another corner point to improve the value of the objective function until it will be reached the optimum.
6)
a)Choose j-th pivot column from the condition: the evaluation γj <0 and at least one element aij >0 ;
b)Choose k-th pivot row from the condition:
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c)Recalculate the elements of the simplex table by using the Jordan-Gauss algorithm.
d)Repeat step 1.
7)
a)Choose j-th pivot column from the condition: the evaluation γj >0 and at least one element aij >0 ;
b)Choose k-th pivot row from the condition:
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c)Recalculate the elements of the simplex table by using the Jordan-Gauss algorithm.
d)Repeat step 1.
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Answers for questions and EXERCISES
8)All evaluations of free variables are non-negative
(γj ≥0) .
9)All evaluations of free variables are non-positive
(γj ≤0) .
10)All evaluations of free variables are non-negative, but there is at least one evaluation equal to zero.
11)All evaluations of free variables are non-positive, but there is at least one evaluation equal to zero.
12)All evaluations of free variables are positive (γj >0) .
13)All evaluations of free variables are negative (γj <0).
14)There is a negative evaluation and, in the column, corresponding to this evaluation all the elements are negative.
15)There is a positive evaluation and, in the column, corresponding to this evaluation all the elements are negative.
Exercises
1)xmax =(0;0;2;0;9), fmax =14 .
2)xmax =(0, 3t, 0, 3−3t, 0,5−3t); t [0,1]. fmax =3.
3)xmin =(4;1;0;0;5), fmin =2 .
4)fmax = +∞.
5)xmax =(6;9;2;0;0), fmax =14 .
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Answers for questions and EXERCISES
Section 5
Question
1)The method of artificial variable is used if a linear programming problem is written in the canonicals form and the system of main constraints is not reduced to the unit basis (it means that the basic and free variables are not determined and there is no basic solution).
2)We add an artificial variable in equation where there is no basic variable.
3)F = −y1 − −ym
where variables y1, ,ym are the artificial variables.
4) The artificial objective function is equal to zero
F=0 .
5)The artificial objective function is less than zero
F<0 .
Exercises
1)fmax = +∞.
2)xmax =(2+4t, 3+t, 0, 7 −t, 7t); t [0,1]. fmax =20 .
3)xmax =(1;0;0;4), fmax =3.
4)The problem is infeasible.
5)The problem is infeasible.
6)xmax =(2;3;0;15;0), fmax =3.
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RECOMMENDED LITERATURE
1.Математическая экономика: Учебник / И. Г. Шандра. — М: Прометей, 2018. — 182 с.
2.Методы оптимальных решений в экономике и финансах: Учебник / Под ред. В.М. Гончаренко и В.Ю Попова. — М: КНОРУС, 2017. — 400 с.
3.Методы оптимальных решений в экономике и финансах: Учебное пособие. Серия. Конспект лекций / Под ред. В.М. Гончаренко и В.Ю Попова. — М: КНОРУС, 2022. — 182 с.
4.Методы оптимальных в экономике и финансах: Учебное пособие. Практикум / Под ред. В.М. Гончаренко
иВ.Ю Попова. — М: КНОРУС, 2016. — 298 с.
5.Математические методы в экономике и финансах: Учебник / Под ред. В.М. Гончаренко и В.Ю Попова. — М: КНОРУС, 2016. — 602 с.
6.Математика в экономике. Ч. 1: Учебник / А.С. Солодовников [и др.]. — 3-е изд., перераб. и доп. — М.: Финансы и статистика; Инфра-М, 2011 — 384 с.
7.Сборник задач по курсу «Математика в экономике». Ч. 1: Учебное пособие / Под ред. В.А. Бабайцева
иВ.Б. Гисина. — М.: Финансы и статистика, 2013. — 255 c.
8.Сборник задач по курсу математики: Учебное пособие / Под ред. А.С. Солодовниковa и В.А. Браилова. — М.: Финансовая академия, 2001. — 508 c.
9.Высшая математика для экономического бакалавриата: Учебник и практикум / Н.Ш. Кремер [и др.]; под ред. Н.Ш. Кремера. — 4-е изд., перераб. и доп. — М.: Юрайт, 2012, 2014. — ЭБС Юрайт.
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Recommended literature
10.Математика для экономистов и менеджеров [электронный ресурс]: Учебник / Под ред. Н.Ш. Кремера. — М.: КНОРУС, 2015. — ЭБС: book.ru
11.Математика для экономистов и менеджеров. Практикум [электронный ресурс]: Учебное пособие / Н.Ш. Кремер, Б.А. Путко, М.Н. Фридман; под ред. Н.Ш. Кремера. — М.: КНОРУС, 2015. — ЭБС: Book.ru
12.Высшая математика [электронный ресурс]: Учебник и практикум для академического бакалавриата / М.Б. Хрипунова [и др.]; под общ. ред. М.Б. Хрипуновой,
И.И.Цыганок. — М.: Юрайт, 2017. — ЭБС Юрайт.
13.Математика: Часть 2. Линейная алгебра в LMS Moodle: Учебник для бакалавриата / Л.В. Липагина, Е.Ф. Олехова, А.А. Рылов — М.: Прометей, 2023. — 312 с.
14.Simon C.P. Mathematics for Economists / C.P. Simon, L. Blume. — Norton Company. N.-Y., 1994. — 930 с.
15.Dowling E. Mathematical Economics / E. Dowlin. — Shaum’s Outline Series. N.-Y., 2006. — 850 с.
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Учебное издание
ШАНДРА Игорь Георгиевич
ЛИНЕЙНОЕ
ПРОГРАММИРОВАНИЕ
ЧАСТЬ 1
Учебное пособие для бакалавриата экономических и математических вузов и факультетов
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