IGNOU MTE 12 SOLVED ASSIGNMENT HINDI

Ques 1.

बताइए निम्नलिखित कथनों में से कौन-से कथन सत्य हैं और कौन-से असत्य। अपने उत्तर की पुष्टि एक संक्षिप्त उपपत्ति या प्रत्युदाहरण द्वारा कीजिए
a) परिमित संख्या में अवमुख समुच्चयों का सर्वनिष्ठ अवमुख नहीं होता है।
b) यदि  आव्यूह खेल  का मान 4 हो, तो ।
c) यदि किसी  नियतन समस्या में लागत आव्यूह की प्रत्येक प्रविष्टि में 10 जोड़ा जाए, तो परिवर्तित लागत आव्यूह के लिए इष्टतम नियतन की कुल लागत में 10 की वृद्धि हो जाएगी।
d) किसी अधिकतमीकरण रैखिक प्रोग्रामन (LP) निदर्श में, जब सभी मान  हों, तो एकधा विधि सम्पन्न हो जाती है।
e) एक परिवहन समस्या में अपभ्रष्ट हल से बचने के लिए काल्पनिक (dummy) स्रोत या गंतव्य जोड़ा जाता है।

Ques 2.

) निम्नलिखित दो खिलाड़ी शून्य योग खेल को प्रमुखता सिद्धांत द्वारा  खेल में समानीत कीजिए और इस प्रकार खेल को हल कीजिए।

Ques 3.

निम्नलिखित आद्य रैखिक प्रोग्रामन (एलपी) समस्या की द्वैती प्राप्त कीजिए :

 का अधिकतमीकरण कीजिए
जबकि 

Ques 4.

 एक कम्पनी दो प्रकार की चमड़े की बेल्ट बनाती है। बेल्ट A उच्च कोटि की व बेल्ट B निम्न कोटि की है। दोनों बेल्टों A और B पर लाभ क्रमशः ₹ 4 और ₹ 3 प्रति बेल्ट है। A प्रकार की बेल्ट को बनाने में B, प्रकार की बेल्ट को बनाने के समय से दुगुना समय लगता है और यदि सभी बेल्ट केवल B, प्रकार की ही हों, तो कम्पनी प्रतिदिन 1000 बेल्ट बना पाती है। चमड़े की पूर्ति भी प्रतिदिन केवल 800 बेल्टों (दोनों A और B प्रकार की मिलाकर) के लिए ही उपलब्ध है।

A प्रकार की बेल्ट के लिए एक फैन्सी तुकमें (बकल) की आवश्यकता है और प्रतिदिन केवल 400 तुकमें (बकल) ही उपलब्ध हैं। B. प्रकार की बेल्ट के लिए प्रतिदिन 700 तुकमें ही उपलब्ध हैं। प्रत्येक प्रकार की बेल्ट का प्रतिदिन कितना उत्पादन होना चाहिए? इस समस्या को रैखिक प्रोग्रामन (LP) निदर्श में सूत्रित कीजिए और इसे ग्राफीय-विधि द्वारा हल कीजिए। 
b) उत्तर-पश्चिम कोना विधि का प्रयोग करके निम्नलिखित परिवहन समस्या का प्रारम्भिक आधारी सुसंगत हल ज्ञात कीजिए : 

Ques 5.

. State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter example.
a) The intersection of finite number of convex sets is not convex.
b) If value of the  matrix game  is 4, then .
c) If 10 is added to each of the entries of the cost matrix of a  assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
d) For maximization LP model, the simplex method is terminated when all values .
e) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate.

Ques 6.

 Reduce the following two person zero sum game to  game using principle of dominance. And hence solve the game.

Player B

b) Obtain the dual of the following primal LP problem:
Maximize 
Subject to 
$
$

Ques 7.

 A company makes two kinds of leather belts. Belts A is high quality belt and belt B is of lower quality. The respective profits on A and B are ₹ 4 and ₹ 3 per belt. The production of each type A requires twice as much time as a belt. The production of each type of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A

and B combined). Belt A require a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by the graphical method.

b) Find the initial basic feasible solution of the following transportation problem using North-West Corner method.

Ques 8.

wo breakfast food manufacturers ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ. Determine optimal strategies for both the manufacturers and the value of the game.

b) Find all the basic feasible solutions of the following system of linear equations:

$
$
$
Check if any of them is degenerate solution. Justify your answer.

Ques 9.

A department has five employees with five jobs to be performed. The time (in hours) each employee will take to perform each job is given in the following matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?

b) For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:

Ques 10.

 Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

b) Check whether the following set is convex:

Ques 11.

) Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

b) Check whether the following set is convex:

$

Ques 12.

) For what value of k are the following vectors linearly independent?
$
b) Solve the following LP problem using simplex method:
Maximize 
Subject to 
$
$
$

Ques 13.

Write the LPP formulation of the following transportation problem:

b) Solve the following assignment problem for profit maximization:

Ques 14.

Write the LPP formulation of the following assignment problem:

b) Solve the following game graphically:

Ques 15.

Solve the following LPP graphically:

Maximize:

$
subject to the constraints:
$
$
$
b) Find all values of k for which the vectors:
$
are linearly independent.

Ques 16.

बताइए निम्नलिखित कथनों में से कौन-से कथन सत्य हैं और कौन-से असत्य। अपने उत्तर की पुष्टि एक संक्षिप्त उपपत्ति या प्रत्युदाहरण द्वारा कीजिए
a) परिमित संख्या में अवमुख समुच्चयों का सर्वनिष्ठ अवमुख नहीं होता है।
b) यदि  आव्यूह खेल  का मान 4 हो, तो ।
c) यदि किसी  नियतन समस्या में लागत आव्यूह की प्रत्येक प्रविष्टि में 10 जोड़ा जाए, तो परिवर्तित लागत आव्यूह के लिए इष्टतम नियतन की कुल लागत में 10 की वृद्धि हो जाएगी।
d) किसी अधिकतमीकरण रैखिक प्रोग्रामन (LP) निदर्श में, जब सभी मान  हों, तो एकधा विधि सम्पन्न हो जाती है।
e) एक परिवहन समस्या में अपभ्रष्ट हल से बचने के लिए काल्पनिक (dummy) स्रोत या गंतव्य जोड़ा जाता है।

Ques 17.

) निम्नलिखित दो खिलाड़ी शून्य योग खेल को प्रमुखता सिद्धांत द्वारा  खेल में समानीत कीजिए और इस प्रकार खेल को हल कीजिए।

Ques 18.

निम्नलिखित आद्य रैखिक प्रोग्रामन (एलपी) समस्या की द्वैती प्राप्त कीजिए :

 का अधिकतमीकरण कीजिए
जबकि 

Ques 19.

 एक कम्पनी दो प्रकार की चमड़े की बेल्ट बनाती है। बेल्ट A उच्च कोटि की व बेल्ट B निम्न कोटि की है। दोनों बेल्टों A और B पर लाभ क्रमशः ₹ 4 और ₹ 3 प्रति बेल्ट है। A प्रकार की बेल्ट को बनाने में B, प्रकार की बेल्ट को बनाने के समय से दुगुना समय लगता है और यदि सभी बेल्ट केवल B, प्रकार की ही हों, तो कम्पनी प्रतिदिन 1000 बेल्ट बना पाती है। चमड़े की पूर्ति भी प्रतिदिन केवल 800 बेल्टों (दोनों A और B प्रकार की मिलाकर) के लिए ही उपलब्ध है।

A प्रकार की बेल्ट के लिए एक फैन्सी तुकमें (बकल) की आवश्यकता है और प्रतिदिन केवल 400 तुकमें (बकल) ही उपलब्ध हैं। B. प्रकार की बेल्ट के लिए प्रतिदिन 700 तुकमें ही उपलब्ध हैं। प्रत्येक प्रकार की बेल्ट का प्रतिदिन कितना उत्पादन होना चाहिए? इस समस्या को रैखिक प्रोग्रामन (LP) निदर्श में सूत्रित कीजिए और इसे ग्राफीय-विधि द्वारा हल कीजिए। 
b) उत्तर-पश्चिम कोना विधि का प्रयोग करके निम्नलिखित परिवहन समस्या का प्रारम्भिक आधारी सुसंगत हल ज्ञात कीजिए : 

Ques 20.

. State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter example.
a) The intersection of finite number of convex sets is not convex.
b) If value of the  matrix game  is 4, then .
c) If 10 is added to each of the entries of the cost matrix of a  assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
d) For maximization LP model, the simplex method is terminated when all values .
e) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate.

Ques 21.

 Reduce the following two person zero sum game to  game using principle of dominance. And hence solve the game.

Player B

b) Obtain the dual of the following primal LP problem:
Maximize 
Subject to 
$
$

Ques 22.

 A company makes two kinds of leather belts. Belts A is high quality belt and belt B is of lower quality. The respective profits on A and B are ₹ 4 and ₹ 3 per belt. The production of each type A requires twice as much time as a belt. The production of each type of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A

and B combined). Belt A require a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by the graphical method.

b) Find the initial basic feasible solution of the following transportation problem using North-West Corner method.

Ques 23.

wo breakfast food manufacturers ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ. Determine optimal strategies for both the manufacturers and the value of the game.

b) Find all the basic feasible solutions of the following system of linear equations:

$
$
$
Check if any of them is degenerate solution. Justify your answer.

Ques 24.

A department has five employees with five jobs to be performed. The time (in hours) each employee will take to perform each job is given in the following matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?

b) For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:

Ques 25.

 Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

b) Check whether the following set is convex:

Ques 26.

) Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

b) Check whether the following set is convex:

$

Ques 27.

) For what value of k are the following vectors linearly independent?
$
b) Solve the following LP problem using simplex method:
Maximize 
Subject to 
$
$
$

Ques 28.

Write the LPP formulation of the following transportation problem:

b) Solve the following assignment problem for profit maximization:

Ques 29.

Write the LPP formulation of the following assignment problem:

b) Solve the following game graphically:

Ques 30.

Solve the following LPP graphically:

Maximize:

$
subject to the constraints:
$
$
$
b) Find all values of k for which the vectors:
$
are linearly independent.

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