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Ans. Industrial Applications1. Product Mix problem2. Blending problem3. Production Scheduling problem4. Trim loss problem5. Sub contract problem.

Management Problems1. Media selection problem2. Transportation problem3. Assignment problem4. Man power scheduling problemAgricultural ApplicationsMilitary Applications.

Q. 2.5 What are the advantages and limitations of linear programmingmethods?

Ans. Advantages1. It helps in attaining the optimum use of productive factors.2. It improves the quality of decisions. The individual who makes use of linearprogramming methods becomes more objective than subjective.


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3. It also helps in providing better tools for adjustment to meet changingconditions.4. It highlights the bottlenecks in the production processes.

5. Most business problems involve constraints like raw materials availability, marketdemand etc. which must be taken into consideration. Just we can produce so manyunits of product does not mean that they can be sold. Linear programming canhandle such situation also.Limitations1. In some problems objective functions and constraints are not linear. LPP undernon linear condition usually results in an incorrect soIution2. LPP deals with problems that have a single objective. Real life problem mayinvolve multiple objectives.3. Parameters appearing in the model are assumed to be constant. But in real lifesituation they are neither constant nor deterministic.4. It is applicable to only static situations since it does not take into account theeffect of time.5. LPP can not be used efficiently for large scale problems, the computationaldifficulties are enormous, even when the large digital computer is available. 6. LPP may get fractional valued answers for the decision variables, whereas it may

happen that only integer values of the variable are logical.

Q. 2.6. Define:(i) Solution to LPP(ii) Feasible solution(iii) Basic solution(iv) Basic feasible solution(v) Optimum basic feasible solution

(vi) Unbounded Solution.


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Ans. (i) Solution to LPP- A set of variable [x1x2,....x,J is called a solution to LI?if it satisfy the given constraints(ii) Feasible solutionA set of variable is called a feasible solution to LPP if it satisfythe given constraints as well as non negativity restrictions(iii) Basic solution-A solution obtained by setting any n variables equal to zero and solvmg for remainmg m variables is called basic solution(iv) Basic feasible solution-It is a basic solution that also satisfy the non negativity restrictions.(v) Optimum basic feasible solutionIt is the basic feasible solution thafalsooptimize the objective function.(vi) Unbounded solution-If the value of the objective function can be increased or decreased indefinitely the solution is called unbounded solution

Q 2.7 What is redundant constraint?

Ans A constraint, which does not affect the solution space, is known as redundent Constraint

Q. 2.8. Name three basic parts of simplex techniques.

Ans. 1. Basis2. Body matrix


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3 Identity matrix

Q 2.9 Distinguish between slack, surplus and Artificial variable

Ans SlackThe positive variable, which is added to the left hand side of theconstraint to convert it into equation, is called the slack variable Surplus-The positive variable, which is subtracted from the left hand side of the constraint to convert it into equation, is called the surplus variable.Artificial VariableLPP in which one or more constraints are of or = type (afterensuring that all all bs are non negative)in such cases we introduce another type ofvariable known as artificial variables in order to get the initial basic feasiblesolution.

Q 2.10 What is the physical meaning of artificial variable?

Ans These variables are fictitious and have no physical meaning They are merely adevice to get the starting basic feasible solution so that simplex algorithm may beapplied. These variables are required because in such problems the basic matrix isnot obtained as identity matrix in the starting simplex table.

Q. 2.11. Define pivot element. (Or key element).

Ans. It is intersection element of key column and key row.


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Q. 2.12. While finding the solution by simplex method, when does theproblem have infinite optimal solution, unbounded solution and infeasiblesolution?

Ans. Infinite optimal solution-If a non basic variable have zero coefficient in the C-.Z row there exist an infinite optimal solution.Unbounded solution-An unbounded solution occurred when it is not possible to determme the basic variable that should leave the basis This happens when there isno minimum non negative replacement ratio.Infeasible solution-When optimality condition is satisfied and artificial variable ispresent at some positive level The problem does not have any feasible solution

Q. 2.13. Distinguish between the Big M and two phase method.

Ans. 1. The big M method solves the problem in one pass, whereas the two phase method solve it in two stages.2. The big M method is computionally inconvenient due to existence of the largenumber M The two phase method does not involve M during computation3. The big M method present a difficulty when the problem is solved on digital

computer, but there is no such type of problem in two phase method.

Q 2.14 Give outlines of a simplex method in linear programming


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Ans. 1. Express the problem in standard form.2. Find initial basic feasible solution.3. Construct the starting simplex table which consists of basis, body matrix, identitymatrix and quantity column4. Perform optimality test.(z) Compute Z (=cB a7) elements for variable column(ii) Find index row C-.Z

It may be resolved by perturbation method.(iii) (Minimization Problem) if all the elements of CZ row are non negative thenoptimahty condition is satisfied(Maximization problem) If all the elements of CZ row are zero or negative thenoptimahty condition is satisfied5. Iteration towards an optimal solution.(z) Selecting of the entering variable(ii) Selecting of outgoing variable(iii) Preparing the new simplex table.6. Repeat step 4 and 5 till the optimal solution is obtained.

Q. 2.15. What do you understand by degeneracy?

Ans Degeneracy linear programming is said to occur when one or more basicLables have zero value. It may occur at initial stage or at subsequent iterationstage.


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Q. 2.16. Define basic and non basic variables.

Ans. Basicvariable under the basic column are known as basic variable.Non Basicvariables not entered under basic column are known as non basicvariable.

Q. 2.17. What do you mean by shadow prices?

Ans. The elements in C-Z row (index row) under slack variable columns are knownas shadow prices or imputed values of the resources.

Problems

Problem. 2.18. An advertising company wishes to plan its advertising

strategy in three different media-television, radio and magazines. Thepurpose of advertising is to reach as large a number of potential customersas possible. Following data have been obtained from market survey-


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The company wants to spend not more than Rs. 450000 on advertising.Following are the further requirements.1. at least 1 million exposures take place among female customers.2. advertising on magazines be limited to Rs 1, 50000 3. at least 3 advertising units to be bought on magazine 1 and 2 units onmagazine II.4. The number of advertising units on television and radio should each bebetween 5 and 10.

Formulate an LPP model for the problem.

Solution : Let x1- no. of advertising unit of televisionno. of advertising unit of radiox3- no. of advertising unit of Magazine Ix4 no. of advertising unit of Magazine IIObjective functionMaximize Z = i05 (2x1+ 6x2 + 1.5x3 + x4)Constraints are


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Problem 2.19. A person requires 10, 12 and 12 units of chemicals A, B andC respectively for herbal garden. A liquid product contains 5, 2 and 1 unitsof A,B and C respectively per Jar. A dry product contains 1, 2 and 4 units ofA, B and C per cartoon. If the liquid product sells for Rs. 3 per Jar and dryproduct sells for Rs. 2 per cartoon, how many of each should be purchasedto minimise the cost and meet the requirements.


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Problem. 2.20. A firm manufactures pain relieving pills in two sizes A andB, size A contains 4 grains of element a, 7 grains of element b and 2 grainsof element c, size B contains 2 grains of element a, 10 grains of element b

and 8 grains of c. It is found by users that it requires at least 12 grains ofelement a, 74 grains of element b and 24 grains of element c to provideimmediate relief It is required to determine that least no. of pills a patientshould take to get immediate relief. Formulate the problem as standard

LPP.


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Problem. 2.21. An automobile manufactuEer makes automobiles and trucksin a factory that is divided into two shops. Shop A which perform the basicassy operation must work 5 man days on each truck but only 2 man dayson each automobile. Shop B which perform finishing operations must work

3 man days for each automobile or truck that it produces. Because of menand machine limitations shop A has 180 man days per week available whileshop B has 135.man days per week. If the manufacturer makes a profit ofRs. 300 on each truck and Rs. 200 on each automobile; how many of eachshould be produced to maximize his profit?


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Problem. 2.22 On completing the construction of house a person discoversthat J square feet of plywood scrap and 80 square feet of white pine scrapare in use!able m for the construction of tables and book cases. It takes 16square feet of plywood 8 square feet of white pine to make a table, 12square feet of plywood and 16 Llare feet of white pine are required to

contruct a book case. By selling the finishing duct to a local furniture storethe person can realize a profit of Rs. 25 on each table d Rs. 290 on eachbook case. How may the man most profitably use the left over ood ? Usegraphical method to solve problem.


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Problem 2.23. A truck can carry a total of 10 tons of product. Three typesof produts are available for shipment. Their weight and values aretabulated. Assuming that at. least one of each type must be shipped.Determine the loading which will maximize the total value. Formulate theproblem.


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