QT-II Assignment

Ayush Agarwal I Section B I 20130120099

1. What is the meaning of dual of linear programming problem? Inwhat way solution of primal problem and dual problem are relatedto each other?

Answer:

Whenever we solve an LP problem, we implicitly solve twoproblems: the primal resource allocation problem, and the dualresource valuation problem. Every linear programming problem,referred to as a primal problem, can be converted into a dualproblem, which provides an upper bound to the optimal value ofthe primal problem. In simple terms, a linear programmingequation which is defined by constraints and also has amaximizing function can be converted into its dual problem byredefining the constraints and making the problem a minimizingproblem.

Example: A firm wants to maximize profit by minimizing cost. Bothprofits and cost are dependent variables.

A LP equation would be to say maximize profits with constraintson cost.

The duality of the LP would be minimizing costs with constraintson profits.

The dual model is derived by construction from the standardinequality form of linear programming model as shown below.

(P) Maximize zP = cxsubject to Ax £ bx ³ 0(D) Minimize zD = bsubject to A ³ c³ 0

Thus, the dual variables arise from a problem minimizing themarginal value of the resource endowment subject to constraintsrequiring that the marginal value of the resources used inproducing each product must be at least as great as the marginalvalue of the product. This can be viewed as the problem of aresource purchaser in a perfectly competitive market. Under suchcircumstances, the purchaser would have to pay at least as muchfor the resources as the value of the goods produced using thoseresources. However, the purchaser would try to minimize the totalcost of the resources acquired.

From an algorithmic point of view, solving the primal problemwith the dual simplex method is equivalent to solving the dualproblem with the primal simplex method. When written ininequality form, the primal and dual models are related in thefollowing ways.

When the primal has n variables and m constraints, the dualhas m variables and n constraints.

The constraints for the primal are all less than or equalto, while the constraints for the dual are all greater thanor equal to.

The objective for the primal is to maximize, while theobjective for the dual is to minimize.

All variables for either problem are restricted to benonnegative.

For every primal constraint, there is a dual variable. Associatedwith the ith primal constraint is dual variable πi. The dualobjective function coefficient for πi is the right-hand side ofthe ith primal constraint, bi.

For every primal variable, there is a dual constraint.Associated with primal variable xj is the jth dualconstraint whose right-hand side is the primal objectivefunction coefficient cj.

The number aij is, in the primal, the coefficient of xj inthe ith constraint, while in the dual, aij is thecoefficient of πi in the jth constraint.

References:1.Wikipedia2.University of Texas Austin lecture notes: The Dual LinearProgram

2. Explain interpretation of ‘Cj-Zj’ values of slack variables.

• Cj is the contribution factor/unit or the coefficient of eachvariable in the Objective Function. Zj can be found out bymultiplying each variable in that column with the correspondingcoefficient of the basic solution variable in the Objectivefunction and then adding them together.

• The Zj values indicate the opportunity cost of bringing oneunit of a variable into the solution mix.

• The Cj - Zj value is the net change in the value of theobjective function that would result from bringingone unit of the corresponding variable into the solution.

• For example,Z=6x+4y,3x+7y<=50,5x+2y<=60Implies3x+7y+s1=50, 5x+2y+s2=60 & z=6x+4y+0s1+0s2

• Here S1 and S2 are the slack variables. And they form theInitial basic feasible solution. The coefficient of both S1 andS2 are zero in the Objective function.

o Cj = (6 4 0 0) o Z1=3X0+5X0=0o Z2=7X0+2X0=0o Z3=1X0+0X0=0o Z4=0X0+1X0=0o Thus, Zj = (0 0 0 0)

• The difference Cj - Zj provides a measure of possible net gainor loss (relative to the current basis) which may be obtainedby bringing a vector associated with such a value into thebasis and replacing one of the current members of the basisset.

• By this means, a criterion is supplied for judging thesolutions at any stage. For maximization, if all Cj - Zj arenon-negative an optimum has been achieved. For minimization,if all Cj - Zj are negative, an optimum has been achieved. Ifnot, further improvement is possible.

• Since all c's alongside the initial basis set are zero, thefirst simplex table is completed by transferring the c'sappearing at the top into corresponding positions in thebottom row and reversing their signs.

• A systematic procedure is provided for changing the basis set(and thus the program) and continuing the calculations untilan optimum has been achieved:

i. One of the negative Cj - Zj (usually the most negativeor most positive) is employed to designate an incomingvariable to come into the basis.

ii. To determine the vector to be removed all of the ratiosXi/Yik, for are calculated by dividing the elementsunder Po by their corresponding (positive) elementsunder Pk.

iii. The smallest of these ratios is then used to designatethe outgoing variable to be removed.

iv. A new basis set is written.

• Maximization and minimization problems are quite similar inthe application of the simplex method. Minimization problemsusually include constraints necessitating arti cial andfisurplus variables. In terms of technique, the Cj - Zj row is themain difference. In maximization problems, the greatest positiveCj - Zj indicates the new pivot column; in minimizationproblems, it’s the smallest negative Cj - Zj.

• The Zj entry in the “quantity” column stands for pro tficontribution or cost, in maximization and minimizationproblems, respectively.

• An optimal solution will still be reached if any positive (formaximization) or negative (for minimization) Cj - Zj value ischosen. This procedure will result in a better (morepro table) solution at each iteration, but it may take morefiiterations before the optimum is reached.

Answer 3:

1. The product type that has been chosen by me for thisassignment is Automobiles.

2. The demographic data collected for the consumers ofAutomobiles is depicted in the table provided on the nextpage. Assumptions considered in this stage are as follows:

The Automobiles would be advertised through the print adsin a local brand and will be advertising in Ahmedabadnewspapers only.

Consumption statistics have been provided per month.These will be extrapolated linearly to estimate theconsumption over the entire year.

Based on the consumption trends observed, the mostrelevant figures corresponding to demographic variableshave been depicted in the table on the next page.

3. Demographic Profile of Major Consumers who use the product Assumptions:

The Automobile concentrate brand that will be advertisedis a brand that focuses on a masculine brand image andthus targets itself to the male population only.

The brand also wants to popularize itself in the highersocio-economic strata and thus wants to focus on SEC Aand B levels specifically.

On the basis of a brief analysis of data gathered, I havenarrowed down to the following demographic profile for theconsumers who use the automobiles:i. Males, Ages 16 to 19, SEC A and B, staying in Ahmedabad

ii. Males, Ages 20 to 29, SEC A and B, staying in Ahmedabadiii. Males, Ages 30 to 39, SEC A and B, staying in Ahmedabad

4. Newspapers Read From the data available on IRS analyser, the following fourmajor newspapers are read by people in Ahmedabad:

Gujarat Samachar The Times of India - Ahmedabad Ahmedabad Mirror Divya Bhaskar

The following table depicts the readership data for the targetgroups consisting of the major consumers in :

The data in the previous table has been selected as a productof language and periodicity, thus it encompasses newspapers ofall languages and periodicity. This helps in essentiallynormalizing the figures across newspapers of differentperiodicity and languages.

The final 3 columns of the table highlight the readership ofthe three target groups that have been identified earlier,corresponding to the four newspapers.

5. Cost of Advertisements The costs of half-page ads in the newspapers chosen wereobtained as follows from sources such as releasemyad.com andpublicitas.com:

Times Of India – Rs.391,875 Ahmedabad Mirror – Rs.169,125 Gujarat Samachar – Rs.495,000 Indian Express – Rs.300,000

Framing the Linear Programing Problem

The objective is to maximize readership for our target audience.Let us consider the three target groups of audience as chosen:

TGA: Males, Ages 16 to 19, SEC A and B, staying inAhmedabad

TGB: Males, Ages 20 to 29, SEC A and B, staying inAhmedabad

TGC: Males, Ages 30 to 39, SEC A and B, staying inAhmedabad

To maximize readership, we must know the total target groups’readership for each of the four newspaper brands. From the datain figure 2, this is calculated as depicted below:

Newspaper Readership

Times ofIndia

21000

AhmedabadMirror

9000

GujaratSamachar

103000

DivyaBhaskar

2000

Decision VariablesThe decision variables in this problem are:

1. X1: Number of ads published per year by the AutomobilesCompany in Times of India

2. X2: Number of ads published per year by the AutomobilesCompany in Ahmedabad Mirror

3. X3: Number of ads published per year by the AutomobilesCompany in Gujarat Samachar

4. X4: Number of ads published per year by the AutomobilesCompany in Divya Bhaskar

Objective FunctionThe objective function for maximizing reach of the advertisementscan be written as follows:

Z = 21000X1 + 9000X2 + 103000X3 + 2000X4

Constraints1. The annual advertisement budget is Rs. 50 lacs2. Minimum 2 ads per week are required in each newspaper (2 x

52) = 104 ads per year in each newspaper, assuming the yearselected has 52 weeks.

Thus the constraints are as follows:

391875X1 + 169125X2 + 495000X3 + 300000X4 ≤ 50,00,000

X1 ≥ 104 and X2 ≥ 104

X3 ≥ 104 and X4 ≥ 104

X1 ≥ 0, X2 ≥ 0, X3 ≥ 0, X4 ≥ 0 (Non-negative constraint).Thus theobjective function, decision variables and constraints of the LPproblem have been identified.