DSC2605 Assignment 2 Semester 1 2024 (225272)

DSC2605 Assignment 2 Semester 1 2024 (225272) - DUE 12 April 2024 ... 100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on +/ 2/ 5/ 4 /7 /7 /9 /5 /4 /0 /1 /3 /2 . Assignment 2 DSC2605/103/1/2024 Part 1: Representation of LP models Question 1 [15] Consider the following mathematical programming model: Minimise z = −x 1 − x 2 − x 3 subject to −x 2 + x 3 ≥ −1 x3 x 1 +x 2 + x 3 ≤ 0, 1 x1 = x 3 − 3 x2 − 2x 3 ≤ x1 − 4 ≤ x 2 + 2x 3 and x1 , x2 , x3 ≥ 0. 1.1 Transform the model to a standard LP model. (5) 1.2 Transform the obtained LP model obtained in 1.1 to an augmented model by adding slack and/or surplus variables. (5) 1.3 Put the obtained LP model in 1.1. in matrix form. (5) Question 2 [13] A marketing company is planning a marketing campaign in three different marketing platforms: telephone, social media and emails.The purpose of the marketing campaign is to reach as many potential clients as possible.The results of the market study are given below (values in thousands): Telephone Social Media Email AM PM Cost per marketing unit Number of potential clients reached per marketing unit Number of female customers reached per marketing unit The company does not want to spend more than R870 000 on marketing. It further requires that − at least two million women be reached − advertising on telephone be limited to R430 000 Page 3 of 6 Assignment 2 DSC2605/103/1/2024 − at least five marketing units be bought on morning telesales, and at least two units during afternoon telesales. − the number of marketing units on social media and email should be between four and nine. 2.1 Define all decision variables of this problem clearly. [4] 2.2 Formulate the objective function to maximise the total number of potential customers reached. [2] 2.3 Formulate all the functional constraints. [7] Page 4 of 6 Assignment 2 DSC2605/103/1/2024 Part 2: Solving LP models Question 3 [15] Consider the following LP model: Minimize z = −50x + 20y subject to 2x − y ≥ −5 3x + y ≥ 3 2x − 3y ≤ 12 and x, y ≥ 0. 3.1 Represent all the constraints of the LP model on a graph.Represent x on the horizontal axis and y on the vertical axis. Label all relevant lines on the graph and indicate the feasible region clearly. [5] Important note: Use the necessary tools to draw a reasonably accurate graph. A rough sketch is not acceptable. 3.2 Find all the corner points of the feasible region and evaluate the objective function at each identified corner point. [6] Important note: Half a mark willbe deducted for each incorrect corner point given. 3.3 Deduce the optimal solution. If the LP problem is infeasible or unbounded, give the reason for this. If the problem has multiple solutions, find the general optimal solution. [3] 3.4 State clearly the redundant, binding and nonbinding constraints of this LP problem. [3] Question 4 [12] The owner of a shop producing automobile trailers wishes to determine the best mix for the three products: flat-bed trailer, economy trailers and luxury trailers. His shop is limited to working 24 days per month on metalworking and 60 days per month on woodworking for these products. The following LP model indicates production data for the trailer, Maximise z = 6x 1 + 14x2 + 13x3 subject to 0, 5x1 + 2x 2 + x3 ≤ 24 x1 + 2x 2 + 4x 3 ≤ 60 and x1 , x2 , x3 ≥ 0. Use the simplex method to obtain the optimal solution for the LP model presented above. (12) Page 5 of 6 Assignment 2 DSC2605/103/1/2024 Question 5 [18] Consider the following algebraic LP model for a blending problem: Maximise z = X4 i=1 X4 j=1 sj xij − X4 i=1 X4 j=1 ci xij − X4 j=1 aj subject to X4 i=1 (xij − r j aj ) = d j . X4 i=1 ϑi xij − uj X4 i=1 xij ≥ 0 X4 i=1 ζi xij − vj X4 i=1 xij ≤ 0 X4 j=1 xij ≤ qi X4 i=1 X4 j=1 xij ≤ 24 000 and xij , aj ≥ 0 (i = 1, 2, 3, 4; j = 1, 2, 3, 4) , where xij and aj are decisions variables and where the parameters of the model are given by [sj ] =        1200 1000 900 1100        , [ci] =        900 700 500 800        , [dj ] =        4000 3000 2000 3500        , [ϑi] =        16 10 12 14        , [ζi] =        0.005 0.018 0.020 0.016        , [uj ] =        10 8 6 8        , [vj ] =        0.01 0.02 0.01 0.02        , [qi] =        6000 4000 5000 6000        and [r j ] =        13.2 14.0 13.8 13.0        . (a) Write Lingo codes that will solve the LP models. Run these codes and attach the solution report. (12) (b) Deduce the optimal solution. (6) © UNISA 2024 Page 6 of