” 37242程序 辅导、 写作Java，CS37242 Optimisation in QuantitativeManagementAssignmentStudents can do this Assignment either individually or in group. Thenumber of students in a group cannot exceed four.QUESTION 1This question is based on the material in the pdf file Pineapple Cannersthat has been emailed to you.You must formulate a linear programming problem that determines the optimalproduction plan for the Pineapple Canners; solve this linear Program using LINGO; present the results of your work as a written report.The written report must clearly describe each variable and each constraint; present the Entire linear programming formulation; present the LINGO code which was used to solve the linear program; present the LINGO printouts with the results; present the optimal production plan.The LINGO Code (the linear program in LINGO) must use sections SETS and DATA; use the commands @FOR and @SUM.1QUESTION 2This question is based on the material in the section 5.5.2 The Big-MMethod in S.G. Nash and A. Sofer, Linear and Nonlinear Programming.McGraw-Hill, 1996. A copy of this section has been emailed to you as thepdf file Nash and Sofer Linear and Nonlinear Programming. Study the section 5.5.2 The Big-M Method in S.G. Nash and A. Sofer,Linear and Nonlinear Programming. McGraw-Hill, 1996. Using the Big-M Method, solve the linear programming problemmin 4×1 5×2 + 3x3subject tox1 + 2×2 + x3 = 10×1 x2 6×1 + 3×2 + x3 14×1 0, x2 0, x3 0.Show your working.QUESTION 3Let A be an mn matrix of rank m, m n, and b Em.and z and w Be the optimal values of the objective functions of the linearprogramsmin xnsubject toAx = bx 0andmax xnsubject toAx = bx 0respectively. Prove that for any a [z, w] there existsx {y : Ay = b, y 0, y En}such that xn = a.2QUESTION 4Consider the linear programming problemAfter introducing slack variables x3 and x4, the simplex method producedthe following final tableau(a) Find d, e, f, g, h, k, and w. Show your working.(b) Find c1, c2, b1, b2, a1,1, a1,2, a2,1, a2,2. Show your working.QUESTION 5Consider the linear programmin cT xsubject toAx bx 0where c En is a Nonzero vector, b Em, m n, and A is m n matrix ofrank m. Prove that ifAx0 b and x0 0,then x0cannot be an optimal solution.3QUESTION 6Consider the linear Programming problemmin cT xsubject to Ax = bx 0where A is an m n matrix of rank m, m n, b Em, c En. Supposethat in the optimal Basic feasible solution, obtained according to the PhaseI of the two-phase Method, all basic variables are non-artificial variables.(a) What is the Value of the Objective function for this solution? Justifyyour answer.(b) What is the Rreduced cost of each basic variable in this solution? Justifyyour answer.(c) What is the Reduced cost of each artificial variable for this solution?Justify your answer.(d) What is the reduced cost of each non-artificial nonbasic variable in thissolution? Justify your answer.QUESTION 7Consider the linear Programming problem(a) Prove that the feasible region of this linear program has no extremepoints.(b) Convert this linear Program into an equivalent linear programmingproblem in standard form.(c) Show that the feasible region of the linear program obtained in (b) hasextreme points.如有需要，请加QQ：99515681 或WX：codehelp
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