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Formulate and solve the following optimum design problem. EcoCarnifex, a large chemical company, produces titanium dioxide from ilmenite. Titanium dioxide, which is bright white, is used as a pigment for paint and is also added to many food products to turn an otherwise yucky color into something more appealing (such as the filler for Twinkies). The reduction process to turn ilmenite ore into pure titanium dioxide uses big vats of hot chlorine gas. EcoCarnifex has two reduction facilities which each produce two grades of titanium dioxide (High grade and Low grade). Finished products utilizing the titanium dioxide are produced at EcoCarnifex’s three fabrication plants. There are two finished products made at the plants: paint pigments and food additives. Due to new requirements from the Environmental Protection Agency, EcoCarnifex must minimize the amount of ore processed in its reduction plants while maintaining its production and demand constraints. Production and demand constraints: 1. The total tonnage of ilmenite ore processed by both reduction facilities must equal the total tonnage processed into the two grades of titanium dioxide for shipment to the fabrication plants. 2. The total tonnage of ilmenite ore processed by each reduction facility cannot exceed its capacity. 3. The total tonnage of titanium dioxide manufactured into products at each fabrication plant must equal the tonnage of titanium dioxide shipped to it by the reduction facilities. 4. The total tonnage of titanium dioxide manufactured into products at each fabrication plant cannot exceed its available capacity. 5. The total tonnage of each product must equal its demand. Nomenclature: T(r,s) = Tonnage yield of titanium dioxide stock s (High or Low grade) from 1 ton of ilmenite ore processed at reduction plant r. Y(s,f,p) = Total yield from 1 ton of titanium dioxide stock s shipped to fabricating plant f and manufactured into product p. C(r) = Ilmenite ore processing capacity in tonnage at…

Formulate-and….pdf

1. Make a sketch of the feasibility region defined by the following constraints.  Label the edges of the region with numbers; label the extrema with letters.  Find and present the coordinates of the extrema. Assume that x and y are both equal to or greater than zero.

   Version B:  2y<=2x, 2x+3y<=15, 3y>=x, x>=1

   The constraints on a particular manufacturing process are shown on the right.  The extrema of the feasibility region have been calculated and plotted.

   Using the profit function given below, calculate the profit (value of P) at each extrema.

   P=x+2y

   At which extremum is the profit the maximum?  The minimum?  (A negative profit is a loss. The minimum profit is either the smallest positive profit, or the largest loss.)

   3. Eye-Full Optics assembles astronomical telescopes (x), premium binoculars (y) and student-grade microscopes (z) from imported parts. Each telescope takes one hour to assemble, each pair of binoculars two hours, and each microscope three hours; the availability of skilled labor limits assembly work to 1000 hours per day. Eye-Full has a contract with FedEx, and must ship no less than 400 items per day. A contract with a major retailer requires them to deliver a minimum of 100 telescopes, 250 binocs, and 50 microscopes per day. But there are supply limitations. The telescopes and binocs are shipped with the same eyepieces;  each scope has one, and each pair of binocs has two. The subcontractor who supplies the eyepieces can only furnish 800 per day. Similarly, both the binocs and the microscopes use the same prisms; each pair of binocs needs two, and each microscope needs four. The prism supplier can only ship Eye-Full 1600 per day.

   If Eye-Full makes a profit on $150 on each scope, $220 on each pair of binocs, and $300 on each microscope, how many of each should the company manufacture each day?  What is its daily profit?

   (Since the feasibility region is a volume in three-dimensional space, a sketch is not required.)

   (HINT:  Use an online app to solve Problem 3. Submit a screen shot of the output, plus an explanation.)

   Please read and heed the hints given in Cases 1, 2, and 3.  The same general advice is applicable in this Case, in particular:

   Read the source materials before beginning.
   The Case is about linear programming, not the wider topic of production planning.
   Follow standard format.  A cover page, a short discussion, references and citations are all required.
   In case you missed it:  Use an application to work out Problem 3.  Insert a screen shot of the output.  Be sure to provide a reference and a citation for the app you use.
   Assignment Expectations
   Graphics must be neat, clear and complete.  A graphics app can be used, but a freehand sketch is also acceptable.
   All calculations should be shown.
   All answers must be clearly stated.
   Relevant theory should be cited as necessary to explain which procedures were used to arrive at the answers, and why.
   Follow the instructions in the BSBA Writing Style Guide (July 2014 edition), available online at https://mytlc.trident.edu/files/Writing-Guide_Trident_2014.pdf.
   There are no guidelines concerning length.  Write what you need to write – neither more, nor less.
   Clearly demonstrate your understanding of both the theory covered in the Module, and the particulars of the Case.
   Provide references and citations.  At a minimum, you should reference the course materials. These are referenced in APA format on the Background Info page.

   Unless otherwise indicated, all the sources listed below are accessible via the Trident Online Library.

   Where given, URLs were checked and validated  on the dates shown.  If you are citing an online source in an assignment, you should enter the date that you consulted that source.

   Sources marked with an asterisk (*) are a part of this Course.  Direct links are provided.

   Feel free to browse the Web for additional sources.

   Required Reading
   Staple, E.  (2012).  Linear programming; Introduction.  Retrieved on 13 May 2014 from http://www.purplemath.com/modules/linprog.htm.

   *Rensvold, R. (2014e). Finding feasibility regions using Relplot [Word document]. Available in OPM300 Introduction to Operations Management at Trident University International, 5757 Plaza Drive, Suite 100, Cypress, CA 90630.

   *Rensvold, R. (2014f). Linear programming [course page]. Available on Module 4 home page in OPM300 Introduction to Operations Management at Trident University International, 5757 Plaza Drive, Suite 100, Cypress, CA 90630.

   Additional Reading
   Discovery (2014).  Solve a simultaneous set of two linear equations.  Retrieved on 6 June 2014 from http://www.webmath.com/solver2.html.

   Khan (2014a).  Khan academy video:  Plotting (x,y) relationships. Retrieved on 6 June 2014 from https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing_solutions2/v/plotting–x-y–relationships.

   Khan (2014b).  Khan Academy video:  Solving linear equations by substitution.  Retrieved on 6 June 2014 fromhttps://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/v/solving-linear-systems-by-substitution.

   Myers, A. (2014).  Relplot: A general equation plotter.  Retrieved on 6 June 2014 from http://www.cs.cornell.edu/w8/~andru/relplot/.

   Staple, E.  (2014a).  Introduction to the x,y-Plane (The “Cartesian” plane).  Retrieved on 6 June 2014 from http://www.purplemath.com/modules/plane.htm

   Staple, E. (2014b). Systems of linear equations:  Solving by substitution.  Retrieved on 6 June 2014 from http://www.purplemath.com/modules/systlin4.htm.

   Waner, S. (2010).  Finite mathematics utility;  Simplex method tool.  Retrieved from http://www.zweigmedia.com/RealWorld/simplex.html on 28 May 2014.

   WyzAnt (2014).  WyzAnt resources: Graphing linear equations.  Retrieved on June 2014 from http://www.wyzant.com/resources/lessons/math/algebra/graphing_linear_equations