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Repetition Blindness Order Description lab report research study about repetition Blindness. use the data sheet for the write up Using The Excel Solver To Solve Mathematical Programs summarize "Chapter 8: Using The Excel Solver To Solve Mathematical Programs." I want pictures. I want animations in each slide for written language, pictures and slid. Chapter Overview 8.1 Introduction 8.2 Formulating Mathematical Programs 8.2.1 Parts of the Mathematical Program 8.2.2 Linear, Integer, and Nonlinear Programming 8.3 The Excel Solver 8.3.1 The Solver Steps 8.3.1.1 Standard Solver 8.3.1.2 Premium Solver 8.3.2 A Solver Example 8.3.2.1 Product Mix 8.3.2.2 Infeasibility 8.3.2.3 Unboundedness 8.3.3 Understanding Solver Reports 8.4 Applications of the Solver 8.4.1 Transportation Problem 8.4.2 Workforce Scheduling 8.4.3 Capital Budgeting 8.4.4 Warehouse Location 8.5 Limitations and Manipulations of the Solver 8.6 Summary 8.7 Exercises Chapter 8 Using The Excel Solver To Solve Mathematical Programs Chapter 8: Using The Excel Solver To Solve Mathematical Programs 2 8.1 Introduction This chapter illustrates how to use the Excel Solver as a tool to solve mathematical programs. We review the basic parts of formulating a mathematical program and present several examples of how the Solver interprets these parts of the program from the spreadsheet. We give examples of linear, integer, and non-linear programming problems to show how the Solver can be used to solve a variety of mathematical programs. We also give an overview of the Premium Solver and its benefits. This chapter is important for the reader to understand as many DSS applications involve solving optimization problems, which are mathematical programs. The reader should be comfortable with preparing the spreadsheet for use with the Solver. In Chapter 19, we revisit the Solver using VBA commands. We have several examples of DSS applications which use the Solver to solve optimization problems, su ch as Portfolio Management and Optimization. 8.2 Formulating Mathematical Programs The Excel spreadsheet is unique because it is capable of working with complex mathematical models. Mathematical models transform a word problem into a set of equations that clearly define the values that we are seeking, given the limitations of the problem. Mathematical models are employed in many fields, including all disciplines of engineering. In order to solve a mathemat ical model, we develop a mathematical program which can numerically be solved and re translated into a qualitative solution to the mathematical model. 8.2.1 Parts of the Mathematical Program A mathematical program consists of three main parts. The first is the decision variables . Decision variables are assigned to a quantity or response that we must determine in a problem. For example, if a toy manufacturer wants to determine how many toy boats and toy cars to produce, we assign a variable to represent the quantity of toy boats produced, x 1 , and the quantity of toy cars produced, x 2 . Decision variables are defined as negative , non-negative, or unrestricted . An unrestricted variable can be either negative or non-negative. These variables represent all other relationships in a mathematical program, including the objective, the limitations, and the requirements. The second part of the math program, called the objective function , is an equation that states the goal, or objective, of the model. In the same example of the toy manufacturer, we want to know the quantities of toy boats and toy cars to produce. However, the goal of the manufacturing plant’s production may be to increase profit. If we know that we can profit $5 for every toy boat and $4 for every toy car, then our objective function is: Maximize 5x 1 + 4x 2 In other words, we want profit to drive us in determining the quantity of boats and cars to produce. Objective functions are either maximized or minimized ; most applications involve maximizing profit or minimizing cost. The third part of the math progam, the constraints , are the limitations of the problem. That is, if we want to maximize our profit, as in the toy manufacturer example, we could produce as many toys as possible if we di d not have any limits. However, in most Chapter 8: Using The Excel Solver To Solve Mathematical Programs 5 8.3 The Excel Solver We will now discuss how to operate these two versions of the Solver. In general, the Solver must understand the problem’s mathematical program parts, which we take care of by preparing our spreadsheet to contain distinct cells for the decision variables, constraints, and objective function. We must then tell the Solver if we want to minimize or maximize the problem, or if we want to solve it for a particular value of the objective function. There are also several options that we can apply to give more specific instructions to the Solver for solving the problem. (Note: To find the Solver, go to Tools > Solver from the menu options. If you do not see Solver in the Tools menu, you must first choose the Solver Add-In . To do so, select the Add-In option from the Tools menu. A small dialog box will appear; from there, select Solver Add-In from the list. If you do not see Solver Add-In in the Add-In list, click Browse and look for the Solver.xla file from the following directory: C Drive > Program Files > Microsoft Office > Office (or Office10 ) > Library > Solver . Double-click this file. Now you should find Solver Add-In in the list; check the box next to it. Restart Excel. If you do not find the Solver.xla file, go to the Add-Ins window as explained above; select Solver Add-in and press OK . Insert the MS Office CD in CD-ROM drive when asked.) 8.3.1 The Solver Steps To operate the Solver , we must follow a short sequence of steps: 1) read and interpret the problem; 2) prepare the spreadsheet; and 3) solve the model and review the results. We will now describe these steps in detail for both the Standard Solver and the Premium Solver. The Standard Solver STEP 1: Read and Interpret the Problem We must first determine the type of problem that we are dealing with (linear programming, Integer Programming, or nonlinear programming) and outline the model parts. Whether the problem is an LP, IP, or NLP model does not affect the model parts but does affect the Options that we specify for the Solver. They may also require some additional constraint specifications. In each case, we still need to determine the decision variables, the objective function, and the constraints. We need to write these mathematically, with the objective function and constraints in terms of the decision variables. STEP 2: Prepare the spreadsheet Next, we transfer these parts of the model into our Excel spreadsheet, clearly defining each part of our model in the spreadsheet. The Solver interprets our model according to the location of these model parts on the spreadsheet. Chapter 8: Using The Excel Solver To Solve Mathematical Programs 6 STEP 2.1 : Place the Input Table Usually the input for the problem is provided for us. We just need to place it on the spreadsheet in the form of a table. We reference this input when forming our constraint and objective function formulas. STEP 2.2 : Set the Decision Variables Cells Next, we list the decision variables in individual cells with an empty cell next to each one. The Solver places values in these cells for each decision variable as it solves the model. We recommend naming the range of decision variables for easier reference in constraint and objective function formulas. STEP 2.3 : Enter the Constraint Formulas Now we place the constraint equations in the spreadsheet; we enter them separately using formulas, with an optional description next to each constraint. As each constraint is in terms of the decision variables, all of these formulas must be in terms of the decision variable cells already defined. Another important consideration when laying out the constraints in preparation for the Solver is that there must be individual cells for the right-hand side (RHS) values as well. We should also place all inequality signs in their own cells. This organization will become clear once we explain how the Solver interprets our model. Another advantageous way to keep our constraints organized as we use the Solver is to name cells. We can also group constraints that have the same inequality signs. The benefit of this habit will become apparent once we input the model parts for the Solver . STEP 2.4 : Enter the Objective Function Formula We can now place our objective function in a cell by transforming this equation into a formula in terms of the decision variables. The spreadsheet is now prepared for the Solver with all three parts of the model clearly displayed. STEP 3: Solve the Model with the Solver The Solver can now interpret this information and use algorithms to solve the model. The Solver receives the decision variables, constraint equations, and objective function equation as input into a hidden programming code that applies the algorithm to the data. We will explain in more detail how this programming works when we discuss VBA. To use Solver , we choose Tools > Solver from the menu; the window in Figure 8.3 then