Solver • Finding maximum, minimum, or value by changing other cells • Can add constraints • Don’t need to “guess and check”
Using Solver, Excel’s Solver Using Solver. Excel’s Solver 1. EXCEL’SSOLVER The utility Solver is one of Excel’s most useful tools for business analysis. This allows us to maximize, minimize, or find a predetermined value for the contents of a given cell by changing the values in other cells. Moreover, this can be done in such a way that it satisfies extra constraints that we might wish to impose. Example1. The size limitations on boxes shipped by your plant are as follows. (i) Their circumference is at most 100 inches. (ii) The sum of their dimensions is at most 120 inches. You would like to know the dimensions of such a box that has the largest possible volume. Let H, W, and L be the height, width, and length of a box; respectively; measured in inches. We wish to maximize the volume of the box, V = HWL, subject to the limitations that the circumference C = 2H + 2W 100 and the sum S = H + W + L 120. This problem is set up in the Excel file Shipping.xls. We will outline its solution with screen captures and directions. First, enter any reasonable values for the dimensions of the box in Cells B7:D7. Shipping.xls T C I (material continues)
Using Solver, Solver FRAGILE Crush slowly Enter cell that computes volume. Select Max. Enter cells that contain dimensions H W Click on Add. L Using Solver. Excel’s Solver: page 2 To use Solver, click on Data, then Solver in the Analysis box. In older versions of Excel select Tools in the main Excel menu, then click on Solver. To use Solver, click on Data, then Solver in the Analysis box. In older versions of Excel select Tools in the main Excel menu, then click on Solver. Computer Problem? (material continues) Shipping.xls T C I
Using Solver, Solver Using Solver. Excel’s Solver: page 3 The requirement that the circumference be at most 100 inches is called a constraint. We want to have the contents of Cell E7 be at most 100. Enter cell that computes circumference. Select <=. Click on OK. Enter the limiting number. Repeat the above process to add the constraint F7 <= 120, then click on Solve. Shipping.xls T C I (material continues)
Using Solver, Solver Using Solver. Excel’s Solver: page 4 Click on Solve. Click on Keep Solver Solution. Click on OK. Shipping.xls T C I (material continues)
Using Solver, Solver Using Solver. Excel’s Solver: page 5 The dimensions that maximize volume are now shown in Cells B8:D8. The maximum volume, the value of the circumference and the sum of the dimensions are now displayed. For a maximum volume of 43,750 cubic inches, the box should be 25 inches high, 25 inches wide, and 70 inches long. In rare cases; such as very large or small initial values of H, W, or L; you may need to add the constraints B7 >= 0, C7 >= 0 and D7 >= 0. Shipping.xls T C I (material continues)
Using Solver, Solver Using Solver. Excel’s Solver: page 6 Show ex3-sep14-shipping.xls Rush! shipping company limits the size of the boxes that it accepts by limiting their volume to at most 16 cubic feet (27,648 cubic inches). For it to ship a box, each dimension must be between 3 and 54 inches. (i) Modify Shipping.xls and use Solver to find the dimensions of a Rush! box which will accept the longest possible item. Hint: Use different initial values for each dimension. (ii) What is the maximum length of such an item? Note that the longest item which can be shipped in a box has a length of Exercise 3 Shipping.xls T C I (material continues)
Solver • Sensitive to initial value • Use graphical approximation to help solve project • Use to verify/solve Questions 1 - 3 • Use to solve Questions 6 - 8
Demand Function D(q) Revenue D(q) q q Integration • Revenue as an area under Demand function
Demand Function Total Possible Revenue Integration • Total possible revenue-The total possible revenue is the money that the producer would receive if everyone who wanted the good, bought it at the maximum price that he or she was willing to pay. This is the greatest possible revenue that a seller or producer could obtain when operating with a given demand function
Demand Function Consumer Surplus D(q) Revenue Not Sold q Integration • Consumer surplus – revenue lost by charging less/ Some buyers would have been willing pay a higher price for the good than we charged. The total extra amount of money that people who bought the good would have paid is called the consumer surplus • Producer surplus – revenue lost by charging more/ some potential customers do not buy the good, because they feel that the price is too high. The total amount of this lost income, which we will call not sold, is represented by the area of the region under the graph of the demand function to the right of the revenue rectangle.
Integration • Approximating area under graph - estimating areas of rectangles (by hand) - Using Midpoint Sums.xls (using Excel) - Using Integrating.xls (using Excel)
Integration • Approximating area (Midpoint Sums) - Notation - Meaning
Integration • Approximating area (Midpoint Sums) - Process Find endpoints of each subinterval Find midpoint of each subinterval
Integration • Approximating area (Midpoint Sums) - Process (continued) Find function value at each midpoint Multiply each by and add them all This sum is equal to
Integration • Approximating area (Midpoint Sums) Ex1. Determine where .
Integration • Approximating area (Midpoint Sums) Ex1. (Continued)
IntegrationApproximating area (Midpoint Sums.xls) =6*x-4*x^2 Ex1. (Continued)
EXAMPLE 2 - Modify sheet n = 20 in Area Example.xls, so that it computes the sum S100(f, [0, 4]), with 100 subintervals, for f(x) = 2x x2/2. • Show • ex2-n-100Area Example.xlsm
Integration • Approximating area (Integrating.xls) - File is similar to Midpoint Sums.xls - Notation: or or …
Integrationshow ex3-Integrating.xlsm • Approximating area (Integrating.xls) Ex3. Use Integrating.xls to compute
Integration • Approximating area (Integrating.xls) Ex3. (Continued) So . Note that is the p.d.f. of an exponential random variable with parameter . This area could be calculate using the c.d.f. function .
Integration • Approximating area (Integrating.xls) Ex3. (Continued)
Integration • Approximating area - Values from Midpoint Sums.xls can be positive, negative, or zero - Values from Integrating.xls can be positive, negative, or zero
Integration, Applications Integration. Applications: page 6 Revenue computations for an arbitrary demand function work in the same way as those for the buffalo steak dinners. Let D(q) give the price per unit for a good,that would result in the sale of q units, and let qmax be the maximum number of units that could be sold at any price. That is, D(qmax) = 0. The total possible revenue is given by If qsold units are sold, then the revenue will be qsoldD(qsold). The following formulas give consumer surplus and lost revenue from units not sold. It is clear that revenue + consumer surplus + not sold = total possible revenue. T C I (material continues)
Integration • Ex4. Suppose a demand function was found to be . Determine the consumer surplus at a quantity of 400 units produced and sold.
Integration • Ex. (Continued) Calculate Revenue at 400 units
Integration • Ex. (Continued) $107,508.80 – $83,569.60 = $23,939.20 So, the consumer surplus is $23,939.20
Integration, Evaluation Integration. Evaluation: page 6 The study of differentiation and integration is called calculus. It is evident that a relationship between these two branches of calculus is a major accomplishment. First Fundamental Theorem of Calculus Let f and F be well behaved functions (continuous in the sense defined in the Help Me Understand link on page 82) that are defined on the closed interval [a, b]. Assume that f is the derivative of F on the open interval (a, b). In this case, The expression F(b) -F(a) is given a standard notation. This is read as “F(x) evaluated from a to b.” C T I (material continues)
Integration, Evaluation Integration. Evaluation: page 13 FORMULAS If the First Fundamental Theorem of Calculus is to be of any use in business problems, we must be able to find antiderivatives. The only available tools come from what we know about differentiation. Every differentiation formula translates into a formula for antidifferentiation. We will start with our four rules for the differentiation of specific types of functions. C T I (material continues)
Integration, Evaluation Products of functions do not work well with differentiation or antidifferentiation. Caution! Integration. Evaluation: page 14 Differentiation formulas that allowed us to split up functions into smaller parts yield antidifferentiation formulas that can be used to split up indefinite integrals. Suppose that a is a constant and that both f and g are differentiable functions with antiderivatives. C T I (material continues)
Integration, Evaluation Integral Additivity. If a function f is integrable on some closed interval containing the numbers a, b, and c(in any order), then all three of the following integrals exists and a b c Integration. Evaluation: page 21 C T I (material continues)