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Digital cameres are characterized by two key dimensions: the optical zoom ratio and the number of megapixels of resolution in the charge coupled device (CCD) (or other technology) used to record the image, Thus, we can think of all potential buyers as being distributed in this space. We might discretize this space so that we consider the following combinations of zoom ratio and megapixels: Zoom 6 none 3x 5x 8x 12x 200 500 300 200 150 Megapixels 7 8 100 50 600 750 350 400 400 500 200 250 10+ 300 400 500 350 200 The table estimates the number of weekly sales we would get from each of 20 combinations of the zoom ratio and megapixels. In addition, we can estimate the number of sales we would get of one manufactured camera model from people wanting to buy a different model, as shown below. In essence, we will try to locate the cameras to be manufactured in the space of camera characteristics as described by the zoom ratio and the number of megapixels. 6 10+ Megapixels 7 8 10% 25% 20% Zoom none 3x 5x 8x 12x 15% This table applies to people who want to buy a camera with 3x optical zoom and 7 megapixels. If such a camera is made, then 100 percent of the market will buy that camera. If it is not made, the 10% of the market will downgrade to the next lower level of zoom (in this case no zoom), 15% will downgrade to the next lower level of megapixels. 25% will upgrade to the next higher level of megapixels, and 20% will upgrade to the next higher level of zoom ratio, Note that 30% of the demand for the 3x/7 camera is lost if this model is not made. This information is also shown in the table below, for all combinations of desired camera (shown in rows) and manufactured camera (shown in the columns). In essence, this table gives the maximum fraction of the demand for a given camera (in a row) that can be accommodated or assigned to a manufactured camera (in a column). Again, if a particular camera id manufactured, then all potential buyers will buy that comera, even if having them buy some other comero might be more profitable for the manufacturer 3x / 10+ 5x / [6 Will buy Zoom / Pixel 5x / 7 5x / 8 5x / 10+ 8x / 6 |8x/7 /7 8x / 8 8x / 10+ 12x / 6 12x / 7 12x/ 12x / 10+ 0.2 0.2 0.2 0.25 0.2 1 0.2 1 0.25 0.2 Want Zoom / Pixe none / 6 none / 7 none / 8 none / 10 3x / 6 |3x/7 3x/ mone / 6 1 0.25 0.2 none / 7 0.15 1 0.251 0.2 mone / 8 0.15 1 0.25 0.2 mone / 10+ 0.15 1 Bx / 6 0.1 1 0.25 Bx / 0.1 0.15 1 0.25 Bx / 8 0.1 0.15 1 Bx / 10+ 0.1 0.15 5x / 6 0.1 5x/7 0.1 5x / 8 27 0.1 5x / 10+ Bx/6 EX / 8x/7 8x / 8 Ex / 10+ 12x / 6 12x / 7 12x / 8 12x / 10+ 0.15 1 0.25 0.2 0.15 0.25 0.2 1 0.15 0.1 1 0.2 0.1 0.2 1 0.15 0.1 0.25 1 0.15 0.2 0.1 0.25 1 0.15 0.2 0.25 1 0.1 0.2 0.1 1 0.1 0.15 0.25 1 0.15) 0.1 0.25 1 0.15) 0.25 1 0.1 Finally, we can estimate the profit per sale of each type of camera as shown below: Camera Combination Zoom / Megapixel none / 6 none / 7 none / 8 none / 10-3x / 6 3x / 7 3x / 8 3x / 10+ 5x / 6 5x / 7 5x / 8 5x / 10+ 8x / 6 Profit per camera 20 25 28 30 25 30 33 35 30 35 38 40 8x/7 8x / 8 35 40 43 8x / 10+ 12x / 6 12x / 7 12x / 8 12x / 10+ 45 40 45 48 50 If we have only a limited number of possible camera combinations that we can produce, we may want to identify the combinations that maximize the total profit. Define the following notation: Inputs and Sets J J hi set of desired camera (zoom ratio / megapixel) combinations set of possible camera (zoom ratio / megapixel) combinations or models that can be produced demand for camera combination i maximum fraction of the demand for camera combination i \in I that can be assigned to camera combination j \in] profit per camera sold for camera combination ; \in] maximum number of camera models that can be produced p Decision variables x 1 if camera model j \in J is produced; 0 if not fraction of demand for camera combination i \in Ithat is staisfied by camera model j \in] Y Questions a) b) c) d) e) f) g) Using the notation above, formulate the objective of maximizing the profit of all cameras sold. Formulate the constraint that at most 100 percent of the demand for camera combination i \in I can be satisfied. Formulate the constraint that the fraction of demand for zoom/megapixel combination i \in I must be less than or equal to f, if we produce camera model j \in J and O if we do not Formulate the constraint that stipulates that we produce exactly p models Formulate the constraint that if camera model j \in J is produced, then 100 percent of the pople wanting that combination will buy that model Assume that it is possible to produce every desired camera combination, i.e., I=J. Solve the model and report the solution with p=6 camera models Assume that we want to produce cameras with either 5x zoom or 8x zoom but not both. Formulate this additional constraint and resolve the model with p=6 camera models Digital cameres are characterized by two key dimensions: the optical zoom ratio and the number of megapixels of resolution in the charge coupled device (CCD) (or other technology) used to record the image, Thus, we can think of all potential buyers as being distributed in this space. We might discretize this space so that we consider the following combinations of zoom ratio and megapixels: Zoom 6 none 3x 5x 8x 12x 200 500 300 200 150 Megapixels 7 8 100 50 600 750 350 400 400 500 200 250 10+ 300 400 500 350 200 The table estimates the number of weekly sales we would get from each of 20 combinations of the zoom ratio and megapixels. In addition, we can estimate the number of sales we would get of one manufactured camera model from people wanting to buy a different model, as shown below. In essence, we will try to locate the cameras to be manufactured in the space of camera characteristics as described by the zoom ratio and the number of megapixels. 6 10+ Megapixels 7 8 10% 25% 20% Zoom none 3x 5x 8x 12x 15% This table applies to people who want to buy a camera with 3x optical zoom and 7 megapixels. If such a camera is made, then 100 percent of the market will buy that camera. If it is not made, the 10% of the market will downgrade to the next lower level of zoom (in this case no zoom), 15% will downgrade to the next lower level of megapixels. 25% will upgrade to the next higher level of megapixels, and 20% will upgrade to the next higher level of zoom ratio, Note that 30% of the demand for the 3x/7 camera is lost if this model is not made. This information is also shown in the table below, for all combinations of desired camera (shown in rows) and manufactured camera (shown in the columns). In essence, this table gives the maximum fraction of the demand for a given camera (in a row) that can be accommodated or assigned to a manufactured camera (in a column). Again, if a particular camera id manufactured, then all potential buyers will buy that comera, even if having them buy some other comero might be more profitable for the manufacturer 3x / 10+ 5x / [6 Will buy Zoom / Pixel 5x / 7 5x / 8 5x / 10+ 8x / 6 |8x/7 /7 8x / 8 8x / 10+ 12x / 6 12x / 7 12x/ 12x / 10+ 0.2 0.2 0.2 0.25 0.2 1 0.2 1 0.25 0.2 Want Zoom / Pixe none / 6 none / 7 none / 8 none / 10 3x / 6 |3x/7 3x/ mone / 6 1 0.25 0.2 none / 7 0.15 1 0.251 0.2 mone / 8 0.15 1 0.25 0.2 mone / 10+ 0.15 1 Bx / 6 0.1 1 0.25 Bx / 0.1 0.15 1 0.25 Bx / 8 0.1 0.15 1 Bx / 10+ 0.1 0.15 5x / 6 0.1 5x/7 0.1 5x / 8 27 0.1 5x / 10+ Bx/6 EX / 8x/7 8x / 8 Ex / 10+ 12x / 6 12x / 7 12x / 8 12x / 10+ 0.15 1 0.25 0.2 0.15 0.25 0.2 1 0.15 0.1 1 0.2 0.1 0.2 1 0.15 0.1 0.25 1 0.15 0.2 0.1 0.25 1 0.15 0.2 0.25 1 0.1 0.2 0.1 1 0.1 0.15 0.25 1 0.15) 0.1 0.25 1 0.15) 0.25 1 0.1 Finally, we can estimate the profit per sale of each type of camera as shown below: Camera Combination Zoom / Megapixel none / 6 none / 7 none / 8 none / 10-3x / 6 3x / 7 3x / 8 3x / 10+ 5x / 6 5x / 7 5x / 8 5x / 10+ 8x / 6 Profit per camera 20 25 28 30 25 30 33 35 30 35 38 40 8x/7 8x / 8 35 40 43 8x / 10+ 12x / 6 12x / 7 12x / 8 12x / 10+ 45 40 45 48 50 If we have only a limited number of possible camera combinations that we can produce, we may want to identify the combinations that maximize the total profit. Define the following notation: Inputs and Sets J J hi set of desired camera (zoom ratio / megapixel) combinations set of possible camera (zoom ratio / megapixel) combinations or models that can be produced demand for camera combination i maximum fraction of the demand for camera combination i \in I that can be assigned to camera combination j \in] profit per camera sold for camera combination ; \in] maximum number of camera models that can be produced p Decision variables x 1 if camera model j \in J is produced; 0 if not fraction of demand for camera combination i \in Ithat is staisfied by camera model j \in] Y Questions a) b) c) d) e) f) g) Using the notation above, formulate the objective of maximizing the profit of all cameras sold. Formulate the constraint that at most 100 percent of the demand for camera combination i \in I can be satisfied. Formulate the constraint that the fraction of demand for zoom/megapixel combination i \in I must be less than or equal to f, if we produce camera model j \in J and O if we do not Formulate the constraint that stipulates that we produce exactly p models Formulate the constraint that if camera model j \in J is produced, then 100 percent of the pople wanting that combination will buy that model Assume that it is possible to produce every desired camera combination, i.e., I=J. Solve the model and report the solution with p=6 camera models Assume that we want to produce cameras with either 5x zoom or 8x zoom but not both. Formulate this additional constraint and resolve the model with p=6 camera models

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