1. In this problem we are going to come up with a linear optimization model for...

  
 

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1. In this problem we are going to come up with a linear optimization model for the following problem in management. Suppose that a company mamufactures two types of tents for outdoor events: the VIP tent, and the Delux tent. Over the past four weeks, here is the demand for each model: week 1 2 3 4 Dehux 23 27 34 40 VIP 11 13 15 16 Table 1: Demand of VIP and Delux tents for the past 4 weeks (a) (10 points) We first need to predict the demand for both tents during the next week in order to start production (and avoid producing too many tents). First predict the demand for Delux tents. For that, use the method from problem 1 of the last homework: fit a polynomial through the relevant data points (explain what you need to solve, but you can solve it using a computer) and find its value on week 5. Repeat the process to find the demand of Delux tents. You should find demands of 41 Delux tents, and 15 VIP tents. (b) (10 points) To produce cach tent two machines are required: machine A, and machine B. To produce a tent, machine A needs to be used, and then machine B (they are not used at the same time, but sequentially). Each Delux tent that is produced requires 48 minutes processing time on machine A and 25 minutes processing time on machine B. Each unit of VIP tent that is produced requires 7 minutes processing time on machine A and 45 minmutes processing time on machine B. The available time on machine A in week 5 is forecast to be 20 hours and on machine B in week 5 is forecast to be 15 hours. Each Delux tent sold in week 5 gives a contribution to profit of $150 and each VIP tent sold in week 5 gives a contribution to profit of $1000. It may not be possible to produce enough to meet your forecast demand for these tents in week 5 and each unsatisfied demand for Delux tents costs $50, cach unsatisfied demand for VIP tents costs $300. Our gonl is to determine how many tents to produce to maximize profits, without exceeding the demand. To solve this problem, we first set up two variables: • d: the number of Delux tents produced for week 5 • v: the number of VIP tents produced for week 5 Since we need to maximize profits, we need to determine the profit made by manufacturing d Delux tents, and v VIP tents, taking into account the penalties for not reaching the demand. Show that it is equal to 200d + 1300v – 6550. Since machine A cannot be used more than 20 hours, which is 1200 minutes, and that each Delux tent and VIP tent requires respectively 48 and 25 minutes of time on the machine, then 7d + 25u S 1200, Come up with a similar inequality constraint for the use of machine B. Finally, since we do not want to exceed the demand of Dehux tents, then 0s ds 41. Come up with a nimilar constraint for the production of Delux tents. Putting all the above functions and constraints together, determine the linear optimization problem that you end up with. It should maximize the profits, and include 4 inequalities (coustraints), in addition to e 2 0 and d 2 0, which are the positivity constraints. In total, you should have 6 inequalities (and a funetion to maximize of course). Make sure to write it in the form max (write a function here) subject to (write a list of inequalities) v20, d20 2. (30 points) Solve (by hand) the previous linear optimization problem using the Simplex Method shown in class. Apart from the initial dictionary, two more dictionaries will be necessary. Throughout the method, the variables d, v and the slack variables have integer values, and if you get fractions you know that you have made a mistake (some other coefficients are fractions however). Write a sentence which determines how many VIP and Delux tents should be produced, and the total profits made. 1. In this problem we are going to come up with a linear optimization model for the following problem in management. Suppose that a company mamufactures two types of tents for outdoor events: the VIP tent, and the Delux tent. Over the past four weeks, here is the demand for each model: week 1 2 3 4 Dehux 23 27 34 40 VIP 11 13 15 16 Table 1: Demand of VIP and Delux tents for the past 4 weeks (a) (10 points) We first need to predict the demand for both tents during the next week in order to start production (and avoid producing too many tents). First predict the demand for Delux tents. For that, use the method from problem 1 of the last homework: fit a polynomial through the relevant data points (explain what you need to solve, but you can solve it using a computer) and find its value on week 5. Repeat the process to find the demand of Delux tents. You should find demands of 41 Delux tents, and 15 VIP tents. (b) (10 points) To produce cach tent two machines are required: machine A, and machine B. To produce a tent, machine A needs to be used, and then machine B (they are not used at the same time, but sequentially). Each Delux tent that is produced requires 48 minutes processing time on machine A and 25 minutes processing time on machine B. Each unit of VIP tent that is produced requires 7 minutes processing time on machine A and 45 minmutes processing time on machine B. The available time on machine A in week 5 is forecast to be 20 hours and on machine B in week 5 is forecast to be 15 hours. Each Delux tent sold in week 5 gives a contribution to profit of $150 and each VIP tent sold in week 5 gives a contribution to profit of $1000. It may not be possible to produce enough to meet your forecast demand for these tents in week 5 and each unsatisfied demand for Delux tents costs $50, cach unsatisfied demand for VIP tents costs $300. Our gonl is to determine how many tents to produce to maximize profits, without exceeding the demand. To solve this problem, we first set up two variables: • d: the number of Delux tents produced for week 5 • v: the number of VIP tents produced for week 5 Since we need to maximize profits, we need to determine the profit made by manufacturing d Delux tents, and v VIP tents, taking into account the penalties for not reaching the demand. Show that it is equal to 200d + 1300v – 6550. Since machine A cannot be used more than 20 hours, which is 1200 minutes, and that each Delux tent and VIP tent requires respectively 48 and 25 minutes of time on the machine, then 7d + 25u S 1200, Come up with a similar inequality constraint for the use of machine B. Finally, since we do not want to exceed the demand of Dehux tents, then 0s ds 41. Come up with a nimilar constraint for the production of Delux tents. Putting all the above functions and constraints together, determine the linear optimization problem that you end up with. It should maximize the profits, and include 4 inequalities (coustraints), in addition to e 2 0 and d 2 0, which are the positivity constraints. In total, you should have 6 inequalities (and a funetion to maximize of course). Make sure to write it in the form max (write a function here) subject to (write a list of inequalities) v20, d20 2. (30 points) Solve (by hand) the previous linear optimization problem using the Simplex Method shown in class. Apart from the initial dictionary, two more dictionaries will be necessary. Throughout the method, the variables d, v and the slack variables have integer values, and if you get fractions you know that you have made a mistake (some other coefficients are fractions however). Write a sentence which determines how many VIP and Delux tents should be produced, and the total profits made.

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