numerical methods

You are working with a group of five engineers to study the traffic congestion in a town which require you to consider a T-junction in the area. Figure 1 shows the T-junction and the important amenities around the area that affect the congestion at the junction From Town from From Town Figure 1 T-junction A study on the traffic of the junction is made to record the number of vehicles pass the T- junction for 5 toutes shown in Figure 1 during peak hour at four times from 8.00 am to 9.00 am. Table I recorded the average number of vehicles pass through 5 routes for 1 month during the peak hour period: Table 1: Average Number of Vehicles Time (O), min 15 Route 32 46 60 3 5 10 6 1 Route 8 9 10 7 Average number of vehicles 7 8 6 3 Route 3 Route 4 Route 5 4 7 10 6 5 0 6 8 Your group were asked to solve following problems related to traffic congestion. Please use mathematical software such as Matlab or Excel for all the numerical methods calculations below. Question 1 Based on the data given, estimate the route with highest number of vehicles at t = 30 min. Develop equations for 3rd order polynomial for your calculation. Question 2 From the equations developed in Question 1, find the rate of change on the number of vehicles pass through all five routes using step size, h = 10 min. Use numerical differentiation of (h2) for your calculation and use formula of centered finite divided difference for intermediate data points. Question 3 Find the time for route(s) that has a maximum rate of change on the number of vehicles between 30 mins and 60 mins. (Hint: maximum happen when the first derivative of a function, = 0) Question 4 A new traffic light manufacturing company, Trafficm plan to focus their production to one type of traffic light only in order to achieve the best profit per month by considering the constraints related to the production. Traffic firstly consider three types of traffic light which are time-based light, queue-based light and artificial intelligence-based (Al- based) light Each type of traffic light requires development's cost and different production times. They are also sold in the market at different prices as stated in the following table: Al-based light Time-based light Queue-based light Selling price ('000) RM 35/unit RM 130/unit Development's cost 44/unit 200/unit (*000) Production Time 1.5 days/unit 6 days/unit RM 200/unit 250/unit 10 days/unit The company is only able to provide a capital cost of RM 500 thousand for development cost per month. Due to working hours restrictions, the plant is only able to operate for 21 days per month. Since the company is new, the company has limited storage size which can store only 10 sets of traffic lights in a month. Other monthly costs required to produce the products amount to RM70000. (a) Based on the information given above, set-up the linear programming to determine the type of traffic light that TrafficM needs to focus on their production to maximize their monthly profit. (b) Using the Simplex method, solve the optimization problem in part (a). Discuss the possibility whether the aim to focus on production of 1 type of traffic light is possible to the company in order to achieve maximum profit per month. You are working with a group of five engineers to study the traffic congestion in a town which require you to consider a T-junction in the area. Figure 1 shows the T-junction and the important amenities around the area that affect the congestion at the junction From Town from From Town Figure 1 T-junction A study on the traffic of the junction is made to record the number of vehicles pass the T- junction for 5 toutes shown in Figure 1 during peak hour at four times from 8.00 am to 9.00 am. Table I recorded the average number of vehicles pass through 5 routes for 1 month during the peak hour period: Table 1: Average Number of Vehicles Time (O), min 15 Route 32 46 60 3 5 10 6 1 Route 8 9 10 7 Average number of vehicles 7 8 6 3 Route 3 Route 4 Route 5 4 7 10 6 5 0 6 8 Your group were asked to solve following problems related to traffic congestion. Please use mathematical software such as Matlab or Excel for all the numerical methods calculations below. Question 1 Based on the data given, estimate the route with highest number of vehicles at t = 30 min. Develop equations for 3rd order polynomial for your calculation. Question 2 From the equations developed in Question 1, find the rate of change on the number of vehicles pass through all five routes using step size, h = 10 min. Use numerical differentiation of (h2) for your calculation and use formula of centered finite divided difference for intermediate data points. Question 3 Find the time for route(s) that has a maximum rate of change on the number of vehicles between 30 mins and 60 mins. (Hint: maximum happen when the first derivative of a function, = 0) Question 4 A new traffic light manufacturing company, Trafficm plan to focus their production to one type of traffic light only in order to achieve the best profit per month by considering the constraints related to the production. Traffic firstly consider three types of traffic light which are time-based light, queue-based light and artificial intelligence-based (Al- based) light Each type of traffic light requires development's cost and different production times. They are also sold in the market at different prices as stated in the following table: Al-based light Time-based light Queue-based light Selling price ('000) RM 35/unit RM 130/unit Development's cost 44/unit 200/unit (*000) Production Time 1.5 days/unit 6 days/unit RM 200/unit 250/unit 10 days/unit The company is only able to provide a capital cost of RM 500 thousand for development cost per month. Due to working hours restrictions, the plant is only able to operate for 21 days per month. Since the company is new, the company has limited storage size which can store only 10 sets of traffic lights in a month. Other monthly costs required to produce the products amount to RM70000. (a) Based on the information given above, set-up the linear programming to determine the type of traffic light that TrafficM needs to focus on their production to maximize their monthly profit. (b) Using the Simplex method, solve the optimization problem in part (a). Discuss the possibility whether the aim to focus on production of 1 type of traffic light is possible to the company in order to achieve maximum profit per month