Exam 12: Integer Linear Optimization Models

In order to choose the best solution for implementation, practitioners usually recommend re-solving the integer linear program several times with variations in the

Andrew is ready to invest $200,000 in stocks and he has been provided nine different alternatives by his financial consultant. The following stocks belong to three different industrial sectors and each sector has three varieties of stocks each with different expected rate of return. The average rate of return taken for the past ten years is provided with each of the nine stocks. Stock Industry Annual Return 1 Airlines 18.24\% 2 Airlines 28.75\% 3 Arrlines 11.08\% 4 Banking 20.12\% 5 Banking 14.00\% 6 Banking 26.17\% 7 Agriculture 23.67\% 8 Agriculture 18.25\% 9 Agriculture 16.50\% ​ The decision will be based on the constraints provided below: o Exactly 5 alternatives should be chosen. o Any stock chosen can have a maximum investment of $55,000. o Any stock chosen must have a minimum investment of at least $25,000. o For the Airlines sector, the maximum number of stocks that can be chosen is two. o The total amount invested in Banking must be at least as much as the amount invested in Agriculture. Formulate and solve a model that will decide Andrew's investment strategy to maximize his expected annual return.

Which of the following is true of rounding the optimized solution of a linear program to an integer?

A manufacturer makes two types of rubber, Butadiene and Polyisoprene. The plant has two machines, Machine-1 and Machine-2, which are used to make the rubber strips. Manufacturing one strip of Butadiene requires 2.75 hours on Machine-1 and 3 hours on Machine-2. Processing one strip of Polyisoprene, takes 3.5 hours on Machine-1 and 4 hours on Machine-2. Machine-1 is available 180 hours per month, and Machine-2 is available 200 hours per month. Formulate an all-integer spreadsheet model that will determine how many units of each type of rubber should be produced to maximize profits if the profit contributions of Butadiene and Polyisoprene are $20 and $26, respectively. How many units of each type of rubber will maximize profits? ​

A coffee manufacturing company has two processing plants (P1 and P2) that roast imported coffee beans. After roasting, the plants produce three types of coffee beans, A, B, and C. The company has contracted with a chain of cafes to provide coffee beans each week in the following quantities - 20 tons of type A, 11 tons of type B, and 18 tons of type C. The two plants have the same capacity, but their diverse operational procedures affect costs per ton as below. ​ $\begin{array} { l c c c c } & { \text { Manufacturing Cost per Ton (\$) } } & \\\hline \text { Plant } & \text { A } & \text { B } & \text { C } & \text { Capacity } \\\hline \text { P1 } & 900 & 1125 & 875 & 25 \\\text { P2 } & 850 & 1200 & 950 & 25 \\\text { Demand } & 20 & 11 & 18 &\end{array}$ ​ Formulate and solve the all-integer model that will determine how many tons of each type of coffee beans are produced in each plant while minimizing the total cost.

To meet excess demand for the pizza home delivery services, ROFiL Pizza is planning to open new stores in various regions. The store locations that are under consideration and their coverage areas are given in the following table. ​ Store Location Potential Areas Covered L1 3,4,6,8 L2 1,5,9 L3 1,4,7 L4 2,6,7 L5 3,4,9 L6 2,7,9 L7 4,8,9 L8 1,2,5,6 L9 3,6,8 ​ Develop an integer optimization model that determines the minimum number of stores to open in order to meet the coverage demand.

Sansuit Investments is deciding on future investments for the coming two years and is considering four bonds. The investment details for the next two years are given in the table below. $\begin{array}{ccc} \text { Investment Requirements (\$) }\\& \text { Year 1 } & \text { Year 2 } \\\hline \text { Bond A } & 25,000 & 30,000 \\\text { Bond B } & 15,000 & 21,000 \\\text { Bond C } & 8000 & 9500 \\\text { Bond D } & 10,000 & 7000\end{array}$ ​ The net worth of these four bonds at maturity is $60,000, $40,000, $25,500, and $18,000, respectively. The firm plans to invest $35,000 and $62,000 in Year 1 and Year 2, respectively. Develop and solve a binary integer programming model for maximizing the net worth

Which of the following is true of the relationship between the value of the optimal integer solution and the value of the optimal solution to the LP Relaxation?

In a fixed-cost model, each fixed cost is associated with a binary variable and a specification of the

Binary variables are identified with the _____designation in the Solver Parameters dialog box.

A binary mixed-integer programming problem in which the binary variables represent whether an activity, such as a production run, is undertaken or not is known as the

The optimal solution to the integer linear program will be an extreme point of the

A manufacturer wants to construct warehouses in six different locations of the city to supply dry cells to his customers on time. The manufacturer wants to construct the minimum number of warehouses such that each warehouse is within 40 miles of at least one other warehouse. The following table provides the distance (in miles) between the locations. To From Location A Location B Location C Location D Location Location Location A 0 35 40 45 60 70 Location B 0 35 40 70 75 Location C 0 45 50 50 Location D 0 40 50 Location E 0 30 Location F 0 Formulate and solve an integer linear program that can be used to determine the minimum number of warehouses needed to be constructed. What are their locations?

Sansuit Investments is deciding on future investment for the coming two years and is considering four bonds. The investment details for the next two years are given in the table below. $\begin{array}{ccc} \text { Investment Requirements (\$) }\\& \text { Year 1 } & \text { Year 2 } \\\hline \text { Bond A } & 25,000 & 30,000 \\\text { Bond B } & 15,000 & 21,000 \\\text { Bond C } & 8000 & 9500 \\\text { Bond D } & 10,000 & 7000\end{array}$ ​ ​ The net worth of these four bonds at maturity is $60,000, $40,000, $25,500, and $18,000, respectively. The firm plans to invest $35,000 and $62,000 in Year 1 and Year 2, respectively. a. Develop and solve a binary integer programming model for maximizing the return on investment (in dollars) assuming that only one of the bonds can be considered. How much money is invested? What is the return on investment (in dollars)? b. Suppose the investment has to be made on Bond B, and only two of the four bonds can be considered for investment. Modify your formulation from Part (a) to reflect this new situation. How much money is invested? What is the return on investment (in dollars)? Based on the ratio of return vs. investment, which of these two options would you recommend? ​

An apparel designing company is planning to enter the women's trousers market. They are in the process of developing a product that will appeal most to customers. The levels - small, medium, and large of the size attribute are modeled using

The objective function for a linear optimization problem is: Max 3x + 2y, with one of the constraints being x and y both only take the values 0, 1. Also x and y are the only decision variables. This is an example of a ​

The objective function for an optimization problem is: Min 3x - 2y, with constraints x ≥ 0, y ≥ 0. x and y must be integers.. Suppose that the integer restriction on the variables is removed. If so, this would be a familiar two-variable linear program; however, it would also be an example of

In a fixed-cost problem, choosing excessively large values for the maximum production quantity will result in

The imposition of an integer restriction is necessary for models where

Which of the following is a likely constraint on the production quantity x associated with a maximum value and a setup variable y in a fixed-cost problem?