Exam 8: Linear Optimization Models

Northwest California Ventures Ltd. has decided to provide capital in five market areas for the start-ups. The investment consultant for the venture capital company has projected an annual rate of return based on the market risk, the product, and the size of the market. The maximum capital provided will be $5 million. The consultant has imposed conditions on allotment of capital based on the risk involved in the market. • The capital provided to retail should be at most 40 percent of the total capital. • The capital for education should be 26 percent of the total of other four markets (Electronics, Software, Logistics, and Retail) • Logistics should be at least 15 percent of the total capital. • The capital allocated for Software plus Logistics should be no more than the capital allotted for Electronics. • The capital allocated for Logistics plus Education should not be greater than that allocated to Retail. Calculate the expected annual rate of return based on the allocation of capital to each market area to maximize the return on capital provided. Also, show the allocation of capital for each market area.

A manager of a quality testing team wanted to test different lots of products using three resources, R1, R2, and R3. Each lot can be tested for quality using any one of the three procedures, P1, P2, or P3. The product once tested will be sent for packaging. The profit contribution per lot for each of these procedures varies and they are $4, $5, and $8, respectively. Also, resource A requires 2 hours, 3 hours, and 4 hours to test a lot using the procedure P1, P2, and P3, respectively. Resource B requires 3 hours, 2 hours, and 3 hours using the procedure P1, P2, and P3, respectively. Lastly, resource C requires 2 hours, 3 hours, and 4 hours using the procedure P1, P2, and P3, respectively. The available times for these three resources are 80 hours, 90 hours, and 65 hours, respectively. Formulate and solve a linear program and solve for the optimal solution for the above scenario by maximizing the profit.a. What will be the change in total profit, if machine M3 is given an extra hour of production time?

The nonnegativity constraints create a feasible region that is:

Problems with infeasible solutions arise in practice because:

The assumption that is necessary for a linear programming model to be appropriate and that ensures that the value of the objective function and the total resources used can be found by summing the objective function contribution and the resources used for all decision variables is known as _____.

Michael has decided to invest $40,000 in three types of funds. Fund A has projected an annual return of 8 percent, Fund B has projected an annual return of 10 percent, and Fund C has projected an annual return of 9 percent. He has decided to invest no more than 30 percent of the total amount in Fund B and no more than 40 percent of the total amount in Fund C. a. Formulate a linear programming model that can be used to determine the amount of investments Michael should allocate to each type of fund to maximize the total annual return. b. How much should be allocated to each type of fund? What is the total annual return?

A Cake & Pastry shop makes 3 types of cakes which require three significant ingredients, given the combination of other ingredients vary. The data for the amount of these ingredients needed to make the cakes are provided in the table below: Develop and solve a linear programming model to maximize the profit. What is the optimal solution for this problem?

A soft drink manufacturing company has 3 factories set up one in each of the three cities - Orland, Tampa, and Port St. Lucie and it supplies the produced soft drink bottles to 3 warehouses located in the city of Miami. The associated per-unit transportation cost table is provided below: The factory at Orlando has a capacity of 15,000 units. The factory at Tampa has a capacity of 18,000 units. The factory at Port St. Lucie has a capacity of 8,000 units. The requirements of the warehouses are: a. Determine how much of the company's production should be shipped from each factory to each warehouse in order to minimize the total transportation cost? b. Find an alternative optimal solution for this transportation problem? Hint: Use the procedure described in section 8.7.

The term _____ refers to the expression that defines the quantity to be maximized or minimized in a linear programming model.

Which algorithm, developed by George Dantzig, is effective at investigating extreme points in an intelligent way to find the optimal solution to even very large linear programs?

Two mining fields, field A and field B of a coal mining company produce Lignite and Bituminous coal. The operating cost per day for field A and field B are $55,000 and $45,000, respectively. The recent records at the company indicate that the mining field A can produce 250 tons of Lignite along with 300 tons of Bituminous coal per day whereas the mining field B can produce 200 tons of Lignite along with 450 tons of Bituminous coal per day. The demand for Lignite is expected to be 120,000 tons and for Bituminous coal, it is expected to be 170,000 tons. The expected demand should be met. To minimize the operating costs of the mining fields, how many days does the company need to operate each of these fields?

Ethan Steel, Inc. has two factories that manufacture steel components for four different rail projects located at four different sites. The demand for the steel components for the four projects, Project A, Project B, Project C, and Project D, are 3220, 3675, 4125, and 2975, respectively. The shipping details are as below: Production Details: Shipping Details (with per-unit shipping cost): What is the optimal (cost minimizing) distribution plan for this transportation problem?

Constraints are:

The _____ for a decision variable is the shadow price of the nonnegativity constraint for that variable.

Reference - 8.2: Use the information given below to answer questions A canned food manufacturer has its manufacturing plants in three locations across a state. Their product has to be transported to three central distribution centers, which in turn disperse the goods to seventy-two stores across the state. -Reference - 8.2: Which of the following is most likely to be the objective function in this scenario?

Reference - 8.1: Use the information given below to answer questions Rob is a financial manager with Sharez, an investment advisory company. He must select specific investments-for example, stocks and bonds-from a variety of investment alternatives. -Reference - 8.1. Restrictions on the type of permissible investments would be a _____ in this case.

Reference - 8.2: Use the information given below to answer questions A canned food manufacturer has its manufacturing plants in three locations across a state. Their product has to be transported to three central distribution centers, which in turn disperse the goods to seventy-two stores across the state. -Reference - 8.2: Which of the following visualization tools could help understand this problem better?

A variable subtracted from the left-hand side of a greater-than-or-equal to constraint to convert the constraint into an equality is known as a(n) _____.

In problem formulation, the:

Three plants P1, P2, and P3 of a gas corporation supply gasoline to three of their distributors located in the city at three different locations A, B, and C. The plants' daily capacities are 4500, 3000, and 5000, gallons respectively, while the distributors' daily requirements are 5500, 2500, and 4200 gallons. The per-gallon transportation costs (in $) are provided in the table below: Because of a failure of expected supply earlier, the distributors - A, B, and C this time have decided to charge a penalty of $0.45, $0.55, and $0.5 per gallon, respectively, to avoid any further delays. Now, determine the optimum supply of gasoline to the distributors, in order to minimize the total transportation cost as well as the charges payable as penalty.