Exam 11: Linear Optimization Models

Clever Sporting Equipment, Inc. makes two types of balls: Soccer balls and Cork balls. The making of each soccer ball and cork ball requires 3 hours and 4 hours of production time, respectively. A total of 500 production hours are available during the next month At least 150 balls, combined, must be produced. The production cost for each Soccer ball is $9 and each Cork ball is $7. Develop a linear programming model to minimize production costs and determine how many of each type of ball should be produced to meet the required demand.

Geometrically, binding constraints intersect to form the

A(n) ___________ solution satisfies all the constraint expressions simultaneously.

The Pat-A-Cake Pastry Shop makes chocolate cake in three sizes - Small, Medium, and Large. The shop has the following amounts of the three main ingredients on hand - 400 ounces of cake flour, 550 ounces of caster sugar, and 150 ounces of cocoa powder. The table below provides details on the amount of each ingredient required for each cake size as well as the profit contributions. Cake Small Medium Large Available Plain flour (Ounce) 8 16 21 400 Caster sugar (Ounce) 18 22 25 550 Cocoa powder (Ounce) 3 5 11 150 Profit/Unit \ 18 \ 25 \ 32 Develop and solve a linear programming model to maximize the profit. What is the optimal solution for this problem?

A ___________ refers to a constraint that can be expressed as an equality at the optimal solution.

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

The shadow price of nonbinding constraints

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

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. ​ Market Area Annual Rate of Return on Capital (\%) Electronics 12 Software 18 Logistics 15 Education 12 Retail 17 ​ 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.

Which of the following error messages is displayed in Excel Solver when attempting to solve an unbounded problem?

The supervisor of a manufacturing plant is trying to determine how many of two parts, Part X and Part Y, are to be produced per day. Each part must be processed in three sections of the plant. The time required for the production along with the profit contribution for each part are given in the following table. Time required (Minutes/Unit) Section 1 Section 2 Section 3 Profit/Unit Part X 50 30 18 \ 2 Part Y 80 45 22 \ 3 Available time (minutes) 3600 2500 1200 No more than 60 units of Part X and up to 70 units of Part Y can be produced per day. The company already has orders for 30 units of Part Y that must be satisfied. ​ a. Develop a linear programming model and solve the model to determine the optimal production quantities of Parts X and Y. b. If more time could be made available in Section 2, how much would the profit increase? (Hint: Generate Answer Report).

Robin Tires, Inc. makes two types of tires, one for SUVs and the other for Hatchbacks. The firm has the following limits - 500 hours for production, 250 hours for packaging, and 150 hours for shipping. The times required per tire type is given in the following table Production Hours Type Production Packaging Shipping Profit/Tire SUV tires 2 1.5 1 \ 22 Hatchbacks tires 1.5 1 0.5 \ 12 ​ Assuming that the company is interested in maximizing the total profit contribution, find the optimal solution using Excel Solver and answer the following: a. How many hours of production time will be scheduled in each department? b. What is the slack time in each department? c. If one more hour is available for packaging, what is the change in profit? d. What is the change in profit if one more hour is available for shipping?

Ethan Steel, Inc. has two factories that manufacture steel components for four different rail projects. 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: Factory Maximum Capacity 1 6500 2 8500 ​ Shipping Details (with per-unit shipping cost): Project Factory A B C D 2 \ 7 \ 7 \ 8 \ 4 2 \ 6 \ 5 \ 7 \ 3 Find an alternative optimal solution for this transportation problem?

The study of how changes in the input parameters of a linear programming problem affect the optimal solution 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 scenario in which the optimal objective function contour line coincides with one of the binding constraint lines on the boundary of the feasible region leads to _____ solutions.

Zen Inc. manufactures two types of products, the G.1 and the T.1 models. The manufacturing process consists of two principal departments: production and assembly. The production department has 58 skilled workers, each of whom works 7 hours per day. The assembly department has 25 workers, who also work a 7-hour shift. On an average, to produce a G.1 model, Zen Inc. requires 3.5 labor hours for production and 2 labor hours for assembly. The T.1 model requires 4 labor hours for production and 1.5 labor hours in assembly. The company anticipates selling at least 1.5 times as many T.1 models as G.1 models. The company operates five days per week and makes a net profit of $130 on the G.1 model, and $150 on the T.1 model. Zen Inc. wants to determine how many of each model should be produced on a weekly basis to maximize net profit. Formulate the problem. ​ Let the number of G.1 product produced each week be G Let the number of T.1 product produced each week be T ​ Maximize 130G + 150T s.t. production's labor constraint 3.5G + 4T <= 2030 assembly's labor constraint 2G + 1.5T <= 875 ​

A soft drink manufacturing company has 3 factories-one in Orlando, one in Tampa, and one in Port St. Lucie-which supply soft drink bottles to 3 warehouses located in the city of Miami. The associated per-unit transportation cost table is provided below. $\begin{array}{l}\text { Transportation Costs (\$) }\\\begin{array} { l c c c } \hline \text { Factories/Warehouse (W) } & \text { W1 } & \text { W2 } & \text { W3 } \\\hline \text { Orlando } & 4 & 3 & 7 \\\text { Tampa } & 7 & 6 & 4 \\\text { Port St. Lucie } & 3 & 6 & 6 \\\hline\end{array}\end{array}$ The factory in Orlando has a capacity of 15,000 units. The factory in Tampa has a capacity of 18,000 units. The factory in Port St. Lucie has a capacity of 8,000 units. The requirements of the warehouses are: Warehouse Requirement (Bottles) W1 18,000 W2 12,000 W3 5,000 ​ 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?

In a linear programming model, the __________ assumption plus the nonnegativity constraints mean that decision variables can take on any value greater than or equal to zero.

Robin Tires, Inc. makes two types of tires, one for SUVs and the other for Hatchbacks. The firm has the following limits - 500 hours for production, 250 hours for packaging, and 150 hours for shipping. The times required per tire type is given in the following table. Production Hours Type Production Packaging Shipping Profit/Tire SUV tires 2 1.5 1 \ 22 Hatchback tires 1.5 1 0.5 \ 12 ​ Assuming that the company is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Develop a spreadsheet model and find the optimal solution using Excel Solver.How many tires of each model should Robin manufacture? c. What is the total profit contribution Robin can earn with the optimal production quantities?