Exam 21: Modeling Using Linear Programming

The Northwest Flower Company owns a greenhouse, which furnishes roses and carnations to florists in Oregon, Washington, and Idaho. The greenhouse can grow any combination of the two flowers. They sell the flowers in "bunches" with 25 blooms to a bunch. They have 10,000 square feet available for planting this year. Each bunch of roses takes about 4 square feet and each bunch of carnations about 5 square feet. Special fertilizer is required for flowers: roses need 5 pounds and carnations 2 pounds. The availability of the fertilizer is limited to 5000 pounds. Sales commitments require the company to grow at least 500 bunches of roses. Profit contributions are $6 per bunch of roses and $8 per bunch of carnations. a. What is the objective function if the company wants to maximize its profits? b. What is the constraint for the square footage assuming X1 corresponds to roses? c. What is the optimal solution point for this problem? d. What is the optimal value of the objective function?

Since price is usually set by market conditions, the blending problem use of linear programming attempts to meet demand at minimum cost.

Excel solver can handle basic linear programming but not the special transportation problem.

(a month's production) + (beginning inventory) - (ending inventory) - (number of lost sales for the month) =

ABC Products produces three products (A, B and C) on three machines. Machines 1 and 2 are available for 40 hours a week and Machine 3 is available for 60 hours a week. Profit contribution and standard production time in hours are given in the following table: Product Profit Contribution/Unit \ 40 \ 60 \ 30 Machine \#1 time/unit) 0.6 2.2 1.8 Machine \#2 time/unit) 1.2 0.8 0.7 Machine \#3 time/unit) 2.4 2.8 2.0 Only one operator per machine is required on Machines #1 and #2. Two operators are required for Machine #3. Therefore, two hours of labor must be scheduled for each hour of Machine #3's time. To restate this requirement, two operators must be scheduled for each hour of Machine #3's operation, as well as one operator for each hour of Machine #1's operation and one operator for each hour of Machine #2's operation. A total of 110 labor hours is available for assignment to the three machines during the coming week. Other production requirements are that Product A cannot account for more than 40% of the units produced and that Product C must account for at least 25% of the units produced. a. Develop the constraint for the capacity limit of Machine #1. b. Develop the constraint for the capacity limit of Machine #3. c. Develop the constraint for the labor capacity limit. d. Develop the constraint for limiting Product A to no more than 40% of the units produced. e. Develop the constraint that ensures Product C accounts for at least 25% of the units produced.

A company has normal capacity for 2,500 units a month. If, for a given month, X = demand, which of the following correctly allows for either overtime or under-time in a given month?

Which of the following factors would generally not be part of a linear programming model for production scheduling?

Given the partial project network below and the fact that F can be crashed three times if X_t = start time of Activity i, and Y_i = amount of crash time used for Activity i. a. Which of the following is correct? X_F  X_D + 10  Y_D or X_F  X_E + 10 + Y_E or Y_D, Y_E  0 or X_F  X_D  X_E + 10  3 b. Which of the following is correct? y_D  3 or y_E  3 or y_F = y_D + y_E or y_F  3

Differentiate between a feasible solution and an optimal solution.

If Activity C directly precedes both D and E on a project network and C can be crashed three( 3) times in using linear programming,

A food processing company makes meatloaf to be sold in the frozen food section of supermarkets. Each week the recipe used changes based on the current cost of ingredients. Ingredients and current costs are as shown below: Inpredient Costpound pork \ 2.87 hamburger \ 1.63 wheat filler \ 0.22 corn filler \ 0.16 F or each batch made, at least 300 pounds of pork and 100 pounds of hamburger are required. No more than 200 pounds of wheat filler can be used per batch and the amount of corn filler has to range between 50 and 150 pounds. Moreover, each batch must contain at least 500 pounds of meat and no more than 200 pounds of filler. a. What is the objective function if the company seeks to minimize costs? b. If Xl is the amount of pork used, what are the constraints associated with it? c. If X3 is the amount of wheat filler used, what are the constraints associated with it?

Ending inventory from the previous month) + current production) - ending inventory this month) =

A cement company has three factories that they identify as Alpha, Beta, and Gamma. They supply cement to three warehouses that they call X, Y, and Z. The company wants to determine how much cement should be shipped from each factory to each warehouse to minimize shipping costs. The cost to ship each 100-pound bag, along with warehouse requirements and factory capacities in bags are shown in the table below: Warehouses Alpha \ .40 \ .60 \ .78 10,000 Beta .65 .78 .59 9,000 Gamma .56 .61 .71 7,000 Requirements bags) 10,000 5,000 10,000 a. How many decision variables are there in this problem? b. What is are) the constraints) corresponding to Factory Alpha? c. How many constraints are required for this problem?

Which of the following is generally correct regarding the transportation problem of linear programming?

A clothing distributor has four warehouses which serve four large cities. Each warehouse has a monthly capacity of 5,000 blue jeans. They are considering using a transportation LP approach to match demand and capacity. The following table provides data on their shipping cost, capacity, and demand constraints on a per-month basis: Warehouse City E City F City G City H A .53 .21 .52 .41 B .31 .38 .41 .29 C .56 .32 .54 .33 D .42 .55 .34 .52 City demand 2,000 3,000 3,500 5,500 a. How many variables are there in this formulation? b. How many constraints are involved in this problem? c. What is the constraint corresponding to City F?

A Singapore company manufactures 50-inch and 75-inch rear projection television sets. Each 50-inch set contributes $200 to profits and each 75-inch set contributes $475 to profits. The company has purchase commitments for 500 50-inch sets and 200 75-inch sets for the next month so they want to make at least that many. Although they think they can sell all the 50-inch sets that they could currently make, they do not think they can sell more than 375 75-inch sets. Their factory capacity allows them to make only 975 sets of both sizes total. They want to know how many of each type to make so as to maximize profits. a. What is the objective function for this LP problem? b. What are the constraints involving X1, assuming that X1 corresponds to 50-inch TV sets? c. What is the optimal solution point for this problem? d. What is the optimal value of the objective function?

The constraint that requires the beginning inventory plus production minus sales to equal the ending inventory is called material-balance equation.

A cargo airline company in South America ferries materials from four different airfields in Brazil called (A, B, C, and D) to two different airfields in Peru numbered (1 and 2). Distances in hundreds of miles between the six different airfields are as shown below: Brazil 1 2 6 17 5 2 12 8 12 9 Capacities 60 40 75 5 Requirements for airfield 1 are 140 tons per year and 80 tons per year for airfield 2. The president of the airline wants to determine how much material should be shipped from each airfield in Brazil to each airfield in Peru so as to minimize total travel distance. a. How many decision variables are there in this problem? b. What is the constraint corresponding to airfield 1? c. What is the constraint corresponding to airfield B?

Power Fuels is developing a new additive for rocket fuel. The additive is a mixture of liquid ingredients A, B and C. For proper performance, the total amount of additive must be at least 18 ounces per gallon of fuel. For safety reasons, the total amount of additive must not exceed 22 ounces per gallon. At least 2 ounces of A must be used for every ounce of C. The amount of B must be greater than one-half the amount of A. a. The cost per ounce for ingredients A, B and C is $50, $40 and $70, respectively. What is the objective function? b. Develop the constraint for limiting the additives for safety reasons. c. Develop the constraint for ensuring the performance requirement is met. d. Develop the constraint that the relationship between Ingredient A and Ingredient C. e. Develop the constraint that states the relationship between Ingredient B and Ingredient C.

Explain the essence of a blending problem.