Linear Programming Problems in Optimization

A farmer in Fayetteville owns 50 acres of land. He is going to plant each acre with cotton or corn. Each acre planted with cotton yields $400 profit; each with corn yields $200 profit. The labor and fertilizer used for each acre are given in the table below. Resources available include 150 workers and 200 tons of fertilizer

                               Cotton                  Corn

Labor (Workers)      5                           3

Fertilizer (Tons)      6                             2

(A) Formulate a linear programming model that will enable the farmer to determine the number of acres that should be planted cotton and/or corn in order to maximize his profit.

(B) Find an optimal solution to the model in (A) and determine the maximum profit.

(C) Implement the model in (A) in Excel Solver and obtain an answer report. Which constraints are binding on the optimal solution?

(D) Obtain a sensitivity report for the model in (A). How much should the farmer be willing to pay for an additional worker?

(E) Suppose the farmer hires 10 additional workers. Can you use the sensitivity analysis obtained for (D) to determine his expected profit? Would his planting plan change? Explain your answer.

(F) Using SAS Code or JMP (no support) to run an optimization problem and print output decision variables, objective function Z, and constraints solutions. You need to provide your Code on a separate file called it SAS_Problem1.sas

Assume that you have decided to enter the candy business. You are considering producing two types of candies: A and B, both of which consist solely of sugar, nuts, and chocolate. At present you have 12,000 ounces of sugar, 3000 ounces of nuts, and 3000 ounces of chocolate in stock. The mixture used to make candy A must contain at least 10% nuts 80% sugar, and 10% chocolate. The mixture used to make candy B must contain at least 10% chocolate, 70% sugar, and at least 20% nuts. Each ounce of candy A can be sold for $0.40 and each ounce of candy for B $0.50. Determine how you can maximize your revenues from candy sales using linear programming approach. Provide a linear programming formulation, notation and solution using EXCEL Solver.

An ABC company is a vertically integrated company (both producing and sellingthe goods in its own retail outlets). After production, the goods are stores in two warehouses until needed by retail outlets. Trucks are used to transport the goodsfrom two plantsto two warehouses, and then from the warehouses to the three retail outlets. Using full truckloads, the following table shows each plant’s monthly output, its shipping cost per truckload sent to each warehouse and the maximum amount it can ship per month to each warehouse

For each retail outlet (RO), the next table shows its monthly demand, its shipping cost from each warehouse, and the maximum amount that can be shipped per month from each warehouse. The management wants to determine a distribution plan that will minimize the total shipping cost.

a. Draw a graph that shows the company’s distribution network. Identify the supply nodes, transshipment nodes and demand nodes. Write down the arc costs and capacities.

b. Formulate this problem as a linear programming formulation with notation.

c. Use a solver to find the optimal solution and report your solution.

a) A boutique chocolatier - Lafayette Village Raleigh has two most sellable products: Its flagship assortment of triangular chocolates, called Pyramide, and the more decadent and deluxe Pyramide Nut and Caramel. 

How much of each should it produce to maximize profits?

Every box of Pyramide has a profit of $1.

Every box of Nut and Caramel has a profit of $6.

The daily demand is limited to at most 200 boxes of Pyramide and 300 boxes of Nut and Caramel.

The current workforce can produce a total of at most 400 boxes of chocolate per day.

Let x1 be # of boxes of Pyramide, x2 be # of boxes of Nut and Caramel

b) What if there is a third and even more exclusive line of chocolates, called Pyramide Luxe. One box of these will bring in a profit of $13.

Nut and Caramel and Luxe require same packaging machinery, except that Luxe uses it three times as much, which imposes another constraint x2 + 3x3 ? 600.