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Use the LINDO LP solver software to solve the following problems. Problems are due Friday of week 4 at class time.

45 points

  1. The Houses are Us Corporation has acquired 100 lots on which it is about to build homes near West Lafayatte. Two styles of homes are to be built, a "Cape Cod" and a "Ranch Home." Houses are Us wishes to build 100 homes over 9 months starting in early spring. During this time, Houses are Us will have 13,000 manhours of bricklayer labor and 12,000 hours of carpenter labor. Each Cape Code house requires 200 hours of carpentry labor and 50 hours of bricklayer labor. Each Ranch Home requires 120 hours of bricklayer labor and 100 hours of carpentry. The profit contribution of a Cape Cod is projected to be $5,100 and that of each Ranch Home is projected to be $5,000.
    1. How many of each type of house would you recommend building?
    2. Houses are Us has an option to by 15 vacant lots adjacent to their planned development for a total of $60,000. Should Houses are Us purchase these lots?
    3. A new salesman for Houses are Us feels certain that she can sell the Cape Cods for $2,000 more each than the current projection. If this is the case, should Houses are Us change the mix of houses that will be built?

  2. Ben McDonald operates 3 farms in Texas. The acreage and irrigation water available for each farm are shown below.

    Farm       Acreage    Water Available (acre feet)
    1 400 1500 2 600 2000 3 300 900

    Three crops can be grown; however, the maximum acreage which can be grown of each crop is limited by the amount of appropriate harvesting equipment available. The three crops that can be grown are described below.

            Total Harvesting   Water Requirements    Expected
    Crop    Capacity (acres)     (acre-ft/acre)   Profit ($/acre)
    Milo 700 6 400 Cotton 800 4 300 Wheat 300 2 100

    Any combination of crops may be grown on a farm. Find the crop mix that will maximize profit.

  3. You have just received a contract to produce animal feed and would like to determine the least cost combination of materials to produce the desired product. The feed must have a minimum of 16% protein and a minimum of 5% fiber. You have identified the following as potential ingredients:
    ingredient       cost ($)/100 kg    protein (%)   fiber (%)
    corn 5.24 8 2 oats 5.71 12 7 corn cob 1.10 10 20 soy meal 8.85 49 3

    What is the least cost ration? What happens if the soybean market is cornered resulting in a doubling of the price of soy meal?

Important Terms and Concepts:

Introduction to LINDO

LINDO (Linear, INteractive, Discrete Optimizer) is a PC-based tool for solving linear programming problems. It is a command oriented program (as opposed to menu oriented). We will be using LINDO to assist with solving of LP problems in this class. The commands that will be required to get started with LINDO are summarized below. LINDO also provides on-line help that provides further details about these and other commands.

Command                       Use

MAX Start input of a maximization problem. MIN Start input of a minimization problem. ST (Subject To) Start input of constraints. END End problem input and return to command level. GO Solve the current problem and print solution. LOOK Print selected portions of current formulation. ALTER Alter an element of the current problem. QUIT Quit LINDO

The following input would be valid input to be entered in LINDO:

MAX 2X + 3Y
4X + 3Y < 10
3X + 5Y < 12

The following screens show the above example entered and solved in LINDO.

Example Application - Optimal Crop Mix

A farmer has 1,000 acres of land and produces corn and soybeans. The production of corn requires 1.25 hours of labor per acre and the production of soybeans requires 0.8 hours per acre. The farmer obtains average corn yields of 150 bu/acre and average soybean yields of 45 bu/acre. Production costs are $150/acre for corn and $95/acre for soybeans. For planning purposes, the farmer would like to assume that the selling price for corn will be $2.25/bu and for soybeans $5.50/bu. There are 2 people available 10 hours per day to provide labor for the 60 day period in which cropping operations must be completed. Find the mix of corn and soybean production to maximize profits.

Follow the LP problem solving approach recommended in the previous course materials to solve the problem.

  1. Define the control or decision variables. The control variables in this problem are the areas to be planted to corn and to soybeans (we'll use the variable C for corn and S for soybeans).
  2. Establish the objective function. The objective in this problem is to maximize profit from the farm. We can write the profit function (objective function):

    Profit= ((2.25)(150)C - 150C) + ((5.50)(45)S - 95S)

  3. Establish constraints placed on the system. For our problem, there are 2 resource constraints - land and labor. The land constraint is: C + S < = 1000. The labor constraint is: 1.25C + 0.8S < = 2(10)(60). Non-negativity of C and S are also assumed.
Next, LINDO was used to solve the above problem. The resulting screens are shown below.

Note the optimum solution is 888.9 acres of corn and 111.1 acres of soybeans.

Next a sensitivity analysis was requested. The results are shown in the figure below.

Examining the allowable coefficient ranges for the objective function before the solution changes, one can determine the changes in corn and soybean prices and production costs before the solution changes. Further examination of the sensitivity results provides information on the changes in labor and land that would be required before the solution changes.

Additional LINDO Commands

In the previous section, enough LINDO commands to get one started in using the software were described. In this section, additional commands that will likely be helpful in LP problem solving are described.

Command                       Use

SAVE Saves the problem formulation in the file specified DELETE Used to delete a row number (constraint) EXTEND Used to add more constraints DIVERT Writes results into the file specified RVRT Close file to which results were being written and cause results to be displayed on the screen RETR Select and load file with problem formulation HELP Provides on-line help COMMANDS Lists available commands

Installation of LINDO software

LINDO runs on a PC or will run under sunpc from the UNIX workstations in Agricultural Engineering. I would recommend running from a PC unless you are familiar with sunpc. You have access to PCs in the AGAD PC laboratory in the basement of the AGAD building. You should be able to download LINDO here.

If you have difficulties with the above, try this process. The following process describes how to install and start LINDO from a PC in the AGAD lab.

  1. Login to the PC and obtain the DOS prompt (the DOS prompt from MS Windows is acceptable).
  2. Use ftp to connect to the ABE file server pasture by typing the following from the DOS prompt: ftp
  3. You will be asked to type your pasture/ecn login (for example mine is engelb).
  4. Next you will be asked to type your pasture password - note that it will not be displayed when you type it.
  5. Next change directories to the location of the LINDO software by: cd ~engelb/565
  6. Set the file type to binary by typing: bin
  7. Get LINDO from pasture by typing: get lindo.exe
  8. Quit ftp by typing: quit
  9. LINDO has now been transferred to the PC at which you are seated. To start LINDO, type lindo at the DOS prompt.