Question

Use the model-building process described in Chapter 2 to analyze the following scenarios. After identifying the problem to be solved using the process, you may find it helpful to answer the following questions in words before formulating the optimization model.

a. Identify the decision variables: What decision is to be made?

b. Formulate the objective function: How do these decisions affect the objective?

c. Formulate the constraint set: What constraints must be satisfied? Be sure to consider whether negative values of the decision variables are allowed by the problem, and ensure they are so constrained if required.

After constructing the model, check the assumptions for a linear program and compare the form of the model to the examples in this section. Try to determine which method of optimization may be applied to obtain a solution.

Q. A Farming Problem—A farm family owns 100 acres of land and has $25,000 in funds available for investment. Its members can produce a total of 3500 work-hours worth of labor during the winter months (mid-September to mid-May) and 4000 work-hours during the summer. If any of these work-hours are not needed, younger members of the family will use them to work on a neighboring farm for $4.80 per hour during the winter and $5.10 per hour during the summer.

Cash income may be obtained from three crops (soybeans, corn, and oats) and two types of livestock (dairy cows and laying hens). No investment funds are needed for the crops. However, each cow requires an initial investment outlay of $400, and each hen requires $3. Each cow requires 1.5 acres of land and 100 work-hours of work during the winter months and another 50 work-hours during the summer. Each cow produces a net annual cash income of $450 for the family. The corresponding figures for the hen are as follows: no acreage, 0.6 work-hour during the winter, 0.3 more work-hour in the summer, and an annual net cash income of $3.50. The chicken house accommodates a maximum of 3000 hens, and the size of the barn limits the cow herd to a maximum of 32 head.

Answer 1

Let, be the acres of soybeans

, be the acres of corn

, be the acres of oats

, be the number of dairy cows.

, be the number of hens.

, be the excess winter work hours.

, be the excess summer work hours respectively

After choosing the decision variables we move on to the constraints of the problems.

The farm family has a total of acres of land with which we can formulate our acres constraint which is given by

.

Since each cow requires 1.5 acres of land whereas hens does not requires any acreage.

The farm family has a total capital of $25000 in fund available for investment.

No investment funds are needed for crops where as cow requires initial outlay of $400 and hen requires $3 as initial investment. Hence our constraint on capital is given by

.

The members of the family can produce a total of 3500 worth work hours during winter

The three crops namely soybeans, corn, oats requires 20,35,10 hours during winter.cow requires 100 hours and hen requires 0.6 hours during winter. Hence our constraint is

.

The members of the family can produce a total of 4000 worth work hours during winter

The three crops namely soybeans, corn, oats requires 30,75,40 hours during winter.cow requires 50 hours and hen requires 0.3 hours during summer. Hence our constraint is

.

The chicken house accommodates a maximum of 3000

Barn can accommodates a maximum of 32 cows

Non negativity constraints

Now our objective function is to maximize the net cash income by determining how much acreage should be planted and how many livestock’s should be kept to maximize the cash income.

Income from crops is respectively.

Income from livestock’s is and respectively.

If any of these work hours are not needed the younger members use them to work them on neighboring farm for per hour during winter and during winter

Maximize.

The mathematical formulation

Maximize

Subject to

The assumptions of linear programming are satisfied the variables related to crops can take integer values also.