AGEC 352: Homework 4
Due October 1, 2012
There is no lab portion this week that must be completed by Wednesday’s class. Do both
parts of the lab and answer the homework questions at the end. There are questions for both
part I and part II in the homework so be sure and look at the last page after you have
worked through part I.
Part I: Extending the 3 Crop Model and Minimization Problems
The objective of this week’s assignment is to get some experience working with nonresource constraints (i.e. a constraint that is not some stock that can be drawn from), adding a hirein activity, and introducing minimization problems. In Part I of the assignment we extend the model
from Lab3: Part II, adding one constraint and one activity to see how they can be implemented in
the spreadsheet and how we might interpret them in the results. In Part II you are asked to build a
simple minimization model.
If you don’t remember how to use Solver in Excel, refer to Lab Handout 2 and 3.
Part I: Extending the Three Crop Model
We will use the three crop model from part II of last week’s assignment as our starting point
in part I this week. You can use the spreadsheet model you created last week or save a working copy
of the spreadsheet from the course website at the link accompanying this handout. The three crop
model in Excel form is shown below with formulas shown where they occur in the spreadsheet cells:
Activity Levels
Profit per acre
Corn Wheat Oats
0
0
0
45
32
20
=SUMPRODUCT(B4:D4,B3:D3)
Constraints
Land
Rowcrop land
Wht Allot
Jan-Apr Labor
May-Aug Labor
Sep-Dec Labor
1
1
0
5
1
3
1
0
1
1
2
3
1
0
0
1
2
3
LHS
=SUMPRODUCT($B$3:$D$3,B7:D7)
=SUMPRODUCT($B$3:$D$3,B8:D8)
=SUMPRODUCT($B$3:$D$3,B9:D9)
=SUMPRODUCT($B$3:$D$3,B10:D10)
=SUMPRODUCT($B$3:$D$3,B11:D11)
=SUMPRODUCT($B$3:$D$3,B12:D12)
<-Obj. Cell
Sign
<=
<=
<=
<=
<=
<=
RHS
500
400
120
1600
2000
1600
Recall that the cells directly under the column headings of Corn, Wheat, and Oats in the
Activity Levels row are the decision variables in this spreadsheet model. Each row under the
constraints heading are equations or inequalities that define the feasible space and that Excel only
uses the columns LHS and RHS to evaluate feasibility. The values under the variable headings in
the constraint rows are the constraint coefficients and represent technical information about the
usage rates of constrained resources in each of the activities.
However, we can use constraints to define relationships that do not reflect actual resource
limits. For example, corn is often grown in rotation with another crop to help control for invasive
species that diminish yield. Our first task with this model will be to introduce a constraint that
requires corn and wheat to be grown in a (Corn-Corn-Wheat) rotation, meaning that two years of
corn planting must be followed by one year of wheat planting. Our model only reflects one planting
season so an accurate way to depict the rotation for a single year would be to require at least one
acre of wheat be planted for every two acres of corn that are planted.
Mathematically this constraint would indicate that the ratio of wheat to corn must be at least
as large as 1:2. In an inequality we could write the following:
W / C  0.5
Unfortunately this is not a linear equation which we are required to use for linear programming
models. We can convert this to a linear equation however by simply multiplying both sides of the
inequality by C giving us:
W  0.5C
Then, we can subtract 0.5C from both sides of the inequality and we have a constraint that looks like
the following:
W  0.5C  0
Add this constraint to the three crop model beneath the Sep-Dec labor constraint. The last row of
your spreadsheet should look like this when you have added the new constraint. Be sure that you
include the minus sign in front of the coefficient on corn and that you fill in the sumproduct
formula under LHS.
C-C-W Rotation
-0.5
1
0
=SUMPRODUCT($B$3:$D$3,B13:D13)
>=
0
This model is ready to solve. Follow the steps for Solver implementation (see previous lab handouts)
being sure that you add the new constraint and paying attention to the fact that you will need to
choose >= in the ‘Add Constraint’ box for this new constraint.
Our next step is to add a labor hiring activity to the model. To do this, we need to insert a
new column to the right of Oats. You can do this by right-clicking on the Excel column heading
that is currently next to Oats (it is E in the spreadsheet on the website) and choosing ‘Insert.’ (See
the figure at the top of the next page). In this new column we will have a decision variable in the
‘Activity Levels’ row, initially set to zero. This is the value that Solver will adjust in its calculation of
the optimal choice set. In the objective equation row (currently labeled ‘profit per acre’ but we
should re-label that profit per unit since labor hired will not be in units of acres) we will need to add
the per unit cost of labor. Since labor is a cost, we can think of it as a negative profit so e.g. if the
price of labor is $6.00 it would appear in the objective equation with a coefficient of -6.
We have three types of labor in the model but recall from last week that only Jan-Apr labor
was limiting so we will assume that any hired labor is used during that period. That means that the
only constraint we will need to modify is the one in the row labeled ‘Jan-Apr Labor.’ The three crop
planting activities have coefficients of 5,1, and 1 respectively and are interpreted as the usage of
available labor. The RHS of this constraint gives all available labor during the period as 1600.
Mathematically we have the following inequality. 5C  W  A  1600 , where C = corn, w = wheat,
and A = oats. If we wanted to add a variable for labor hiring (call it L) to this inequality we could
add it to the right hand side since any labor we hire would increase the available labor amount for
use in crop planting.
Figure 1. Inserting a Column into the Spreadsheet
The new constraint would then be 5C  W  A  1600  L . We would then interpret this as
the LHS giving us the different demands for labor by planting activities and the RHS giving us
supply of labor either from the family’s stock of time or labor that is hired-in. Our spreadsheet
model will not allow us to modify the RHS of the constraint however, so we must convert the
constraint to something we can enter into Excel. This means getting all of the variables on the LHS
of the inequality so we must subtract L from both sides of the inequality giving us the following.
5C  W  A  L  1600
Add the hired activity variable to your model by inserting the column and inputting the coefficients
into the objective and Jan-Apr Labor equations. Assume that the cost of labor is $6.00. Pay close
attention to the following as you proceed to solve the model with the hired labor activity.
1) Before you start Solver you need to check all of your formulas since we have inserted a
new column into the spreadsheet. The objective cell and all of the LHS formulas must now
include the additional column we added for hired labor so edit those accordingly.
2) When you open solver, delete the constraint for the corn and wheat rotation since we want
to consider the changes in profits from the base case that does not assume a rotation
requirement.
3) Be sure you include the hiring labor activity in the ‘By Changing Cells’ box of Solver.
If you have correctly done everything, you are ready to Solve the model. If your model is
correct you will have a hired labor quantity of 420. If yours is not that value recheck your
steps above.
Part II: Building a minimization model
Our problem in part II is a minimization problem. The decision maker in this case wants to
minimize the cost of a meal that meets a set of predefined nutrition requirements. The choices
available to the decision maker are quantities of food to consume as part of the meal. The data for
the problem is given below:
Cereal A
Cereal B
3.8
4.2
0.10
1.00
100
0.25
0.25
125
Cost per oz (cents)
Nutrition
Constraints
Thiamine (mg/oz)
Niacin (mg/oz)
Calories
Total Nutrient
Required (mg)
1.00
5.00
500
We can use the information above to write the following algebraic LP problem where C=total cost
of the meal, A is cereal A, and B is cereal B.
min C  3.8 A  4.2 B
s.t.
Thi : 0.10 A  0.25 B  1.00
Nia : 1.00 A  0.25 B  5.00
Cal : 100 A  125B  500
Non  neg : A  0; B  0;
Note that the constraints are all minimum amounts of the nutrient that must be met for the diet to
be feasible and that we are trying to minimize the cost of those feasibility restrictions. Setup this
model and solve it in Excel for the optimal (minimum) cost level. Save that model and solution to
the third sheet (label it Lab 3) in the Excel workbook that you are preparing for submission (i.e. the
one that has the different models I have asked you to save from labs). Remember that you only need
to submit answers to the questions below next Monday and that your answers should be typed.
Questions for Homework: All questions require typed answers. There is no lab assignment
this week, submit answers to all of these questions in class on October 1.
Part I Questions:
1) The solution we had to the original three crop model was the following:
Corn = 275
Wheat = 120
Oats = 105
Profits = 18,315
What is the new solution when we add the rotation constraint? Do different constraints bind
in the rotation constrained problem (recall that binding means LHS = RHS)?
2) In the last section of part 1 you are asked to add a labor hiring activity and investigate how
this choice changes the solution. How does this solution differ from the 3 crop model from
last week? Do profits increase or decrease?
3) It might be difficult to hire seasonal labor for just the Jan-Apr period. What if to hire
someone for Jan-Apr meant that we had to guarantee them an equal number of hours during
each of the other two periods as well (e.g. if you hire them 200 hours in Jan-Apr you must
guarantee them 200 hours of work in each of the other two periods).
Under this assumption, how would you model the hired labor activity in this model?
Describe the changes made to the algebraic model.
Part II Questions:
4) What is the optimal cost of the meal?
5) What are the optimal quantities of each cereal to consume as part of the meal?
6) Which constraints bind (have LHS = RHS) in the optimal solution?
7) For each cereal type, divide the RHS value by the coefficient of each constraint. Report these
values and determine which is the most limiting requirement for each cereal. Note that we
are doing a minimization problem with >= constraints here so that the interpretation is
exactly opposite of the ‘most limiting resource’ that we calculated for the crop problem.
8) Provide a brief diet recommendation based on the results of your model.
9) If a new food was available to the decision maker that cost 3.5 cents/oz, had 0.25 mg/oz of
Thiamine, had 1.00 mg/oz of Niacin, and contained 125 calories/oz would the decision
maker be able to find a lower cost meal with the same nutrition? (Note that you don’t need
to solve the model to answer this question. Just compare the objective and constraint
coefficients of the new food with the two available cereals.)
10) Assume that the decision maker has a coupon that discounts the per oz price of cereal A by
50 percent. How does this change the best choice for the decision maker. (Note you should
resolve the model for this queston).