#### 1. The hospital gardener

For full credit for your graph:

1. Draw the constraints.
2. Draw at least one iso-cost line.
3. Draw a circle or an arrow pointing out the optimal point.

Some people have trouble graphing the potash constraint. Here is a question to help you with that:

(This applet assumes that, in your graph, the X axis shows the amount of the first fertilizer, the one that costs \$4 per bag, and the Y axis shows the amount of the second fertilizer, the one that costs \$5 per bag.)

How many bags of the first (\$4 per bag) fertilizer do you use?

If you got that right, this one should be right, too, but let's check anyway. How much of the second fertilizer do you use?

What's your cost, when you're satisfying the requirements for the least amount of money?

Now you're told that you must buy at least 6 bags of the first fertilizer. This adds a constraint, another line to your graph.

Now how much of each kind of fertilizer do you use?
Fertilizer 1:

Fertilizer 2:

Now's the time to look at your sensitivity report. There are four shadow prices, one for each constraint, so you need four sentences, one for each shadow price, to answer this question. The four constraints are Nitrogen, Phosphates, Potash, and Brother.

Depending on which software you are using, how you set up the analysis, and what options you picked, the shadow prices may be called Shadow Prices, Lagrange Multipliers, or Opportunity Costs. They may also be called Reduced Costs or Reduced Gradient, and be located in the upper table of the Sensitivity Report, not in the lower table that has the other shadow prices.

To see if you're on the right track, tell me how many of those shadow prices are not 0.

In your write-up, explain the shadow prices that are zero and the shadow prices that are not zero.

#### 2. Scheduling aides

When you solve this one, try telling Solver to assume a linear model and show you each iteration. That way you can see how the simplex method works. You will see it crawl through 14 feasible solutions until it reaches the optimum.

In a problem this big, the simplex method is a big improvement over the enumeration method. That would require evaluating 3432 possible solutions!

Let's check one hiring number. How many aides do you hire for Monday through Friday?

What is your total cost per week?

#### 3. The transportation problem

Let's check one result. After you include the constraints for the bridge and import limits, how much do you send from B to D?

Did you get negative numbers for the shadow prices for the bridge and import limit constraints? It makes sense that you would, because those are less-than constraints in a cost-minimization problem. If you loosen a less-than constraint, that has a negative effect on your cost. You can save money.

If you could pay someone to smuggle units in excess of 100 into Canada, how much would it be worth to you per unit?

(This example is stated this way purely for amusement. It does not endorse illegal activity.)