Problem 1 Suppose that you are the pest control manager of a huge apartment comp
ID: 350518 • Letter: P
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Problem 1 Suppose that you are the pest control manager of a huge apartment complex in Florida. Recently, you have observed that there are too many complaints from the tenants about two types of bugs (spiders and fire-ants) in the apartments therefore, you decided to fight with these bugs. From the complaints, you roughly estimate that the number of spiders is 10,000 and the number of fire-ants is 150,000. There are three types of chemicals that you can use to kill these harmful bugs, but each chemical has different effect on different bugs and different environmental damage (which is measured in terms of toxic meters). Particularly, you have the following data about the chemicals: 1 pound of Chemical A: costs $20, kills 1,000 banana-spiders and 5,000 fire ants in the complex, and causes 5 toxic meters 1 pound of Chemical B: costs $10, kills 600 banana-spiders and 8,000 fire ants in the complex, and causes 10 toxic meters 1 pound of Chemical C: costs $15, kills 750 banana-spiders and 12,000 fire ants in the complex, and causes 2 toxic meters Since you are in Florida, you want to fight with these bugs while keeping the environmental damage minimum because of the chemicals you decide to use. In particular, you want to guarantee the following goals: At least 90% of banana-spiders must be killed At least 60% of the fire-ants must be killed At least 75% of all bugs must be killed * * You can spend at most $1,500 on chemicals In this problem, you are asked to mathematically formulate a linear programming model for the above pest control problem that will minimize the total environmental damage (15 points) Define your decision variables, formulate your objective function and state your objective, formulate your constraints one by one, and combine everything to present the final complete mathematical formulation. (5 points) Instead of minimizing the environmental damage, suppose that you have now decided to minimize the total spending on chemicals such that your environmental damage should not exceed 75 toxic meters. How would you modify your linear programming model in part a? Explain briefly (that is, you do not need to present a whole new model, just discuss what will be added and what will be removed and what will change and show these changes mathematically). a) b)Explanation / Answer
a) The decision variables are the controllable factors. Here the controllable factors that I can adjust are the amount of chemical A, B and C to use to achieve my objectives. These can be denoted by X, Y and Z.
X = pounds of chemical A
Y = pounds of chemical B
Z = pounds of chemical C
X, Y, Z are the decision variables.
The objective here is to minimize the total environmental damage. In other words, achieve the goal with minimum toxicity. However we know that X, Y and Z pounds of chemicals A, B and C respectively will cause a damage of 5, 10 and 2 toxic meters. This objective can be denoted by K. Our objective is to
Minimize K, where K = 5X +10Y +2Z
To setup the constraints, we must check the goals that have to be achieved.
At least 90% of the banana-spiders must be killed. We estimated that number of spiders are 10,000 hence we must kill greater than or equal to 9,000 spiders. Since we are using X, Y and Z units of chemicals we know that the number of spiders they will kill and this can be formulated as
1000X + 600Y +750Z >= 9000
Similarly, to kill 60% (90,000) of fire ants, we can formulate,
5000X + 8000Y + 12000Z >= 90,000
Similarly killing 75% of all bugs means killing 75% of 160,000 bugs or 120,000 bugs. This can be formulated as,
1000X + 600Y +750Z + 5000X + 8000Y + 12000Z >= 120,000 or,
6000X + 8600Y + 12750Z >= 120,000
Spend at the most $1500 on chemicals can be represented as
20X + 10Y +15Z <= 1500
Of course the values of X, Y and Z cannot be negative hence we shall add another constraint
X, Y, Z >= 0
The complete mathematical formulation for this problem is
Decision variable
X, Y, Z that corresponds to pounds of Chemical A, B, and C
Objective function
Minimize K where K = 5X +10Y +2Z
Constraints
1000X + 600Y +750Z >= 9000
5000X + 8000Y + 12000Z >= 90,000
6000X + 8600Y + 12750Z >= 120,000
20X + 10Y +15Z <= 1500
X, Y, Z >= 0
b) If the objective is to minimize the total spending then we will need to change the objective function. We need to minimize J where
J = 20X + 10Y +15Z
Replace the previous objective function K with the new objective function J
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