# Formulation Problems In Mathematical Modeling

For each of the following problems, describe a model. Identify the important factors or quantities, decide on which quantities will be the main variables, which ones should be auxiliary variables. What are the underlying assumptions? Would you try to model this situation by a stochastic method (probability or combinatorics), a dynamic method (calculus or differential equations), by a linear programming method, a statisical method, a geometrical method (topology, symmetry, proportionality) or something else? How are the variables related, what additional information would you need to make a better model? Can you construct any graphs (either in the sense of functions or graph theory), diagrams , charts or other visual representations that might be useful?

In those cases in which some structure is already given (such as problem 2) try to write out some of the equations for the model.

1. What is the Sonoma County Hospital worth?

2. Construct a model for an epidemic; assume that the population is partitioned into four mutually exclusive subgroups:

1. The susceptibles (S), those persons who are currently uninfected, but may become infected;
2. The latently infected (L), those who are currently infected, but not yet capable of transmitting the disease to others;
3. The infectives (I), those who are currently infected and capable of spreading the infection; and
4. The removeds (R), those persons who have had the disease and are dead, or have recovered and are permanently immune, or are isolated until death, recovery or permanent immunity occurs.

3. How would the assumption that some people are naturally immune affect your model in Problem #2? How would the assumption that some people can be reinfected affect your model in Problem #2?

4. The Aztec Refining Company produces two types of unleaded gasoline, regular and premium, which it sells to its chain of service stations for \$12 and \$14 per barrel, respectively. Both types are blended from Aztec's inventory of refined domestic oil and refined foreign oil, and must meet the following specifications:

```        Maximum     Minimum    Maximum   Minimum
Vapor      Octane      Demand    Deliveries
Pressure   Rating      bbl/wk    bbl/wk
Regular    23        88        100000     50000
Premium    23        93         20000      5000
The characteristics of the refined oils in inventory are as follows:
Vapor       Octane    Inventory      Cost
Pressure    Rating      bbl        \$/bbl

Domestic   25          87        40000        8
Foreign    15          98        60000       15
```
What quantities of the two oils should Aztec blend into the two gasolines in order to maximize weekly profit?

(This problem is from Operations Research by Richard Bronson, Schaum Outline Series, McGraw-Hill

5. There was a certain pond. One day, 20 fishes were caught in it and the next day 10. How many would one expect to catch on the third day?

6. Consider cars traveling along a roadway in one direction. Let k be the concentration of cars (then number of cars per 100 ft of roadway) and let q be the rate of flow (cars per minute).

The flow - concentration curve

• Argue that q and k are related as shown in the figure above.
• Describe explicitly the assumptions supporting your argument and defend or criticize these assumptions.
• Analyze this curve in terms of max, slope, curvature, etc. interpreting these quantities as rules for "traffic flow theory".

7. Construct a predator-prey model in which the prey has:

• an exponential growth rate
• a logistic growth rate

8. Derive a number of different models from the general population model:

dP/ dt = F(t,P)

Where P(t) is the population and t is time. Considering F to be a variety of possible functions, e.g. periodic (in P or t), monotonic, autonomous (independent of t), stochastic, etc. and discuss the consequence of each model. What function F might be used in a population harvesting model?

9. A U.S. Coast Guard ship is out in the open sea in a heavy fog. Suddenly the fog lifts and the Captain spots a drug runner about 1 mile away and the fog immediately drops again. The coast guard ship can travel twice as fast as the drug runner; describe the captains best course of action.

10. A cargo plane has three compartments for storing cargo: Front, center, and back. These compartments have capacity limits on both weight and space, as summarized below:

```                 Weight          Space
capacity        capacity
Compartment      (tons)          (cu ft)
Front              12             7000
Center             18             9000
Back               10             5000```
Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weigth capacity to mainatain the balance of the airplane. The following four cargoes have been offered for shipment on an upcoming flight as space is available:

```            Total weight    Space   Profit
Cargo        (tons)         (cu ft/ton)  (\$/ton)
1           20              500        280
2           16              700        360
3           25              600        320
4           13              400        250   ```
Any portion of these cargoes can be accepted. The objective is to determine how much (if any ) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight.

(This Problem is from Operations Research by Hillier & Leiberman)

11. One day it started snowing at a heavy and steady rate. A snow plow starts out at noon and goes two miles in the first hour and one mile in the second hour. When, to the nearest minute, did it start snowing ?

12. Cars A, B, C are lined up with B directly behind A and C directly behind B on a side road, all are waiting at an intersection for a break in the cross traffic on the main road, and all are going to turn left onto the main road. Car A makes it through a gap in the traffic, but there is no time for car B to make it into the same gap, similarly C cannot make the same gap as B. Formulate a mathematical model that will desribe the change in the relationship among these three cars during this process.

13. Devise a scheme for scheduling a round-robin tournament among n players (where n is any integer) and explicity write out the rounds for the case in which n = 10, n = 9 and n = 11.

14. A thermostat is set at A; it continuously reads the room temperature and makes an adjustment in the heat source; this adjustment is proportional to the difference between A and the temperature that it reads at time t (in minutes). After the heat is adjusted, it takes ten minutes for the temperature near the thermostat to change. Write out the necessary assumptions and equations to model this situation. Compare this to a Stock Room model in which the clerk wishes to maintain an inventory at a certain number, say A, and it takes 10 days for supplies be delivered after placing an order.

15. Consider the epidemic epidemic models in problem #2 in which the rate of change in the infected population is proportional to the number of people infected at some time in the past, say for a disease with an incubation period of ten weeks.