1.) For each problem, your responsibility is to formulate the decision environme

Question

1.) For each problem, your responsibility is to formulate the decision environment as a mathematical model. You are not required to solve the problems! Simply write up the appropriate models. This, you should recall, will include complete descriptions of the decision variables, a statement of the objective function, and all relevant constraints. (Note that constraints are not to contain “IF” statements or multiplication or division of variables.) Your models should be written in a form that I can interpret, not one that is meant for Excel.

The town of Lytle must determine how to best deploy four new emergency medical services vehicles among five existing EMS locations. The new vehicles differ in their hardware, and thus differ in their ability to respond to emergencies. The table below gives a “coverage rating: for each pair of vehicle and EMS location. The “coverage rating” is a function of the capability of each vehicle and an estimate of the number of emergencies in the immediate area around each EMS location.

EMS Location

                Vehicle                 A             B             C             D             E

                     1                         12           8              10           6              2

                     2                         6             6              4             8              9

                     3                         10           10           8             4             1

                     4                         12           10           12           10           14

It has been determined that no EMS location would get more than one of the new vehicles. Further, for space considerations, Vehicle 4 cannot be assigned to Location E. Also, Locations C and D cannot both get a new vehicle. Formulate this problem as a mathematical model with the goal of maximizing the “coverage” provided by the four new vehicles.

Explanation / Answer

Decision Variables

Let Xij represents the binary integer such that Xij=1 when vehicle-i is assigned to location-j and Xij=0 otherwise
i=1,2,3,4
j=A, B, C, D, E

Objective function:

Maximize Z = Total coverage rating =
12X1A + 08X1B + 10X1C + 06X1D + 02X1E
+ 06X2A + 06X2B + 04X2C + 08X2D + 09X2E
+ 10X3A + 10X3B + 08X3C + 04X3D + 01X3E
+ 12X4A + 10X4B + 12X4C + 10X4D + 14X4E

Constraints

4i=1 Xij <= 1 for j = A, B, C, D, E (as no location would get more than one of the new vehicles)

X4E = 0 (Vehicle 4 cannot be assigned to Location E)

4i=1 XiC + 4i=1 XiD <= 1 (C and D cannot both get a new vehicle)

4i=1 Ej=A Xij <= 4 (there are four vehicles only)

Xij = {0,1}

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