Assign Four Patients To Five Nurses Using The Follow

Assign Four Patients To Five Nurses Using The Foll

You are asked to assign four patients to five nurses using the following travel distances. We want to minimize total mileage. Determine the optimal solution, the objective function value, the number of decision variables if formulated as a linear program, whether a fifth patient would be assigned to Emily if added, and whether this is an integer program (True/False).

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Assignment problems involving resource allocation, such as assigning patients to nurses to minimize total travel distance, are classical examples of combinatorial optimization problems. These problems are often modeled using linear programming, specifically through the formulation of an assignment problem. In the context of the given scenario, four patients must be assigned to five nurses with the goal of minimizing the total travel distance, which suggests the utilization of assignment problem techniques.

To begin, let's understand the key elements of the problem. The patients are Warren Xavier, Yolanda Zanthia, Amy Brenda, Connor Danielle, and Emily. The nurses are not named explicitly in the problem statement, but it is implied that there are five nurses available. The distances between each patient and each nurse are provided in some matrix, which, while not explicitly listed here, form the basis for the optimization problem.

Optimal Solution and Objective Function

The optimal assignment involves selecting pairs of patient–nurse matches that minimize the total travel mileage. This is typically solved using the Hungarian Algorithm or other combinatorial optimization approaches designed for assignment problems. Although the explicit distance matrix is not shown, this solution process would evaluate possible assignments and select the combination with the minimum total distance.

Suppose, for example, that the optimal solution assigns:

• Warren to Nurse A
• Xavier to Nurse B
• Yolanda to Nurse C
• Zanthia to Nurse D

with the fifth nurse remaining unassigned. The total mileage for this configuration, which can be computed based on the known distances, constitutes the objective function value. This value—the minimal total distance—would be computed after solving the assignment problem.

Decision Variables in the Linear Programming Model

When formulating this as a linear programming problem, the decision variables typically represent whether a particular patient is assigned to a particular nurse. Specifically, each variable, say x_ij, indicates whether patient i is assigned to nurse j (1 if assigned, 0 otherwise). Since there are four patients and five nurses, the total number of possible patient–nurse pairs is 4×5=20. Thus, the linear program would have 20 decision variables.

Adding a Fifth Patient and Its Assignment

If a fifth patient is introduced to the problem, questions arise regarding the assignment of this new patient. Would she be assigned to Emily? Since Emily is one of the nurses who may already be assigned alternatively, adding a fifth patient might lead to assigning her to Emily if this improves the total mileage. However, this depends on the specific distances involved. Given that only four patients are assigned to five nurses, and assuming the constraints are such that each patient gets a nurse, the fifth patient could be assigned to Emily if she does not already have a patient and if this assignment reduces total mileage or meets other constraints depending on the problem's specifics.

Integer Programming Aspect

This problem is an example of an integer programming problem because the decision variables are binary (either a patient is assigned or not). The assignment problem typically involves binary variables, making the problem an integer program. Since fractional assignments are not feasible in this context, the statement referencing the model as an integer program is true.

Conclusion

In summary, solving this assignment problem involves determining the optimal patient–nurse pairings that minimize the total travel distance, computing the total mileage (objective function value), and formulating the problem in terms of decision variables. The decision variables are 20 in total if formulated as a linear program. Adding a fifth patient could lead to her being assigned to Emily if the distances justify it. The problem is inherently an integer program due to the binary nature of assignments.

References

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