Question

PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 85 full-time equivalent pharmacists (combination of full time and part time) and 175 full-time equivalent technicians. Each spring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the next year shows that at least 267 full-time equivalent employees (pharmacists and technicians) will be required to staff the pharmacies. The personnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition and prepare for future growth, management stated that at least 15 new pharmacists must be hired. In addition, PharmaPlus's new service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is $40 per hour and the average salary for technicians is $10 per hour.

How many pharmacists and technicians are needed? What is the Optimal Objective Value? $_______ at (P,T) = (_____)
Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)?
Additional Technicians to hire ______
What will be the impact on the payroll?
The payroll cost using the current levels of 85 pharmacists and 175 technicians is $____ per hour. The payroll cost using the optimal solution in part (a) is $ ___ per hour. Thus, the payroll cost will go up by $ _____

Answer 1

Explanation:

To solve this problem, we can set up a linear programming model. Let P represent the number of pharmacists to be hired, and T represent the number of technicians to be hired. The objective is to minimize the payroll cost while meeting the requirements and constraints.

Objective:

Minimize Z = 40P + 10T (Payroll cost)

Subject to the following constraints:

Total full-time equivalent employees required:

P + T ≥ 267 (To meet prescription load requirements)

Expected attrition:

P - 10 ≥ 15 (At least 15 new pharmacists must be hired)

T - 30 ≥ 0 (At least 30 technicians must be hired)

Technician-to-pharmacist ratio:

T ≤ 2P (No more than two technicians per licensed pharmacist)

Now, let's solve this linear programming problem.

Objective Function:

Z = 40P + 10T

Constraints:

P + T ≥ 267

P ≥ 25

T ≥ 30

T ≤ 2P

Let's find the optimal solution.

Start by considering constraint (1):

P + T ≥ 267

Consider constraint (4):

T ≤ 2P

To satisfy constraint (3), P should be at least 25:

P ≥ 25

Now, we have the following equations:

P + T = 267

T = 2P

P ≥ 25

Solving these equations:

From equation (2), we can substitute T in equation (1):

P + 2P = 267

3P = 267

P = 267 / 3

P ≈ 89

Now that we have the number of pharmacists, we can find the number of technicians using equation (2):

T = 2P

T = 2 * 89

T = 178

So, the optimal solution is to hire approximately 89 pharmacists and 178 technicians.

The optimal objective value is:

Z = 40P + 10T

Z = 40 * 89 + 10 * 178

Z = $3,560 + $1,780

Z = $5,340

Therefore, the optimal objective value is $5,340 at (P, T) = (89, 178).

Now, let's calculate the additional technicians needed:

Additional Technicians to hire = Required Technicians - Current Technicians

Additional Technicians to hire = 178 - 175

Additional Technicians to hire = 3

So, you need to hire 3 additional technicians.

Now, let's calculate the impact on the payroll:

Payroll cost using the current levels of 85 pharmacists and 175 technicians:

Current Payroll = 40 * 85 + 10 * 175

Current Payroll = $3,400 + $1,750

Current Payroll = $5,150 per hour

Payroll cost using the optimal solution in part (a) (89 pharmacists and 178 technicians):

Optimal Payroll = 40 * 89 + 10 * 178

Optimal Payroll = $3,560 + $1,780

Optimal Payroll = $5,340 per hour

The payroll cost will go up by:

$5,340 - $5,150 = $190 per hour

So, the payroll cost will go up by $190 per hour with the optimal solution.