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Solve the previous formulated model (Universal Claims Processors problem) by using graphical analysis. The optimal solution to the model is

Solve the previous formulated model (Universal Claims Processors problem) by using graphical analysis. The optimal solution to the model is

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
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**Linear Programming Model for Hiring Operators: An Educational Resource** Universal Claims Processors processes insurance claims for large national insurance companies. Most claim processing is done by a large pool of computer operators, some of whom are permanent and some of whom are temporary. ### Productivity Rates - **Permanent Operator:** Can process 16 claims per day - **Temporary Operator:** Can process 12 claims per day ### Company Requirements - **Overall Claims:** The company processes at least 450 claims each day. - **Workstations:** The company has 40 computer workstations. - **Error Rates:** - Permanent Operator: Generates about 0.5 claim with errors each day. - Temporary Operator: Averages about 1.4 defective claims per day. - **Error Limitation:** The company wants to limit claims with errors to 25 per day. ### Wages - **Permanent Operator:** Paid $64 per day. - **Temporary Operator:** Paid $42 per day. ### Objective The company wants to determine the number of permanent and temporary operators to hire in order to minimize costs. ### Variables Define: - \( x_1 \) = Number of permanent operators to hire - \( x_2 \) = Number of temporary operators to hire - \( Z \) = Total cost ### Constraints 1. Processing at least 450 claims per day: \[ 16x_1 + 12x_2 \geq 450 \] 2. Limiting errors to 25 claims per day: \[ 0.5x_1 + 1.4x_2 \leq 25 \] 3. Limited to 40 workstations: \[ x_1 + x_2 \leq 40 \] ### Objective Function Minimize the total cost \( Z \): \[ Z = 64x_1 + 42x_2 \] This mathematical model helps in determining the optimal number of permanent and temporary operators to hire to ensure cost-effectiveness while meeting the company's constraints related to claims processing capacity, error rates, and available workstations.
Transcribed Image Text:**Linear Programming Model for Hiring Operators: An Educational Resource** Universal Claims Processors processes insurance claims for large national insurance companies. Most claim processing is done by a large pool of computer operators, some of whom are permanent and some of whom are temporary. ### Productivity Rates - **Permanent Operator:** Can process 16 claims per day - **Temporary Operator:** Can process 12 claims per day ### Company Requirements - **Overall Claims:** The company processes at least 450 claims each day. - **Workstations:** The company has 40 computer workstations. - **Error Rates:** - Permanent Operator: Generates about 0.5 claim with errors each day. - Temporary Operator: Averages about 1.4 defective claims per day. - **Error Limitation:** The company wants to limit claims with errors to 25 per day. ### Wages - **Permanent Operator:** Paid $64 per day. - **Temporary Operator:** Paid $42 per day. ### Objective The company wants to determine the number of permanent and temporary operators to hire in order to minimize costs. ### Variables Define: - \( x_1 \) = Number of permanent operators to hire - \( x_2 \) = Number of temporary operators to hire - \( Z \) = Total cost ### Constraints 1. Processing at least 450 claims per day: \[ 16x_1 + 12x_2 \geq 450 \] 2. Limiting errors to 25 claims per day: \[ 0.5x_1 + 1.4x_2 \leq 25 \] 3. Limited to 40 workstations: \[ x_1 + x_2 \leq 40 \] ### Objective Function Minimize the total cost \( Z \): \[ Z = 64x_1 + 42x_2 \] This mathematical model helps in determining the optimal number of permanent and temporary operators to hire to ensure cost-effectiveness while meeting the company's constraints related to claims processing capacity, error rates, and available workstations.
### Graphical Analysis Solution for Universal Claims Processors Problem In the following section, we explore the optimal solution for the previously formulated model, known as the Universal Claims Processors problem, by employing graphical analysis. Based on the analysis, the optimal solution to the model is: - **(28.125, 0)** Below are the potential solutions considered: - (40, 0) - (34.44, 5.56) - (20.12, 10.67) The optimal solution (28.125, 0) is marked, indicating that this is the point at which the conditions of the model are best satisfied according to the graphical analysis. This analysis helps determine the precise values (in this case, coordinates) that maximize or minimize the defined objective function under given constraints.
Transcribed Image Text:### Graphical Analysis Solution for Universal Claims Processors Problem In the following section, we explore the optimal solution for the previously formulated model, known as the Universal Claims Processors problem, by employing graphical analysis. Based on the analysis, the optimal solution to the model is: - **(28.125, 0)** Below are the potential solutions considered: - (40, 0) - (34.44, 5.56) - (20.12, 10.67) The optimal solution (28.125, 0) is marked, indicating that this is the point at which the conditions of the model are best satisfied according to the graphical analysis. This analysis helps determine the precise values (in this case, coordinates) that maximize or minimize the defined objective function under given constraints.
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