A new client has arrived! They want to know the optimal number of each type of room for their new house to maximise their total value. There are four main room types: Bedrooms, Bathrooms, Living Spaces (includes any open living space) and Kitchens. On average, the additional value a Bedroom will produce is $8,000 per room, a Bathroom will produce $15,000 per room, a Living Space will produce $13,500 per room and a Kitchen will produce $15,000. The client’s budget for all their rooms has been determined to be a total of $100,000 – you cannot exceed this number. To produce the new total value of each room, a Bedroom costs $5,000 per room, a Bathroom costs $10,000 per room, a Living Space costs $9,000 per room and a Kitchen costs $10,000 per room. Also, each room has an average size, a Bedroom is 12 sqm per room, a Bathroom is 8 sqm per room, a Living Space is 20 sqm per room and a Kitchen is 8 sqm per room. The total floor space available to renovate for all the rooms must not exceed 200 sqm. In addition, there is a labour allocation for this project of 400 hours to install the flooring, install the windows and put in all the finishes. Each room has an average labour time for a Bedroom it is 30 hours per room, a Bathroom is 50 hours per room, a Living Space is 20 hours per room and a Kitchen is 100 hours per room. The client has a set of requirements which you need to include in your model based on the number of rooms. The requirements are listed below in the table and an additional requirement is the client requires there is at least one Bathroom for every two Bedrooms. Requirements Minimum Number of Rooms 10 Minimum Number of Bedrooms 2 Minimum Number of Bathrooms 1 Minimum Number of Living Spaces 1 Minimum Number of Kitchens 1 Formulate a linear programming model for this problem, filling in the template over the page. Type up the full mathematical model in Word and include it here. Decision Variables Objective and Objective Function Constraints b. Enter your model from (a) into a Linear Programming template. For the solution obtained in part (b), interpret the shadow price for the available “Floor Space” constraint and for the constraint “Budget”. For each interpretation state: The shadow price, The range of feasibility and The impact of the shadow price on the objective function for changes within this range of feasibility. For the Budget constraint only: calculate the change in Total Value ($) if the Budget available is increased by $4,000. (Re-run Solver and state the new optimal solution in the report body) The client has stumbled onto a good deal on flooring for a Living Space which will increase the value by $20 per sqm at the same cost. Calculate the new predicted Total Value ($) and discuss whether the solution remains optimal. Show calculations to support the predicted changes to maximum total value (if any). Hint: You may need to re-run solver. The client notices that renovating a Kitchen costs a lot of money relative to the space. Suppose the client relaxes the condition on the number of Kitchens to be optional (this means zero kitchens are required to be renovated). Calculate the new predicted Total Value ($) for their new design layout and discuss whether the solution remains optimal. Show calculations to support the predicted change to total value ($) (if any). (Re-run Solver and state the new optimal solution in the report body)
A new client has arrived! They want to know the optimal number of each type of room for their new house to maximise their total value. There are four main room types: Bedrooms, Bathrooms, Living Spaces (includes any open living space) and Kitchens. On average, the additional value a Bedroom will produce is $8,000 per room, a Bathroom will produce $15,000 per room, a Living Space will produce $13,500 per room and a Kitchen will produce $15,000.
The client’s budget for all their rooms has been determined to be a total of $100,000 – you cannot exceed this number. To produce the new total value of each room, a Bedroom costs $5,000 per room, a Bathroom costs $10,000 per room, a Living Space costs $9,000 per room and a Kitchen costs $10,000 per room. Also, each room has an average size, a Bedroom is 12 sqm per room, a Bathroom is 8 sqm per room, a Living Space is 20 sqm per room and a Kitchen is 8 sqm per room. The total floor space available to renovate for all the rooms must not exceed 200 sqm. In addition, there is a labour allocation for this project of 400 hours to install the flooring, install the windows and put in all the finishes. Each room has an average labour time for a Bedroom it is 30 hours per room, a Bathroom is 50 hours per room, a Living Space is 20 hours per room and a Kitchen is 100 hours per room.
The client has a set of requirements which you need to include in your model based on the number of rooms. The requirements are listed below in the table and an additional requirement is the client requires there is at least one Bathroom for every two Bedrooms.
|
Requirements |
|
|
Minimum Number of Rooms |
10 |
|
Minimum Number of Bedrooms |
2 |
|
Minimum Number of Bathrooms |
1 |
|
Minimum Number of Living Spaces |
1 |
|
Minimum Number of Kitchens |
1 |
- Formulate a linear programming model for this problem, filling in the template over the page. Type up the full mathematical model in Word and include it here.
Decision Variables
<list and define the decision variables here>
Objective and Objective Function
<state the objective (min or max) and include the objective function here>
Constraints
<enter 1 constraint per line including a name for each constraint that contains your initials>
b. Enter your model from (a) into a Linear Programming template.
- For the solution obtained in part (b), interpret the shadow price for the available “Floor Space” constraint and for the constraint “Budget”. For each interpretation state:
- The shadow price,
- The range of feasibility and
- The impact of the shadow price on the objective function for changes within this range of feasibility.
For the Budget constraint only: calculate the change in Total Value ($) if the Budget available is increased by $4,000. (Re-run Solver and state the new optimal solution in the report body)
- The client has stumbled onto a good deal on flooring for a Living Space which will increase the value by $20 per sqm at the same cost. Calculate the new predicted Total Value ($) and discuss whether the solution remains optimal.
Show calculations to support the predicted changes to maximum total value (if any). Hint: You may need to re-run solver.
- The client notices that renovating a Kitchen costs a lot of money relative to the space. Suppose the client relaxes the condition on the number of Kitchens to be optional (this means zero kitchens are required to be renovated). Calculate the new predicted Total Value ($) for their new design layout and discuss whether the solution remains optimal.
Show calculations to support the predicted change to total value ($) (if any). (Re-run Solver and state the new optimal solution in the report body)
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