## Pinewood Furniture Company

Pinewood Furniture Company produces chairs and tables from two resources—-labor and wood. The company has 80 hours of labor and 36 board -ft. of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood, whereas a table requires 10 hoursof labor and 6 board-ft. of wood. The profit derived from each chair is $400 and fromeach table $100. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit.(Please note that RHS sensitivity can be applied to ALL constraints, including the non-binding constraints. A non-binding constraints usually has either a lower bound at negative infinity and an actual upper bound when it passes through the original optimal solution, or an upper bound at positive infinity and an actual lower bound when it passes through the original optimal solution. To find the finite bound for a non-binding constraint (when it passes through the original optimal solution), you can plug in the coordinates of the original optimal solution in the left-hand-side of the constraint to calculate the corresponding right-hand-side value. The shadow price for a non-binding constraint is always zero, and it would be sufficient to simply express this fact in your homework answers.)

Question:

1 – Develop a linear programming model (CH2): Define each decision variable, include the objective function, and include all constraints, together with the non-negativity constraints. Put a one to two word explanation next to each constraint to identify each constraint’s role.

2 – Solve the linear program graphically (CH2): On a separate page, draw the graph clearly using the entire page (you can use graphing paper if you prefer). Use a ruler to draw the constraint lines. Show all of your calculations to find the points that constraints intersect with horizontal and vertical axes. Show the feasible region by lightly shading it. Assume one of the vertices of the feasible region to be optimum and draw your objective line passing through that point. Determine the direction of progress and find the last point of contact between the objective line and the feasible region as your optimal solution. If the optimal point is at an intersection of two constraint lines, show all of your calculations for solving the system of the two linear equations to find the values of optimal solution. Calculate and include the optimal objective function value.

3 – Find the sensitivity range for the objective function parameters (CH3): On a separate sheet, determine the upper bound and lower bound for the slope of the objective function value for which the optimal solution stays the same. Then, fix one objective function parameter at a time and find the corresponding upper and lower bounds for the other objective function parameter. See the examples in the lecture notes.

4 – Find the sensitivity range for the right-hand-side values of the constraints (CH3): Determine and include the solution mix and configuration in terms of zero and non-zero variables. For each constraint, determine when the configuration of the optimal solution change when you increase and decrease the right-hand-side, and identify these values as the upper and lower bound of the right-side-value of that constraint. Using these bounds and/or the optimal solution, calculate the shadow price for each constraint. Show all your work in detail. See the examples in the lecture notes and the book.

5 – Solve the linear programming model from using Excel Solver (CH3): Format the spreadsheet according to the examples in the lecture notes; do NOT use the formatting in the book. Use SUMPRODUCT function to enter formulas for the objective function and the constraints (see the examples in the lecture notes). After solving the problem, create the “answers” and “sensitivity” reports. Print out the spreadsheet with your solution on it to a PDF file. Print out the same sheet in formula view (Formulas > Show formulas) to a separate PDF file. Print out the sheets with answers and sensitivity reports, each to a different PDF file. Make each printout fit within one page using View > Page Break Preview.