# The scenario: Park Furnishings manufactures school and university classroom furniture

Prof. Reinford
**The scenario**

Park Furnishings manufactures school and university classroom furniture. The company has two production plants, located at Easton and Weston. The Easton plant produce tables and chairs and the Weston plant produces desks and computer workstations. Park has a working day of 7.5 hours and employs twenty workers at each plant. You may assume that there is a total of 24 working days every month.

All Park products are manufactured using finished pressed wood and polished aluminium fittings. Including scrap, each table and desk uses 20 m2 of pressed wood whereas each chair and workstation needs 12.5 m2 and 30 m2 respectively. A total of 240000 m2 of pressed wood is available each month and is divided equally between the two plants. The aluminium fittings that reinforce the legs of all the production items are purchased from an outside supplier. Each plant can purchase up to 9500 boxes of fittings per month and one box is required for every item of furniture produced. Production times are 72 minutes per table, 18 minutes per chair, 90 minutes per desk and 2 hours per workstation.

The unit profit for chairs and tables are £39 and £72 respectively, whereas each desk and workstation generates profits of £105 and £142 respectively.

Park is considering combining operations of both plants into a single factory. This consolidation will have the advantage of combining all available production resources as well as reducing administration costs by £1420 per month; however they have estimated that the cost of renovating and equipping the factory will be £1100000. Due to the current financial position Park Furnishings is only prepared to go ahead with the combined operation if it saves money in the first year of operation.

Let

· x1 represent the number of *tables* made per month;

· x2 represent the number of *chairs* made per month;

· x3 represent the number of *desks* made per month;

· x4 represent the number of *workstations* made per month;

where x1,x2,x3,x4 ≥0

(a) **Easton**

Enter the constraints for the **Easton** plant and the expression to be optimised.

Maximise: x1 + x2

subject to

x1 + x2≤ [Wood]

x1 + x2≤ [Metal fittings]

x1 + x2≤ [Labour]

Sketch the constraints and hence find the values of

· (i) a,b,c, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x1-axis;

· (ii) d,e,f,the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x2-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter a: Enter b: Enter c: Enter d: Enter e: Enter f:

Now draw a sample profit line on your graph. Choose a value of the profit (P>0) and using this value, find the values of

(i) g, the intersection of your sample profit line with the x1x1-axis;

(ii) h, the intersection of your sample profit line with the x2x2-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter P: Enter g: Enter h:

Determine the optimal solution for x1 and x2 to the nearest integer and the profit that this solution will generate and enter your solution below.

The optimal solution is x1= , x2=

Profit: £

Select the two constraints which intersect to give the optimal solution.

The optimal solution is the intersection of with

(b) **Weston**

Enter the constraints for the **Weston** plant and the expression to be optimised.

Maximise: x3 + x4

subject to

x3 + x4≤ [Wood]

x3 + x4≤ [Metal fittings]

x3 + x4≤ [Labour]

Sketch the constraints and hence find the values of

· (i) a,b,c, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x3-axis;

· (ii) d,e,f, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x4-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter a: Enter b: Enter c: Enter d: Enter e: Enter f:

Now draw a sample profit line on your graph. Choose a value of the profit (P>0) and using this value, find the values of

· (i) g, the intersection of your sample profit line with the x3x3-axis;

· (ii) h, the intersection of your sample profit line with the x4x4-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter P: Enter g: Enter h:

Determine the optimal solution for x3 and x4 to the nearest integer and the profit that this solution will generate and enter your solution below.

The optimal solution is x3= , x4=

Profit: £

Select the two constraints which intersect to give the optimal solution.

The optimal solution is the intersection of with

(c) **Combined**

Enter the constraints for **combining** the plants and the expression to be optimised.

Maximise: x1 + x2 + x3 + x4

subject to

x1 + x2+ x3 + x4≤ [Wood]

x1 + x2+ x3 + x4≤ [Metal fittings]

x1 + x2+ x3 + x4≤ [Labour]

The optimal solution is x1= , x2= , x3= , x4=

Profit: £

Is it economically sensible to combine the two plants?