Question

The Question.

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 nineteen 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 165000 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 6000 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 £32 and £79 respectively, whereas each desk and workstation generates profits of £102 and £144 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 £1160 per month; however they have estimated that the cost of renovating and equipping the factory will be £700000. 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;

(c) Combined Enter the constraints for combining the plants and the expression to be optimised. Maximise: *1 + 32 + 33 + 34 subject to *+ *+ 23+ 50 [Wood) [Metal fittings) *1 + 2+ 33 + 34 (Labour] Determine the optimal solution for 21, 22, 23 and 24 and the profit that this solution will generate and enter your solution below. (Enter the optimal solution correct to 3dp and the profit to the nearest pound.) The optimal solution is t=0 , 12 , 13 , 14 Profit: E Is it economically sensible to combine the two plants? Select

Answer 1

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;

The unit profit from chairs = £32

The unit profit from tables = £79

The unit profit from desk = £102

The unit profit from workstation = £144

We have to maximize profit.

Hence Objective Function is:

Maximise: 79*x1 + 32*x2 + 102*x3 + 144*x4

Constraints:

Production time is 7.5 hours per day and 24 days a month = 180 hours per month = 10800 minutes

There are a total of 19 workers. Hence total worker minutes = 10800*19 = 205200 minutes

Production times:

Table = 72 minutes

Chair = 18 minutes

Desk = 90 minutes

Workstation = 2 hours = 120 minutes

Hence the used time should be less than or equal to the available time.

(Labour)                               72*x1 + 18*x2 + 90*x3 + 120*x4 <= 205200

Wood:

Table uses = 20 m²

Chair uses = 12.5 m²

Desk uses = 20 m²

Workstation uses = 30 m²

Total available wood = 165000 m²

Hence,

(Wood)                                20*x1 + 12.5*x2 + 20*x3 + 30*x4 <= 165000

Metal Fittings:

One box is required for every item.

The company can purchase 24000 boxes (6000 boxes per furniture type) of fittings per month

Hence,

(Metal Fittings)                                x1 + x2 + x3 + x4 <= 24000

None of the production numbers can be negative. Hence,

x1 >= 0

x2 >= 0

x3 >= 0

x4 >= 0

For finding the optimal solution, we will use excel solver:

1/A	B	C	D	E	F	G	H	I	J
2						Tables	Chairs	Desks	Workstations
3						x1	x2	x3	x4
4						0	11400	0	0
5
6
7			Objective
8			Maximise: 79*x1 + 32*x2 + 102*x3 + 144*x4			364800
9
10
11			Constraints		RHS	LHS
12		Labour	72*x1 + 18*x2 + 90*x3 + 120*x4	<=	205200	205200
13		Wood	20*x1 + 12.5*x2 + 20*x3 + 30*x4	<=	165000	142500
14		Metal fittings	x1 + x2 + x3 + x4	<=	24000	11400
15			x1	>=	0	0
16			x2	>=	0	11400
17			x3	>=	0	0
18			x4	>=	0	0
19
20

$wHws3ybTBu7HQAAAABJRU5ErkJggg==$

Hence, Optimal Solution

x1 = 0

x2 = 11400

x3 = 0

x4 = 0

Profit = £ 364800

It is NOT economically sensible to combine the two plants.

Because, the optimal profit is only 364800. The renovating cost is 700000. Hence the company will go into losses.

IF YOU LIKE THE ANSWER, PLEASE GIVE AN UP-VOTE OR THUMB UP. THIS WILL ENCOURAGE ME TO ANSWER MORE!!