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# Linear Programming Problem?

Ive been given homework to solve this linear programming via excel solver and i cant do it, i cant even get my head around how to lay the question out on excel in terms of the labour hours etc. There are too many constraints.... This is the question:

Comfortable Hands is a company that features a product line of winter gloves for the entire family, men, women and children. They are trying to decide what mix of these three types of gloves to produce.

The company’s labour force is unionized. Each full-time employee works a 40-hour week. By union contract, the number of full-time employees can never drop below 20. Non-union part-time workers can also be hired with the following restrictions:

(i) Each part-time worker works 20 hours per week.

(ii) There must be at least 2 full-time employees for each part-time employee.

All three types of gloves are made out of the same genuine cowhide leather. Comfortable Hands has a long-term contract with a leather supplier and receives a 5000 ft2 shipment of the material each week. The material and labour requirements along with the gross profit per glove sold are given in the following table.

Glove Material (ft2) Labour required (minutes) Gross profit per pair (£)

Men’s 2 30 8

Women’s 1.5 45 10

Children’s 1 40 6

Each full-time employee earns £13 per hour, while each part-time employee earns £10 per hour.

The company would like to determine the mix of gloves to produce per week and how many full-time and part-time workers to employ in order to maximize their net profit.

Please Help!!

The answer i get via Excel Solver, is :

produce 2,480 Mens Gloves only. Employ, 25 Full Timers, and 12 Part Timers, Giving a Gross Profit of 19840 and a Net Profit of 4440. Is this correct? Because my teacher is a crafty buggar and i know he wouldnt distribute the whole manufacture to JUST Mens Gloves. Leading me to think i have gone wrong somewhere.

### 1 Answer

- TomLv 58 years agoFavorite Answer
Wow = where to start!

ok: think Input-> process -> output

like x -> function f -> f(x)

the input is the labour (hrs) and the leather material (sq. ft)

the function is the production process ( the system of equations we need to set up)

output is gloves by catagory, profits (no gloves => no profit remember)

let no. of full-time workers be F and part-time be P;

F ≥20, 0≤P≤F/2 (since two full-time workers at least for each part-time worker)

so now we have our first input variable - the number of hours => mins worked, say T mins

T = F*40*60 + P*20*60 mins

Labour £costs (full-time) = (13/60)*F*40*60

Labour £costs (part-time) = (10/60)*P*20*60

total labour £costs L = (13/60)*F*40*60 + (10/60)*P*20*60

Let no. of mens gloves (pairs) = M, Womens W and Children's C

2M + 1.5W + 1C≤ 5000 sq.ft (the available leather) where each M,W,C must be integer.

30M +45W + 40C ≤ T (the total production time in mins as above)

Only required now to work out gross profit!

net profit = gross profit -labour costs .. in this instance

gross profit £G = (8M + 10W + 6C)

net profit £P = G - L which must be maximised.

you have to put all of these formulae and use three+two+one columns to show {M,W,C},{F,P} T

as inputs, remembering to use integer increments of gloves starting at 0,1,2.. for each man,woman,child; subject to restraints given by inequalities derived. Output columns would be G,L,P.

Good luck

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