Question

Arlo Industries manufactures three types of dog collars: a deluxe model with solid leather and upgraded leash rings, a standard model using a "leatherette" compound, and a bargain model that they sell to discount outlets. The profit contributions of these collars are $12 Deluxe, $10 Standard, and $8 Bargain. All collars must be cut, assembled and shipped using three different production lines. The following table shows the time (in minutes) for each operation:

Production Line		Deluxe	Standard	Bargain
Cutting		10	7	6
Assembly		8	7	6
Shipping		5	5	5

Next month, the company estimates there will be 380 hours available for cutting, 370 hours for assembly and 400 hours available for shipping. In addition, up to 80 hours of overtime is available that can be used on either the cutting and/or assembly production lines at a cost of $20 per hour. The company has already received orders for 1200 deluxe collars, 1000 standard collars, and 600 bargain collars that must be filled, but believes they can sell as many collars as they can make. The company is interested in maximizing profit, subject to the constraints listed above.

**Please answer me**

Answer 1

Let no. of deluxe models be D, Std. Model be S and Bargain model be B.

Total Profit = 12D + 10S + 8B

We have to maximize this total profit

Subject to constraints:

10D + 7S + 6B <= 380*60........Constraint for No. of minutes available for cutting

8D + 7S + 6B <= 370*60...........Constraint for No. of minutes available for Assembly

5D + 5S + 5B <= 400*60............Constraint for No. of minutes available for Shipping

Let x, y, z be overtime hours used for Cutting, Assembly, and Shipping respectively. hence, we get

10D + 7S + 6B <= (380 + x)*60

8D + 7S + 6B <= (370 + y)*60

5D + 5S + 5B <= (400 + z)*60

x + y + z <= 80*60......................Constraint for no. of Overtime hours available

Total profit will become:

P = 12D + 10S + 8B - 20(x + y + z)

Also,

D >= 1200..............................Constraint for minimum production based on orders received for Deluxe collars

S >= 1000..............................Constraint for minimum production based on orders received for Standard collars

B >= 600..............................Constraint for minimum production based on orders received for Bargain collars

D, S, B, x, y, z >= 0...............Non-negativity constraint as these cannot be negative

We solve the above LPP in Excel Solver as shown below:

The above solution in the form of formulas along with Excel Solver extract is shown below for better understanding and reference:

As seen from above,

No. of Deluxe Collars = 1200

No. of Std. Collars = 1500

No. of Bargain Collars = 600

Total Profit = $32,600

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