Question

[P3](15pts) A manufacturing plant determines how many units of standard toys and luxury toys to produce using the available machines. The first machine has the capacity to produce 300 standard toys, or 250 luxury toys per hour. The second machine can produce 400 standard toys, or 100 luxury toys. (i.e. it takes 1/400 of an hour to produce a single standard toy, and 1/100 of an hour to produce a single luxury toy from machine 2). Regular toys sell for $50 and costs $20 per item, and luxury toys sell for $120 and costs $45 per item to produce. How would you determine the production schedule to maximize profit? Formulate the problem, do not solve. (for this problem only, assume you can sell all you produce)

Answer 1

Formulating the linear programming problem given.

Decision variables

X_ij is number of units of toys to be produced per hour

where i =1 or 2 depending upon whether toy is produced from first machine or second machine

j = 1 or 2 depending upon whether standard toys or luxury toys are produced

X₁₁ = Number of standard toys produced in first machine

X₁₂= Number of luxury toys produced in first machine

X₂₁= Number of standard toys produced in second machine

X₂₂= Number of luxury toys produced in second machine

Objective function

Objective is to maximize profit. Profit = Total Sales - Total cost

Profit for Standard toys = ($50-$20)*Total number of standard toys = $30 * (X₁₁ + X₂₁)

Profit for Luxury toys = ($120-$45)*Total number of luxury toys = $75 * (X₁₂ + X₂₂)

Total profit = $30 * (X₁₁ + X₂₁) + $75 * (X₁₂ + X₂₂)

MAX $30 * (X₁₁ + X₂₁) + $75 * (X₁₂ + X₂₂)

Constraints

Non-negative constraints

X_ij> = 0 ;

X₁₁ >= 0 ;

X₁₂ >= 0;

X₂₁>= 0 ;

X₂₂>= 0

Capacity constraints

First machine: (1/300)*X₁₁ + (1/250)*X₁₂ <= 1

Second machine: (1/400)*X₂₁+ (1/100)*X₂₂ <= 1

(Note that LP formulation is based on an hour capacity and toys to be produced, this can be changed based on the actual available capacity)

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