BP100
It consist of frame unit, press station and pec-dec station. The labor hour requirements are
Frame = 4 hours of machining and welding + 2 hours of painting and finishing
Press station = 2 hours of machining and welding + 1 hour of painting and finishing
Pec-dec station = 2 hours of machining and welding + 2 hours of painting and finishing
Assembly, testing and packaging = 2 hours
In total,
Machining and welding time = 4+2+2= 8 hours at $20/hour
Painting and finishing time = 2+1+2 = 5 hours at $15/hour
Assembly, testing and packaging = 2 hours at $12/hour
The cost/revenue details are
Cost = 450 + 300 + 250 + 50 = $1050
Selling price (to retailers) = 2400*0.7 = $1680
BP200
It consist of frame unit, press station, pec-dec station and a leg press station. The labor hour requirements are
Frame = 5 hours of machining and welding + 4 hours of painting and finishing
Press station = 3 hours of machining and welding + 2 hour of painting and finishing
Pec-dec station = 2 hours of machining and welding + 2 hours of painting and finishing
Leg press station = 2 hours of machining and welding + 2 hours of painting and finishing
Assembly, testing and packaging = 2 hours
In total,
Machining and welding time = 5+3+2+2 = 12 hours at $20/hour
Painting and finishing time = 4+2+2+2 = 10 hours at $15/hour
Assembly, testing and packaging = 2 hours at $12/hour
The cost/revenue details are
Cost = 650+400+250+200 = $1500
Selling price (to retailers) = 3500*0.7 = 2450
Available time
600 hours of machining and welding
450 hours of painting and finishing
140 hours of assembly, testing and packaging.
First of all, we need to optimize our production so that we can maximize the profit at BFI. Let us assume that our decision (decision variables) is to produce X units of BP100 and Y units of BP200. Then the profit contribution of BP100 model is
1680X – (1050X + 8*20X + 5*15X + 2*12X)
Here 1680X is the revenue generated by selling X units of BP100 to retailers.
The cost of producing X units of BP100 includes material cost (1050X), machining and welding hourly cost (8*20X), painting and finishing hourly cost (5*15X) and assembly cost (2*12X). The above equation gives us the profit. If we simplify this we get
Contribution of BP100 = 371X for X units produced
Similarly the profit for BP200 can be denoted as
2450Y – (1500Y + 12*20Y + 10*15Y + 2*12Y)
Or, 536Y for Y units of BP200 sold to the retailers.
Thus if we want to maximize our profit, the equation will be
Maximize Z, where Z = 371X + 536Y
However, we have a constraint for this problem. That is that our time is limited. Thus our constraints can be written as
8X + 12Y <= 600 (our machining and welding time for the models cannot exceed 600 hours)
5X + 10Y <= 450 (our painting and finishing time cannot exceed 450 hours)
2X + 2Y <= 140 (our assembly, testing and packaging time cannot exceed 140 hours)
Also another specific constraint is that at least 25% of the total production should BP200. This means
Y >= 0.25*(X+Y)
In addition, the number of units we produce cannot obviously be negative and must be an integer whole number. Thus,
X, Y = +Z (positive integers)
Thus our final model formulation should be denoted as
Maximize Z where
Z = 371X + 536Y
Subject to constraints
8X + 12Y <= 600
5X + 10Y <= 450
2X + 2Y <= 140
Y >= 0.25*(X + Y)
X, Y = Z+
We can solve it in various methods. For better flexibility, let’s use Excel to solve this model.
Set up the model as shown below
The formulas used are shown below
The solver parameters are shown below
The result is shown below
The sensitivity analysis (after removing the integer constraints) is shown below.
1.
From a mathematical point of view, our ideal numbers of production will be
BodyPlus 100 = 50
BodyPlus 200 = 16.67
And this will generate a profit of $27483.33
However, we cannot sell fractional value of a unit to the retailers. Due to this, the optimal production should be
BodyPlus 100 = 48
BodyPlus 200 = 18
This will generate a profit of $27456
2.
If we remove the specific constraint that was mentioned (25% must be BP200), then our optimal solution is shown below. We can see that the profit is not 27620. This means the impact of that constraint is 27620 – 27456 = 164.
3.
Among all our constraints, we can clearly see that the constraint that is binding is the first constraint of machining and welding time. It is completely exhausted. This means we have not completely utilized other constraints or limits. Due to this, our effort should be in increasing the machining and welding time at the current point.
Ruby123 Sun, Apr 25, 2021 5:33 PM
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Ruby123 Sun, Apr 25, 2021 5:32 PM