Question

Case Problem 2 Chapter 7 Introduction to Linear Programming Production Strategy

Production Strategy Better Fitness, Inc. (BFI) manufactures exercise equipment at its plant in Freeport, Long Island. It recently designed two universal weight machines for the home exercise market. Both machines use BFI-patented technology that provides the user with an extremely wide range of motion capability for each type of exercise performed. Until now, such capabilities have been available only on expensive weight machines used primarily by physical therapists. At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabilities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the Body- Plus 100 and the BodyPlus 200, require different amounts of resources to produce. The BodyPlus 100 consists of a frame unit, a press station, and a pec-dec station. Each frame produced uses 4 hours of machining and welding time and 2 hours of painting and finishing time. Each press station requires 2 hours of machining and welding time and 1 hour of painting and finishing time, and each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing, and packaging each BodyPlus 100. The raw material costs are $450 for each frame, $300 for each press station, and $250 for each pec-dec station; packaging costs are estimated to be $50 per unit. The BodyPlus 200 consists of a frame unit, a press station, a pec-dec station, and a leg- press station. Each frame produced uses 5 hours of machining and welding time and 4 hours of painting and finishing time. Each press station requires 3 hours machining and welding time and 2 hours of painting and finishing time, each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time, and each leg-press station requires 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing, and packaging each BodyPlus 200. The raw material costs are $650 for each frame, $400 for each press station, $250 for each pec-dec station, and $200 for each leg-press station; packaging costs are estimated to be $75 per unit. For the next production period, management estimates that 600 hours of machining and welding time; 450 hours of painting and finishing time; and 140 hours of assembly, testing, and packaging time will be available. Current labor costs are $20 per hour for machining and welding time; $15 per hour for painting and finishing time; and $12 per hour for assembly, testing, and packaging time. The market in which the two machines must compete suggests a retail price of $2400 for the BodyPlus 100 and $3500 for the BodyPlus 200, although some flexibility may be available to BFI because of the unique capabilities of the new machines. Authorized BFI dealers can purchase machines for 70% of the suggested retail price. BFI’s president believes that the unique capabilities of the BodyPlus 200 can help position BFI as one of the leaders in high-end exercise equipment. Consequently, she states that the number of units of the BodyPlus 200 produced must be at least 25% of the total production. Managerial Report Analyze the production problem at Better Fitness, Inc., and prepare a report for BFI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following items: 1. The recommended number of BodyPlus 100 and BodyPlus 200 machines to produce 2. The effect on profits of the requirement that the number of units of the BodyPlus 200 produced must be at least 25% of the total production 3. Where efforts should be expended in order to increase contribution to profits Include a copy of your linear programming model and graphical solution in an appendix to your report.

Answer 1

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.