Moroney’s Cycles sells, repairs, and customizes motorcycles. They stock Harley D

Question

Moroney’s Cycles sells, repairs, and customizes motorcycles. They stock Harley Davidson, Yamaha Custom, and also have a limited number of KTM Motocross Bikes available for sale. The average Harley costs $8,000, a Yamaha Custom costs $6,000, and a KTM costs $3000. They sell Harleys on average for $10,000, Yamaha Customs for $7500, and KTMs for $3700. The fleet manager has $50,000 this month to bring in new bikes. Based on past sales and the shop’s primary clients, the fleet manager wants to stock at least twice as many Harleys as the other two combined. The main showroom has room for 20 new bikes this month. Each bike must have a service, a maintenance check, and a road test. This takes 8 hours for a Harley, 10 hours for a Yamaha Custom, and 7 hours for a KTM. The service manager estimates that his employees have 125 hours available this month.

Formulate this problem as a Linear Program in standard form.

Solve using Excel Solver and answer the following questions.

1. What is the breakdown of Harleys, Yamahas, and KTMs that should be bought and sold this month?

2. Should Moroney’s try to increase the budget for purchasing bikes, increase space to stock bikes, or increase labor hours for service and testing? Why?

3. If Moroney’s stocked a cheaper motocross bike for $1750 that sold for $3000, would this affect the original solution?

4. Is it worth hiring an additional mechanic for 20 hours at $7 per hour? Why or why not?

Explanation / Answer

Formulation of the problem

Decision variables

We are interested to know the number of three types of motorcycles, let X1 , X2 and X3 represents the number of Harley Davidson, Yamaha custom and KTM brands motorcycles to be purchased and sold.

Objective Function

AS cost of purchase and sates price for each type of motorcycle is given, therefore the objective function has to be maximization of total profit which is given as follows:

Maximize Z = 2000X1 + 1500X2 + 700X3

as per unit profit of Harleys is (10000-8000), Yamahas (7500-6000) and for KTM (3700-3000)

Constraints:

Availability of space for storing:   X1 + X2 + X3 <= 20

Availability of labor hours:   8X1 + 10X2 + 7X3 <= 125

Number of Harleys at least twice the sum of other two brands: X1 >= 2(X2+X3) or X1 - 2X2 - 2X3 >= 0

Availability of funds for purchase: 8000X1 + 6000X2 + 3000X3 <= 50,000

Non-negativity X1, X2 and X3 >= 0 as quantities rather needs to be positive numbers

Input data table for Excel Solver is as follows:

Now the solution table as follows:

50000

In case the variables are not restricted to integers, we get the optimal solution as

Variable   Status   Value
X1   Basic   4.5455
X2   Basic   2.2727
X3   NONBasic   0
slack 1   Basic   65.9091
slack 2   Basic   13.1818
surplus 3   NONBasic   0
slack 4   NONBasic   0
Optimal Value (Z)       12,500

The ranging table corresponding to above solution is as follows:

Variable   Value   Reduced Cost   Original Val   Lower Bound   Upper Bound
X1   4.5455   0   2,000   -750   2,000
X2   2.2727   0   1,500   1,500   Infinity
X3   0   50   700   -Infinity   750
Constraint   Dual Value   Slack/Surplus   Original Val   Lower Bound   Upper Bound
labor hours   0   65.9091   125   59.0909   Infinity
space   0   13.1818   20   6.8182   Infinity
No of Harley   0   0   0   -16.6667   6.25
Funds   .25   0   50,000   0   105,769.2

As the profit per unit is maximum (2000) for Harley, it is prudent to have maximum number of Harleys subject to the constraint'

2. Yes Moroney try to increase the budget for purchasing bikes because there is sufficient room to have more bikes and unutilized labor hours

3 After purchase of six Harleys (6*8000) the funds available are $2,000 and there is room and labor hours for this cheaper motorcycle so it would not affect the original solution

4. As there are unutilized labor hours, no need to hire an additional mechanic

X1 X2 X3 Decision Variables 0 0 0 RHS Objective function 2000 1500 700 0 Constraint space 1 1 1 0 20 Constraint hours 8 10 7 0 125 No. Of Harley 1 -2 -2 0 0 funds 8000 6000 3000 0 50000

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