Answered Essay: The optimal solution is to produce of the men's model in month 1, of the men's model in month 2, units of the women's model in month 1, and of the women's model in month 2.

The Silver Star Bicycle Company will be manufacturing both men’s and women’s models of its Easy-Pedal 10-speed bicycles during the next two months. Management wants to develop a production schedule indicating how many bicycles of each model should be produced in each month. Current demand forecasts call for 160 men’s and 135 women’s models to be shipped during the first month and 220 men’s and 170 women’s models to be shipped during the second month. Additional data are shown:

Production Labor Requirements (hours) Current
Model Costs Manufacturing Assembly Inventory
Men’s $140 1.8 1.7 20
Women’s $110 1.6 1 30

Last month the company used a total of 1000 hours of labor. The company’s labor relations policy will not allow the combined total hours of labor (manufacturing plus assembly) to increase or decrease by more than 100 hours from month to month. In addition, the company charges monthly inventory at the rate of 2% of the production cost based on the inventory levels at the end of the month. The company would like to have at least 25 units of each model in inventory at the end of the two months.

Establish a production schedule that minimizes production and inventory costs and satisfies the labor-smoothing, demand, and inventory requirements. What inventories will be maintained, and what are the monthly labor requirements? If required, round your answers to the nearest whole number.

Production schedule:

The optimal solution is to produce  of the men’s model in month 1,  of the men’s model in month 2,  units of the women’s model in month 1, and  of the women’s model in month 2.

Total Cost = $

Inventory Schedule
Month 1 Men’s Women’s
Month 2 Men’s Women’s
Labor Levels
Previous month hours
Month 1 hours
Month 2 hours

If the company changed the constraints so that monthly labor increases and decreases could not exceed 50 hours, what would happen to the production schedule? If required, round your answers to the nearest whole number.

Production schedule:

The optimal solution is to produce  of the men’s model in month 1,  of the men’s model in month 2,  units of the women’s model in month 1, and  of the women’s model in month 2.

Total Cost = $

How much will the cost increase?

$

Expert Answer

 

Let the following variables be defined:

m1 = amount of men’s model in month 1

w1 = amount of women’s model in month 1

m2 = amount of men’s model in month 2

w2 = amount of women’s model in month 2

im1 = inventory of men’s model at end of month 1

iw1 = inventory of women’s model at end of month 1

im2 = inventory of men’s model at end of month 2

iw2 = inventory of women’s model at end of month

The model formulation for part (a) is given.

Min 140m1 + 110w1 + 140m2 + 110w2 + 2.8im1 + 2.2iw1 + 2.8im1 + 2.2iw2

s.t.

20 + m1 – im1 = 160

or

(1) m1 – im1 = 140 Meet Demand

20 + w1 – iw1 = 135

or

(2) w1 –iw1 = 115 Meet Demand

(3) im1 + m2 – im2 = 220 Meet Demand

(4) iw1 + w2 – iw2 = 170 Meet Demand

(5) Im2 >= 25 Ending Inventory

(6) iw2 >= 25 Ending Inventory

Labor Hours: Men’s = 1.8 + 1.7 = 3.5

Women’s = 1.6 + 1.0 = 2.6

(7)

3.5 m1 + 2.6 w1 >= 900

(8)3.5 m1 + 2.6 w1 <= 1100 Month 1 labor constraints

[9]

3.5 m1 + 2.6 w1 – 3.5 m2 – 2.6 w2 <= 100

(10)-3.5 m1 – 2.6 w1 + 3.5 m2 + 2.6 w2 <= 100 Month 2 labor constraints

m1, m2, w1, w2, im1, im2, iw1, iw2 > 0

The optimal schedule and cost is shown below

If the company changed the constraints so that monthly labor increases and decreases could not exceed 50 hours, then the right hand side of contraints 7 to 10 would be changed and results are as below:

With the revised scenario, the LP problem is not solvable. So with change in constraint it is not possible to achieve an ending inventory of 25.

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