Chapter 9 Question 2 [Anisha Patel]
The management of Hartman Company is trying to determine the amount of each of two products to
produce over the coming planning period. The following information concerns labor availability,
labor utility, and product profitability.
Product (hours/unit)
Department
1
2
Labour-Hours Available
A
1.00
0.35
100
B
0.30
0.20
36
C
0.20
0.50
50
Profit contribution/unit
$30.00
$15.00
a.
b.
c.
Develop a linear programming model of the Hartman Company problem. Solve the model to
determine the optimal production quantities of products 1 and 2.
In computing the profit contribution per unit, management doesn’t deduct labor costs because
they are considered fixed for the upcoming planning period. However, suppose that overtime can
be scheduled in some of the departments. Which departments would you recommend scheduling
for overtime? How much would you be willing to pay per hour of overtime in ach department?
Suppose that 10, 6, and 8 hours if overtime may be scheduled in departments A, B, and C,
respectively. The cost per hour of overtime is $18 in department A, $22.50 in department B, and
$12 in department C. Formulate a linear programming model that can be used to determine the
optimal production quantities if overtime is made available. What are the optimal production
quantities, and what is the revised total contribution to profit? How much overtime do you
recommend using in each department? What is the increase in the total contribution to profit if
overtime is used?
Chapter 9 Question 2 Solution
Solution:
a.
X = 77.89, Y = 63.16; $3,284.21
b.
Department A $15.78947; Department B $47.36842
c.
X = 87.21, Y = 65.12, $3341.34
Department A 10 hours; Department B 3.2 hours
Chapter 9 Question 17 [Anisha Patel]
2. Frandec Corporation manufactures, assembles, and rebuilds material handling equipment used in warehouses and distribution centers. One
product, called a Liftmaster, is assembled from four components: a frame, a motor, two supports, and a metal strap. Frandec’s production schedule
calls for 5,000 Liftmasters to be made next month. Frandec purchases the motors from an outside supplier, but the frames, supports, and straps may
either be manufactured by the company or purchased from an outside supplier. Manufacturing and purchase costs per unit are shown.
Component
Frame
Support
Strap
Manufacturing Cost
$38.00
$11.50
$6.50
Purchase Cost
$51.00
$15.00
$7.50
Three departments are involved in the production of these components. The time (in minutes per unit) required to process each component in each
department and the available capacity (in hours) for the three departments are as follows:
Component
Frame
Support
Strap
Capacity (hours)
Department
Cutting
3.5
1.3
0.8
350
Milling
2.2
1.7
0.0
420
Shaping
3.1
2.6
1.7
680
a. Formulate and solve a linear programming model for this make-or-buy application. How many of each component should be manufactured and
how many should be purchased?
b. What is the total cost of the manufacturing and purchasing plan?
c. How many hours of production time are used in each department?
d. How much should Frandec be willing to pay for an additional hour of time in the shaping department?
e. Another manufacturer has offered to sell frames to Frandec for $45.00 per unit. Could Frandec improve its position by pursuing this opportunity?
Why or why not?
Chapter 9 Question 17 Solution
Minimize 38.00xf + 11.50xu + 6.50xt + 51.00yf + 15.00yu + 7.50yt
Demand
Capacity
xf + yf = 5000
3.5xf + 1.3xu + 0.8xt ≤ 350 ｷ 60
x+ y= 10000
2.2x+ 1.7x+ 0.0x≤ 420 ｷ 60
x+ y= 5000
3.1x+ 2.6x+ 1.7x≤ 680 ｷ 60
Chapter 9 Question 17 Solution Cont.
Chapter 9 Question 17 Solution Cont.
a.
The optimal solution is to manufacture 5,000 frames, 2,692.31
supports, and 0 straps; and to purchase 0 frames, 7307.69
supports, and 5,000 straps.
b.
The total cost is $368076.9
c.
The solution calls for 350 hours of cutting time, 259.62 hours of
milling time, and 375 hours of shaping time
d.
Because this last requirement is not binding, Frandec should not
pay anything for an additional hour of time in the shaping
department.
e.
The shadow price of the frame constraint is $47. Therefore,
Frandec would benefit from purchasing frames if they are
available for $45. That would be true up to a maximum
purchase of 2714:3 units, which is the allowable decrease
on that shadow price.