Question

A linear programming computer package is needed.

Frandec Company 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 5,500 Liftmasters to be made next month. Frandec purchases the motors from an outside supplier, but the frames, supports, and straps may be either manufactured by the company or purchased from an outside supplier. Manufacturing and purchase costs per unit are shown.

Component Manufacturing Cost Purchase Cost

Frame $37.00 $50.00

Support $10.50 $14.00

Strap $5.50 $6.50

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 Department

Cutting Milling Shaping

Frame 3.5 2.2 3.1

Support 1.3 1.7 2.6

Strap 0.8 — 1.7

Capacity (hours) 350 420 680

(a)

Formulate and solve a linear programming model for this make-or-buy application. (Let FM = number of frames manufactured, FP = number of frames purchased, SM = number of supports manufactured, SP = number of supports purchased, TM = number of straps manufactured, and TP = number of straps purchased. Express time in minutes per unit.)

Min

Cutting constraint

Milling constraint

Shaping constraint

Frame constraint

Support constraint

Strap constraint

FM, FP, SM, SP, TM, TP ≥ 0

How many of each component should be manufactured and how many should be purchased? (Round your answers to the nearest whole number.)

(FM, FP, SM, SP, TM, TP) =

(b)

What is the total cost (in $) of the manufacturing and purchasing plan?

$

(c)

How many hours of production time are used in each department? (Round your answers to two decimal places.)

Cutting hrs

Milling hrs

Shaping hrs

(d)

How much (in $) 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 each. Could Frandec improve its position by pursuing this opportunity? Why or why not? (Round your answer to three decimal places.)

. The reduced cost of ______ indicates that the solution be improved.

Answer 1

solution:

Based on the decision variables defined in the question,

Total Cost = 37*FM + 10.5*SM + 5.5*TM + 50*FP + 14*SP + 6.5*TP

We have to minimize this cost

Subject to constraints

3.5*FM + 1.3*SM + 0.8*TM <= 350 hours * 60 minutes i.e. 21000..........Constraint for availability of minutes of Cutting time

2.2*FM + 1.7*SM + 0*TM <= 420 * 60 = 25,200 .....................................Constraint for availability of minutes of Milling time

3.1*FM + 2.6*SM + 1.7*TM <= 680*60 = 40,800.....................................Constraint for availability of minutes of Shaping time

Total Demand of final product = 4500 units

FM + FP >= 4500..........Constraint for demand of Frames

SM + SP >= 9000........Constraint for demand of Support (2 supports per 1 Liftmaster)

TM + TP >= 4500..........Constraint for demand of Strap

Hence, we get formulation as

Minimize Total Cost C = 37*FM + 10.5*SM + 5.5*TM + 50*FP + 14*SP + 6.5*TP

Subject to Constraints:

3.5*FM + 1.3*SM + 0.8*TM <= 21000

2.2*FM + 1.7*SM + 0*TM <= 25,200

3.1*FM + 2.6*SM + 1.7*TM <= 40,800

FM + FP >= 4500

SM + SP >= 9000

TM + TP >= 4500

FM, SM, TM, FP, SP, TP >=0

We solve in Excel using Excel solver as shown below:

The above solution in the form of formulas along with Excel Solver extract is shown below for better understanding and reference:

As seen from the above solution, we get

b) Total cost = $307615.38

c) Cutting = 21000

Milling = 16765.38

Shaping = 24450

d) For part d and e, we generate a sensitivity report as shown below:

As seen from the above sensitivity report,

The shadow price for Shaping Department (Cell B15 in Constraints table) is 0. Hence, any increase in the number of hours in the shaping department will lead to a 0 decrease in total cost. Further, there is a slack of 40800 - 24450 = 16,350. Hence, Frandec should be willing to pay $0 because there is slack of 16,350 hours.

e) As seen from the above sensitivity report,

The Reduced cost for Frames to be purchased (Cell B5 in Variables table) is 3.58. In order to make the optimal solution such that Frames are purchased, the price of frames should be reduced by at least $3.58 i.e. 50 - 3.58 = $46.42. Since the supplier is supplier at $45 which is less than minimum reduced price, Frandec can improve by pursuing this opportunity. Hence, the reduced cost of 3.577 indicates the solution can be improved.

******
Please up vote....Its very usefull for us .....Thank you .....So Much......If any qurries comment below....I will resolve ASAP....Until u got.....