Question

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 5000 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	$33.00	$47.00
Support	$10.50	$14.00
Strap	$7.50	$8.50

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:

Department
Component	Cutting	Milling	Shaping
Frame	3	2	2.6
Support	1.3	1.6	2.1
Strap	0.9	—	1.6
Capacity (hours)	340	400	650

1. 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?
   

   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
   	TP = number of straps purchased

   Min

   FM

   +

   FP

   +

   SM

   +

   SP

   +

   TM

   +

   TP

   s.t.

   3FM

   +

   SM

   +

   TM

   ≤

   ______

   FM

   +

   1.6SM

   ≤

   _____

   FM

   +

   SM

   +

   1.6TM

   ≤

   _____

   FM

   +

   FP

   ≥

   _____

   SM

   +

   SP

   ≥

   ______

   TM

   +

   TP

   ≥

   ______

   FM, FP, SM, SP, TM, TP ≥ 0

   If required, round your answers to the nearest whole number.

   	Manufacture	Purchase
   Frames
   Supports
   Straps

2. What is the total cost of the manufacturing and purchasing plan? When required, round your answer to the nearest dollar.
   
   $ ________
   
3. How many hours of production time are used in each department? If required, round your answer to two decimal places.
   

   Department	Hours of production time used
   Cutting	____hours
   Milling	____hours
   Shaping	____ hours

4. How much should Frandec be willing to pay for an additional hour of time in the shaping department? If required, round your answer to two decimal places.
   
   $ ______ because there is slack  of ________ hours.
   
5. Another manufacturer has offered to sell frames to Frandec for $41 each. Could Frandec improve its position by pursuing this opportunity?
   
   Yes
   
   Why or why not? If required, round your answers to the nearest cent.
   
   Because the current purchase price is $_________ . The reduced cost is $___________ which means that the solution may be improved if the cost is $ _________ or below.

Answer 1

a),b) and c)

A F F Requirements Milling Cost to purchase Shaping Cost to manufacture Component Cutting 2 33 47 3 Frame 3 2 2.6 1.3 1.6 2.1 10.5 4 Support Strap Capacity(hours 14 8.5 5 0.9 1.6 7.5 650 400 6 340 Decision Variables 7 Numer of units of each part to be manufactures and to be purchased Manufacture Purchase Component 10 Frame 11 Support Strap 12 Objective function 13 Minimise the total cost 14 15 SUMPRODUCT (B10:C12,E3:F5) Constraints 16 17 SUMPRODUCT(B10:B12, B3: B5) 18 SUMPRODUCT(B10:B12,C3:C5) 19SUMPRODUCT(B10:B12, D3 : D5) 20 B10+C10 =B6 くニ C6 =D6 -5000 1 E5000 2 21 B11+C11 22 B12+C12 -5000*1 co co o

A B C D F G Requirements Cost to Shapingmanufacture purchase Cost to Milling Component Cutting 2 3 Frame 2 2.6 33 47 4 Support 1.3 1.6 2.1 10.5 14 5 Strap 0.9 1.6 7.5 8.5 6 Capacity(hours 340 400 650 Decision Variables 7 Numer of units of each part to be manufactures and to be purchased 8 Manufacture Purchase Component 10 Frame 11 Support 12 Strap Objective function 13 Minimise the total cost 14 15 0 Constraints 16 17 340 <= 400 18 0 19 0 650 20 0 5000 21 0 10000 22 0 5000 OOO O

Solver Parameters SAS15 Set Objective: To: 0 Min Value Of Max By Changing Variable Cells: SBS10:SC$12 Subject to the Constraints: SAS17:SAS19 = SCS17:SCS19 SAS20:SA$22= SC$20:SC$22 SBS10:SC$12>= 0 Add Change Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help Solve Close

Decision Variables 7 Numer of units of each part to be manufactures and to be purchased 8 Purchase Component Manufacture 10 Frame 113 4887 11 Support Strap Objective function 10000 12 5000 13 Minimise the total cost 14 15 415913 Constraints 16 17 340 340 18 227 400 19 295 650 20 5000 5000 10000 21 10000 22 5000 5000 23

a)

Component	Manufacture	Purchase
Frame	113	4887
Support	0	10000
Strap	0	5000

b)

Total cost

415913

c)\

Cutting	340
Milling	227
Shaping	295

d)

Shaping department has excess hours(slack)

Slack = 650-295 = 355 hours

Therefore , there is no need to by hours for shaping

$0  because there is slack  of 355 hours

e)

Variable Cells Objective Final Reduced Allowable Allowable Cell Value Coefficient Name Cost Increase Decrease $B$10 Frame Manufacture 113.3333333 0 33 5.923076923 1E+30 $C$10 Frame Purchase 4886.666667 C 47 1E+30 5.923076923 $B$11 Support Manufacture $C$11 Support Purchase $B$12 Strap Manufacture $C$12 Strap Purchase 0 2.566666667 10.5 1E+30 2.566666667 10000 14 2.566666667 14 3.2 7.5 1E+30 3.2 5000 0 8.5 3.2 8.5

Reduced cost is 0, if more frames are available it would not negatively affect the optimal solution.

Therefore, the frame price can be less than $47.

Yes

Because the current purchase price is $47. The reduced cost is $0 which means that the solution may be improved if the cost is $ 47 or below.