Question

Problem 4-17 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 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) Cutting 3.5 1.3 0.8 Department Milling 2.2 1.7 Shaping 3.1 2.6 1.7 680 350 420 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? 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 + ТР s.t. FM + SM TM s FM 1 + + SM FM + SM + TM S FM + FP SM + + SP 2 TM + TP 2 FM, FP, SM, SP, TM, TP > 0

Answer 1

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a) 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 Objective is to minimize cost = Min 38FM + 51FP + 11.5SM + 15SP + 6.5TM + 7.5TP Subject to, 3.5FM+1.35M+0.8TM <= 21000 (capacity of cutting department in minutes) 2.2FM+1.7SM <= 25200(capacity of Milling department in minutes) 3.1FM+2.6SM+1.7TM <= 40800 (capacity of Shaping department in minutes) FM+FP >= 5000 (Number of frames manufactured and purchased) SM+SP >= 10000(Number of Supports manufactured and purchased) TM+TP >= 5000(Number of Straps manufactured and purchased) FM, FP, SM, SP, TM, TP >= 0 (Non - negativity constraint) Solving in excel solver we get, Manufacture Purchase Frames 5000 0 Supports 2692 7308 Straps 0 5000

Solver screenshot A с F G H J 1 B FM FP 5000 D E SM SP TM 0 2692.308 7307.692 TP 0 2 5000 Objective function 368076.9231 0.8 3.5 2.2 3.1 1 1.3 1.7 2.6 3 Constraints 4 capacity of cutting department 5 capacity of Milling department 6 capacity of Shaping department 7 Number of frames 8 Number of Supports 9 Number of Straps 10 1.7 21000 15576.92 22500 5000 >= 10000 >= 5000 > 21000 25200 40800 5000 10000 5000 1 1 1 1 1 Sensitivity report F G H A B C D E 1 Mlcrosoft Excel 16.0 Sensitivity Report 2 Worksheet: [Refl.xlsx]Sheet35 3 Report Created: 4/26/2019 10:49:00 PM 4 5 6 Variable Cells 7 Final Reduced 8 Cell Name Value Cost 9 $B$2 FM 5000 0 10 $C$2 FP 0 3.576923077 11 $D$2 SM 2692.30769 0 12 $E$2 SP 7307.69231 0 13 $F$2 TM 0 1.153846154 14 $G$2 TP 5000 0 15 16 Constraints 17 Final Shadow 18 Cell Name Value Price 19 SH$4 capacity of cutting department 21000 -2.69230769 20 SH$5 capacity of Milling department 15576.9231 0 21 SH$6 capacity of Shaping departmer 22500 0 22 SH$7 Number of frames 5000 47.42307692 23 SH$8 Number of Supports 10000 15 24 SH$9 Number of Straps 5000 7.5 25 Objective Allowable Allowable Coefficient Increase Decrease 38 3.57692308 47.4230769 51 1E+30 3.57692308 11.5 1.875 1.32857143 15 1.32857143 1.875 6.5 1E+30 1.15384615 7.5 1.15384615 7.5 Constraint Allowable Allowable R.H. Side Increase Decrease 21000 7358.82353 3500 25200 1E+30 9623.07692 40800 1E+30 18300 5000 1000 2714.28571 10000 1E+30 7307.69231 5000 1E+30 5000

b) total cost of the manufacturing and purchasing plan 368076.9231 = 368077 (Rounded to nearest dollar)

c) Department Hours of production time used Cutting 350 hours Milling 259.62 hours Shaping 375 hours Hours of production time used for Cutting = 21000/60 = 350 hours Hours of production time used for Milling = 15576.92/60 = 259.615 = 259.62 hours (Rounded to nearest whole number) d) Frandec Should not be willing to pay for an additional hour of time in the shaping department as shadow price is 0 for the constraint because of not using even available existing hours Answer is 0