Question 2

A machine parts manufacturing company produces five different products. The next month forecast for the products is provided by the marketing manager. The parts are manufactured in two shifts, each shift is eight hour long. The whole 30 days of the next month are available for production.

Two different machine types are available for production; a universal machine for manufacturing all products ( M1 ) and a special machine which can't produce product 1 and 2. The unit manufacturing times are different on the two machine types. There are 12 machines of type 1 ( M1 ) and 45 machines of type 2 ( M2 ).

The plan's objective is to satisfy all demand of the market either by producing the parts or buying them from a subcontractor. If the manufacturing capacity is not sufficient, or it is more economic to buy the parts from the subcontractor and selling them to the customers, then the subcontractor is flexibly available to produce the required quantity.

Demand for the products, unit selling prices, unit manufacturing costs, unit manufacturing time on the different machines, and the unit purchase cost from the subcontractor is given in the table below.

Answer the following question using the information provided without the optimal solution and the sensitivity analysis results:

a) How much must be paid to the subcontractor according to the optimal solution?

b) Currently it is not recommended to buy P2 from the subcontractor. The subcontractor, however, is planning to offer a purchase price discount to motivate the manufacturer for ordering P2 as well. Which is the highest purchase price at which it is worth buying P2 from the subcontractor.

c) The increase in manufacturing cost of P1 is expected. How much is the smallest increase in the unit manufacturing cost of P1 at which the manufacturing quantity of this product should be reduced and the increase of the purchase quantity from the subcontractor is recommended?

d) The customer would like to order an additional 13000 units of P5 over the originally ordered quantity. To motivate the customer for this additional order, the manufacturer offers a 30000 Euro discount. What is the financial consequence of this discount for the manufacturer?

e) To increase production, the management of the plant is planning to increase production capacity with overtime on the universal machines (M1). A two - hour long overtime at the end of the second shift in each day of the month has a total of 100000 Euro costs. What is the financial result of this overtime?

f) What is the financial result of the overtime if it is organized for the special machines (M2) with the same conditions described in point e)?

Product Selling p. Variable c. [Euro/unit] [Euro/unit] [unit] 16500 P1 99 66 4,63 P2 22000 55 4,63 86 110 P3 62000 49 5,23 5,23 P4 7500 124 51 5,23 5,23 P5 62000 70 50 4,17 4,17 Linear programming model of the problem: Max: (19B1+16B2+50B3+54B4+0B5+33U+31U2+61U3+73U4+20U5+61S3+73S4+20S5) U4 U5 5760 5,23 4,17 U U U3 Universal Machines: + + + 4,63 4,63 5,23 S3 S4 S5 + + 21600 5,23 5,23 4,17 Special Machines: Demand of P1: B+U=16500 Demand of P2: B+U=22000 Demand of P3: B3+U3+S3=62000 Demand of P4: B4+U4+S4=7500 Demand of P5: B5+U5+S5=62500 B- quantity of product /bought from the subcontractor (i=1, ...., 5). U- quantity of product/produced on M1 (i=3, ...., 5). S- quantity of product/produced on M2 (i=1, ...., 5). Demand S Manuf. time Universal Special Purchase p. [Euro/unit] 80 70 60 70 70 + Optimal solution provided by Excel: B4 B5 0.0 0.0 50 54 0 1 B1 B2 B3 Variables 11831.2 0.0 34292.2 OFC 19 16 Universal Special Demand 1 P1 Demand P2 Demand P3 Demand P4 Demand P5 1 O 1 1 U1 4668.8 33 0.216 1 U3 U4 U5 0.0 U2 22000.0 0.0 31 61 73 0.216 0.191 0.91 0.240 1 1 S3 S4 $5 7500.0 62000.0 0.0 27707.8 61 73 20 0.191 0.191 0.240 1 1 240 1 1 1 6253149 5760 21600 16500 22000 62000 7500 62000 Cell $B$18 $C$18 $D$18 $E$18 $F$18 $G$18 $H$18 $I$18 $J$18 $K$18 $L$18 $M$18 $N$18 Cell $0$21 $0$20 $0$22 $0$23 $0$24 $0$25 $0$26 Name Variables B1 Variables B2 Variables B3 Variables B4 Variables B5 Variables U1 Variables U2 Variables U3 Variables U4 Variables U5 Variables S3 Variables S4 Variables S5 Name Special Universal Demand P1 Demand P2 Demand P3 Demand P4 Demand P5 Final Value 11831,20 0,00 34292,19 0,00 0,00 4668,80 22000,00 0,00 0,00 0,00 27707,81 7500,00 62000,00 Final Value 21600 5760 16500 22000 62000 7500 62000 OFC Sensitivity Reduced Cost 0,00 -1,00 0,00 -8,00 -6,20 0,00 0,00 -1,39 -1,39 -1,75 0,00 0,00 0,00 RHS Sensitivity Shadow Price 57,53 64,82 19,00 17,00 50,00 62,00 6,20 OFC 19 16 50 54 0 33 31 61 73 20 61 73 20 RHS Value 21600 5760 16500 22000 62000 7500 62000 Feasible Increase 1,575 1,000 11,000 8,000 6,204 1,000 Infinite 1,394 1,394 1,748 1,394 Infinite Infinite Feasible Feasible Decrease 1,000 Infinite 1,394 Infinite Infinite 1,575 1,000 Infinite Infinite Infinite 11,000 1,394 1,748 Feasible Decrease rease 6556,82 2555,33 5297,86 1008,38 11831,20 Infinite 4668,80 11831,20 Infinite 34292,19 7500,00 27707,81 22092,08 27341,96 Product Selling p. Variable c. [Euro/unit] [Euro/unit] [unit] 16500 P1 99 66 4,63 P2 22000 55 4,63 86 110 P3 62000 49 5,23 5,23 P4 7500 124 51 5,23 5,23 P5 62000 70 50 4,17 4,17 Linear programming model of the problem: Max: (19B1+16B2+50B3+54B4+0B5+33U+31U2+61U3+73U4+20U5+61S3+73S4+20S5) U4 U5 5760 5,23 4,17 U U U3 Universal Machines: + + + 4,63 4,63 5,23 S3 S4 S5 + + 21600 5,23 5,23 4,17 Special Machines: Demand of P1: B+U=16500 Demand of P2: B+U=22000 Demand of P3: B3+U3+S3=62000 Demand of P4: B4+U4+S4=7500 Demand of P5: B5+U5+S5=62500 B- quantity of product /bought from the subcontractor (i=1, ...., 5). U- quantity of product/produced on M1 (i=3, ...., 5). S- quantity of product/produced on M2 (i=1, ...., 5). Demand S Manuf. time Universal Special Purchase p. [Euro/unit] 80 70 60 70 70 + Optimal solution provided by Excel: B4 B5 0.0 0.0 50 54 0 1 B1 B2 B3 Variables 11831.2 0.0 34292.2 OFC 19 16 Universal Special Demand 1 P1 Demand P2 Demand P3 Demand P4 Demand P5 1 O 1 1 U1 4668.8 33 0.216 1 U3 U4 U5 0.0 U2 22000.0 0.0 31 61 73 0.216 0.191 0.91 0.240 1 1 S3 S4 $5 7500.0 62000.0 0.0 27707.8 61 73 20 0.191 0.191 0.240 1 1 240 1 1 1 6253149 5760 21600 16500 22000 62000 7500 62000 Cell $B$18 $C$18 $D$18 $E$18 $F$18 $G$18 $H$18 $I$18 $J$18 $K$18 $L$18 $M$18 $N$18 Cell $0$21 $0$20 $0$22 $0$23 $0$24 $0$25 $0$26 Name Variables B1 Variables B2 Variables B3 Variables B4 Variables B5 Variables U1 Variables U2 Variables U3 Variables U4 Variables U5 Variables S3 Variables S4 Variables S5 Name Special Universal Demand P1 Demand P2 Demand P3 Demand P4 Demand P5 Final Value 11831,20 0,00 34292,19 0,00 0,00 4668,80 22000,00 0,00 0,00 0,00 27707,81 7500,00 62000,00 Final Value 21600 5760 16500 22000 62000 7500 62000 OFC Sensitivity Reduced Cost 0,00 -1,00 0,00 -8,00 -6,20 0,00 0,00 -1,39 -1,39 -1,75 0,00 0,00 0,00 RHS Sensitivity Shadow Price 57,53 64,82 19,00 17,00 50,00 62,00 6,20 OFC 19 16 50 54 0 33 31 61 73 20 61 73 20 RHS Value 21600 5760 16500 22000 62000 7500 62000 Feasible Increase 1,575 1,000 11,000 8,000 6,204 1,000 Infinite 1,394 1,394 1,748 1,394 Infinite Infinite Feasible Feasible Decrease 1,000 Infinite 1,394 Infinite Infinite 1,575 1,000 Infinite Infinite Infinite 11,000 1,394 1,748 Feasible Decrease rease 6556,82 2555,33 5297,86 1008,38 11831,20 Infinite 4668,80 11831,20 Infinite 34292,19 7500,00 27707,81 22092,08 27341,96