LP1
Pat’s Problem
A manufacturing company makes two products. The profit estimates are $1000 for each unit of product 1 sold and $1200 for each unit of product 2 sold. The labor-hour requirements for the products in each of the three production departments are summarized below:
(hrs.)
Department A Department B Department C
Product 1
8
3
10
Product 2
12
3
5
The production supervisors in the departments have estimated that the following number of labor-hours will be available during the next month: 48 hours in department A, 18 hours in department B, and 40 hours in department C. Assuming that the company is interested in maximizing profits, formulate a linear program for this problem.
PAR, Inc.
Par, Inc. is a small manufacturer of golf equipment whose management has decided to move into the market for medium and high-price golf bags. Par's distributor is enthusiastic about the new product line and has agreed to buy all the golf bags Par produces over the next the next three months. Two types of golf bags will be produced standard and deluxe. The manufacturing of each bag will require the following operations and the corresponding times:
Production Operations and Production Requirements
(per bag) (in hours)
Bag
Cutting and
Dyeing
Sewing
Finishing
Inspection and
Packaging
Standard
7/10
½
1
1/10
Deluxe
1
5/6
2/3
1/4
The accounting department assigned all the relevant variable costs and arrived at prices for both bags that will result in a profit (from an accounting viewpoint this is contribution margin per bag, e.g. overhead has not been allocated) contribution of $10 for every standard bag and $9 for every deluxe bag produced. The workload projections for the next three months estimates that 630 hours of cutting and dyeing time, 600 hours of sewing time, 708 hours of finishing time and 135 hours of inspection and packaging time will be available.
Formulate a linear program to maximize profit contribution for the next three months.
LP EXAMPLE
MAX 25 X1 + 30 X2
SUBJECT TO
2) 1.5 X1 + 3 X2 <= 450
3) 2 X1 + X2 <= 350
4) 0.25 X1 + 0.25 X2 <= 50
END
LP OPTIMUM FOUND AT STEP 1
OBJECTIVE FUNCTION VALUE
1) 5500.0000
VARIABLE VALUE REDUCED COST
X1 100.000000 .000000
X2 100.000000 .000000
ROW SLACK OR SURPLUS DUAL PRICES
2) .000000 3.333333
3) 50.000000 .000000
4) .000000 80.000000
NO. ITERATIONS= 1
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 25.000000 5.000000 10.000000
X2 30.000000 20.000000 5.000000
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 450.000000 150.000000 75.000000
3 350.000000 INFINITY 50.000000
4 50.000000 4.166667 12.500000
PAR
MAX 10 X1 + 9 X2
SUBJECT TO
2) 0.7 X1 + X2 <= 630
3) 0.5 X1 + 0.8333 X2 <= 600
4) X1 + 0.6667 X2 <= 708
5) 0.1 X1 + 0.25 X2 <= 135
END
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 7667.9420
VARIABLE VALUE REDUCED COST
X1 539.984300 .000000
X2 252.011000 .000000
ROW SLACK OR SURPLUS DUAL PRICES
2) .000000 4.374566
3) 120.007100 .000000
4) .000000 6.937804
5) 17.998820 .000000
NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 10.000000 3.499325 3.700000
X2 9.000000 5.285715 2.333000
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 630.000000 52.358850 134.400000
3 600.000000 INFINITY 120.007100
4 708.000000 192.000000 127.986000
5 135.000000 INFINITY 17.998820