Lafayette Corp. manufactures aluminum patio furniture. There are four different
ID: 380633 • Letter: L
Question
Lafayette Corp. manufactures aluminum patio furniture. There are four different pieces in a new design line of furniture. A Linear Programming model has been developed (see the given program solution output) to determine the number of each piece to produce this week. The four variables (CH, DT, LG, ET) represent, respectively, the number of chairs, dining tables, lounges, and end tables to be produced. The objective function measures the total profit (assume all units produced will be sold). The first three constraints, respectively, measure the aluminum (in pounds), the amount of fabrication time (in hours), and the amount of finishing time (in hours) required for production. The fourth and the fifth constraints are marketing restrictions. The sixth and seventh constraints specify relative production levels.
Linear Program listed:
Maximize 18CH+24DT+45LG+18ET
Subject to
Const1) 6CH+18DT+15LG+4ET<=2700
Const2) 0.4CH+0.3DT+0.6LG+0.4ET<=200
Const3) 0.2CH+0.16DT+0.25LG+0.12ET<=110
Const4) DT>=30
Const5) LG <=60
Const6) CH-6DT>=0
Const7) -LG+2ET<=0
Program Solution Output listed:
OBJECTIVE FUNCTION VALUE 7380.000
VARIABLE VALUE REDUCED COST
CH 190.000000 0.000000
DT 30.000000 0.000000
LG 60.000000 0.000000
ET 30.000000 0.000000
ROW SLACK OR SURPLUS SHADOW(DUAL) PRICES
Const1) 0.000000 3.000000
Const2) 67.000000 0.000000
Const3) 48.599998 0.000000
Const4) 0.000000 -30.000000
Const5) 0.000000 3.000000
Const6) XXX 0.000000
Const7) 0.000000 3.000000
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJECTIVE FUNCTION COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE COEF INCREASE ALLOWABLE COEF DECREASE
CH 18.000000 1.058824 10.000000
DT 24.000000 30.000000 INFINITY
LG 45.000000 INFINITY 3.000000
ET 18.000000 INFINITY 6.000000
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE RHS INCREASE ALLOWABLE RHS DECREASE
Const1 2700.000000 1004.999939 60.000000
Const2 200.000000 INFINITY 67.000000
Const3 110.000000 INFINITY 48.599998
Const4 30.000000 1.111111 30.000000
Const5 60.000000 3.529412 60.000000
Const6 0.000000 10.000000 INFINITY
Const7 0.000000 30.000000 60.000000
Use the above given outputs to answer the following questions. There is no need to run the program with any software. The above given output contains all the needed analysis information.
(C-1) How much total time will be actually used in the fabrication area?
(C-2) What is the optimal arrangement for the production of these four pieces of furniture?
(C-3) How much profit can be generated by all the produced dining tables?
(C-4) If someone offers you 20 pounds of aluminum for the price of $55, will you accept the offer? Why or why not?
(C-5) What is the required finishing time to produce a dining table?
(C-6) If the newly adjusted unit profit for a chair is now $15, what will be the adjusted total profit?
(C-7) The SLACK or SURPLUS information for constraint 6 (line Const6) is missing and currently shown as “XXX”. What is the actual value of “XXX”? Is this a slack or a surplus outcome? Explain.
Explanation / Answer
(C-1)
Fabrication time is the second constraint - Current value is 200 hours (this the total time that will be actually used in the fabrication area)
(C-2)
CH=190; DT=30; LG=60; ET=30 (appearing in the first part of the output)
(C-3)
Profit from the Dining tables = 30 x $24 = $720
(C-4)
The shadow price of the Aluminium Constraint (i.e. Constraint 1) is $3. So, for one pound increase in capacity, a profit of $3 can be generated.
You pay $55/20 = $2.75 per pound which is less than $3.0. So, this is a good deal.
(C-5)
Const3) 0.2CH+0.16DT+0.25LG+0.12ET<=110
So, the required finishing time to produce a dining table = 0.16 hours
(C-6)
The 'Allowable decrease' for the profit per CH is 10. Here the decrease is $3 only. So, the optimal solution will not change. The profit will, however, change as -
Z = 15 x 190 + 24 x 30 + 45 x 60 + 18 x 30 = $6810
(C-7)
The allowable increase is 10. The current value is 0. So, XXX=10
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