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Problem C (35 points / 5 points for each
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.