This question is from a Quantitative Business Analysis graduate course. This is

Question

This question is from a Quantitative Business Analysis graduate course. This is one question with multiple parts.

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) How much total time will be actually used in the fabrication area?

As Shown the Slack is 67, and total allowable is 200 so actual used is 200-67 = 133

(C-2) What is the optimal arrangement for the production of these four pieces of furniture?

The optimal arrangement is

(C-3) How much profit can be generated by all the produced dining tables?

The profit generated by DT is:

$720

(C-4) If someone offers you 20 pounds of aluminum for the price of $55, will you accept the offer? Why or why not?

Yes, as the shadow price is 3, so for 20 pounds 60 is the minimum cost, so if it is available at 55 there will be additional profit. And as that is the constraint that is binding we can utilize the increase capacity.

(C-5) What is the required finishing time to produce a dining table?

Coeff is 0.16 per hour

(C-6) If the newly adjusted unit profit for a chair is now $15, what will be the adjusted total profit?

As the allowable decrease is 10, so the solution does not change for the value of 15.

(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.

Actual value of XXX is 190-6*30 = 10. It is a slack, as it can be decreased by further 10 units.

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