Interpret each of the shadow prices

1.   Nooner Appliance Producers (NAP), a small appliance manufacturing company that specializes in clocks, must decide what types and quantities of output to manufacture for each week’s sale.  Currently Nooner makes only two kinds of clocks, regular clocks and alarm clocks, from which the product mix is selected.  Next week’s product mix can only be produced with the labor, facilities, and parts currently on hand.  These supplies are as follows:

Number of labor hours                  1,600

Number of processing hours         1,800

Number of alarm assemblies            350

The resources are related to the two alternative manufactured outputs, regular clocks and alarm clocks, in the following way:  each regular clock produced requires 2 hours of labor and 6 hours of processing, while each alarm clock produced requires 4 hours of labor and 2 hours of processing.  The profit per unit for regular clocks is $3.00 while the company makes $8 per unit for alarm clocks.  Additionally, at least 300 clocks in total must be produced.  How many of each type of clock should Nooner produce to maximize profit?  The LP structure and solution are shown below.

LINEAR PROGRAMMING PROBLEM:  Nooner Appliance Producers (NAP)

MAX 3X1+8X2

S.T.

1)  2X1+4X2<1600

2)  6X1+2X2<1800

3)  1X2<350

4)  1X1+1X2>300

OPTIMAL SOLUTION

Objective Function Value =        3100.000

Variable             Value             Reduced Costs

————–     —————      ——————

X1                   100.000                   0.000

X2                   350.000                   0.000

Constraint        Slack/Surplus           Dual Prices

————–     —————      ——————

1                      0.000                   1.500

2                    500.000                   0.000

3                      0.000                   2.000

4                    150.000                   0.000

 

OBJECTIVE COEFFICIENT RANGES

Variable       Lower Limit       Current Value     Upper Limit

————   —————    —————  —————

X1                  0.000              3.000            4.000

X2                  6.000              8.000   No Upper Limit

RIGHT HAND SIDE RANGES

Constraint      Lower Limit       Current Value     Upper Limit

————   —————    —————  —————

1               1400.000           1600.000         1766.667

2               1300.000           1800.000   No Upper Limit

3                300.000            350.000          400.000

4         No Lower Limit            300.000          450.000

Given the Nooner  scenario:

(a)     Solve using the graphical solution procedure and identify all extreme points of the feasible region.

(b)     How much of each clock should be produced and what is the associated profit??

Using the output from the Management Scientist, answer the remaining questions.

(c)     What are the values and interpretations of all slack and surplus variables?

(d)    Determine (compute manually) and interpret the range of optimality for the objective function coefficients.

(e)     Interpret each of the shadow prices.

(f)     If the profit on an alarm clock decreased to $6 per unit, what parts of the optimal solution would change and how, and what parts would not change?  Explain your rationale.

(g)     Suppose Nooner can get an additional 100 alarm assemblies at a premium price of $1 more than the current price.  Should Nooner take advantage of this offer?   Explain your rationale.

Solution: