Question

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 where X1 represents regular clocks and X2 represents alarm clocks.  

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

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

(a)    What are the values of the primary variables and the objective function?

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

(c)    Determine (compute manually) and interpret the range of optimality for the objective function coefficient for regular clocks.

(d)    Interpret each of the shadow prices.

Answer 1

(a)

The values of primary variable and the objective function

are x1 = 100.00 and and x2 = 350.00 and Objective Function Value = 3100.00

(b)

Values of slack (unused) variables are:

Slack = RHS - Resources used

Surplus = Total output - RHS

This is directly given in the LINGO report

Number of labor hours = 1600 - 1600 = 0

Number of processing hours = 1800 - 1300 = 500

Number of alarm assemblies = 350 - 350 = 0

Total number of clocks produced = (100+350) - 300 = 150

Interpretation: Number of labor hours used is equal to the number of labor hours available. Number of processing hours used is 500 less than the processing hours available. Number of alarm assemblies used is equal to the total number of alarm assemblies available.

Total number of clocks produced exceeds the required number of 300

(c)

Range of optimality for the coefficients of objective function are:

Range of optimality for X1: 0 to 4

Range of optimality for X2: 6 to  infinity

Interpretation: As long as the coefficient of X1 is within the range of 0 to 4, the current optimal solution remains unchanged.

As long as the coefficient of X2 is within the range of 6 to infinity, the current optimal solution remains unchanged.

(e)

Shadow price or dual price of labor hours constraint is 1.5, which means increasing the available labor hours by one unit, increases the total profit by $ 1.5, similarly, decrease in the available labor hours by one unit, decreases the total profit by $ 1.5

Shadow price or dual price of processing hours constraint is 0, which means increasing the available processing hours by one unit, increases the total profit by $ 0, similarly, decrease in the available labor hours by one unit, decreases the total profit by $ 0

Shadow price or dual price of alarm assemblies constraint is 2, which means increasing the available alarm assemblies by one unit, increases the total profit by $ 2, similarly, decrease in the available alarm assemblies by one unit, decreases the total profit by $ 2

Shadow price or dual price of total number of alarm clocks to be produced constraint is 0, which means increasing the required total number of alarm clocks to be produced by one unit, increases the total profit by $ 0, similarly, decrease in the required total number of alarm clocks to be produced by one unit, decreases the total profit by $ 0

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