Question

1. United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade, as follows:

Mill Aluminum Grade 1 2 High 6 2 Medium 2 2 Low 4 10

The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminum, 9 tons of medium-grade aluminum, and 5 tons of low-grade aluminum. It costs United $6,000 per day to operate mill 1 and $7,000 per day to operate mill 2. The company wants to know the number of days to operate each mill in order to meet the contract at the minimum cost.

1. Formulate a linear programming model for this problem.
2. If the optimal solution is to run Mill 1 for 4.5 days and Mill 2 for 0 days, what’s the minimum cost? How much extra (i.e., surplus) high-, medium-, and low-grade aluminum does the company produce at the optimal solution?
3. If a computer solution is used, please copy and paste both the “set-up” and “results” snapshot, and interpret the results.
4. The process (steps) is required.

Answer 1

a.

Number of days mill 1 is open = x and number of days mill 2 is open = y

Objective function

Minimize Z where

Z = 6000x + 7000y

Subject to constraints

6x + 2y >= 12

2x + 2y >= 9

4x + 10y >= 5

b.

If the optimal solution is x = 4.5 and y =0 then the minimum cost is 4.5*6000 = 27000

Similarly high grade production is 6*4.5 = 27 tons, medium grade is 2*4.5 = 9 tons and low grade is 4*4.5 = 18 tons. The surplus will be 15 tons of high grade and 13 tons of low grade aluminum.

c. and d.

The set up on excel is shown below

2 No of days 0 4 Cost 6000 7000 0 12 6 High 7 Medium 8 Low 10 0 10 12 13 14 15 16 17 18 19 Sheet55 (2) Sheet828 Sheet829Sheet830 Sheet831 Sheet827Sheet832 Sheet833 回凹 + 14596

The formulas are shown below

2 No of days 0 0 4 Cost 6000 7000 SUMPRODUCT(B4:C4,SB$2:$C$2) 6 High 7 Medium 8 Low 9 10 -SUMPRODUCT(B6C6,SB$2:$C$2) SUMPRODUCT(B7:C7,SB$2:$C$2) SUMPRODUCT(B8:C8,SB$2:$C$2) 6 12 10 12 13 15 16 17 18 19 ...Sheets5 (2) Sheet828 Sheet829 Sheet830 Sheet83 Sheet827 Sheet832 Sheet833 14596

The solver parameters are shown below

Solver Parameters Set Objective: 2 No of days 0 Min O Yalue O 4 Cost 6000 7000 By Changing Variable Cells: SBS2:SCS2 6 High 7 Medium 8 Low 12 Subject to the Constraints: 0 4 10 10 Beset All 12 13 Make Unconstrained Variables Non-Negative Select a Solving Method Simplex LP 15 Sohing Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. 16 18 19 4... Sheet55 (2) Sheet828Sheet829 Sheet830 Sheet831 Sheet827 Sheet832 Sheet833 囲回凹

The result is shown below

2 No of days 4.5 0 Solver Results 4 Cost 6000 7 7000 27000 Solver found a solution. All Constraints and optimality conditions are satisfied. Keep Solver Solution Sensitivity Limits 6 High 7 Medium 8 Low 27 12 ORestore Original Values 4 10 18 Return to Solver Parameters Dialog outline Reports Cancel Save Scenario... 10 Solver found a solution. All Constraints and optimality conditions are 12 When the GRG engine is used, Solver has found at least a local optimal solution. When Simplex LP is used, this means Solver has found a global optimal solution 13 15 16 18 Sheet55 (2) Sheet828 Sheet829Sheet830 Sheet831 Sheet827Sheet832 Sheet833 + 145%