Question

A paper recycling company converts newspaper, mixed paper, white office paper, and cardboard into pulp for newsprint, packaging paper, and print stock quality paper. The following table summarizes the yield for each kind of pulp recovered from each ton of recycled material.

Recycling Yield

Newsprint                        Packaging          Print Stock

Newspaper                      85%                                    80%                     ____

Mixed Paper                    90%                                    80%                     70%

White Office Paper         90%                                    85%                     80%

Cardboard                        80%                                    70%                     ­___       

For instance, a ton of newspapers can be recycled using a technique that yields 0.85 tons of newsprint pulp. Alternatively, a ton of newspapers can be recycled using a technique that yields 0.80 tons of packaging paper. Similarly, a ton of cardboard can be recycled to yield 0.80 tons of newsprint or 0.70 tons of packaging paper pulp.

Note that newspaper and cardboard cannot be converted to print stock pulp using

the techniques available to the recycler.

The cost of processing each ton of raw material into the various types of pulp is summarized in the following table along with the amount of each of the four raw materials that can be purchased and their costs.

Processing Costs per Ton                           Purchases Cost      Tons

Newsprint         Packaging          Print Stock         Per Ton                      Available           

Newspaper                      $6.50                    $11.00                  ------                    $15                             600

Mixed Paper                    $9.75                    $12,25                  $9.50                    $16                             500

White Office Paper         $4.75                    $7.75                    $8.50                    $19                             300

Cardboard                        $7.50                    $8.50                    ------                    $17                             400

The recycler wants to determine the least costly way of producing 500 tons of newsprint pulp, 600 tons of packaging paper pulp, and 300 tons of print stock quality pulp.

a. Create a spreadsheet model for this problem and solve it.

b. What is the optimal solution?

Answer 1

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Answer:-

A paper recycling company needs to determine the least costly way to produce 500 tons, 600 tons, and 300 tons of newsprint pulp, packaging paper pulp, and stock quality pulp, respectively.

a. Prepare the spreadsheet model for this problem as shown below: Decision variables: C = cost of purchasing 1 ton of raw material i P, = processing cost for the raw material i to convert into productſ * = tons of raw material i to convert into product j Here, 1 for newspaper 2 for mixed paper 3 for white office paper 4 for cardboard

(1 for newsprint j={2 for packaging 3 for print stock Objective function: Minimize Z- ΣΣΡ,κ, +Σ ΣΣ«-Σ

Subject to, , Σ, Σ 600 . Σκη, 2500 Σκ, Σ 300 Σχ., 2 400 Σ,

Now, prepare a spreadsheet using given values and formulas : Figure 1: Formulas used in the spreadsheet 1 Recycling Yield Packaging Newsprint Print Stock 2 3 Newspaper 0.85 4 Mixed Paper 0.9 5 White Office Paper 0.9 6 Cardboard 0.8 7 0.8 0.8 0.85 0.7 0.7 0.8 Processing Costs per Ton Packaging Purchase Cost per Ton Tons Available 9 Newsprint Print Stock 16.5 9.75 11 12.25 1s 16 600 500 9.5 Newspaper 10 11 Mixed Paper White Office Paper 12 14.75 13 Cardboard 7.5 14 8.5 119 7.75 8.5 300 400 17

15 16 Newsprint 17 Newspaper 18 Mixed Paper 19 White Office Paper 20 Cardboard 21 Total SUMPRODUCT(B3:B6,B17:B20) 22 23 500 24 25 Processing cost ESUMPRODUCT(B10:013,817:020) 26 Purchase cost =SUMPRODUCT(E17:E20, E10:13) 27 Total cost =B25+826 Recycling Yield Packaging Print Stock Total -SUM(B17:017) SUM(B18:018) =SUM(819:019) SUM(B20:020) SUMPRODUCT(C3:06,017:C20) SUMPRODUCT(D3:06,017:D20) 600 300

Figure 2: Formulated spreadsheet E F A B с D 1 Recycling Yield 2 Newsprint Packaging Print Stock 3 Newspaper 85% 80% 4 Mixed Paper 90% 80% 70% 5 White Office Paper 90% 85% 80% 6 Cardboard 80% 70% 7

7 CO Processing Costs per Ton Tons Purchase Newsprint Packaging Print Stock Cost per Ton Available 9 10 Newspaper 11 Mixed Paper 600 $6.50 $9.75 $11.00 $12.25 $15 $16 $9.50 500 White Office Paper $4.75 $7.75 $8.50 $19 300 12 $7.50 $8.50 $17 400 13 Cardboard 14

16 Total 0.00 14 15 Recycling Yield Newsprint Packaging Print Stock 17 Newspaper 18 Mixed Paper 19 White Office Paper 20 Cardboard 21 Total 0.00 0.00 0.00 0.00 0.00 0.00 22 = = 23 500.00 600.00 300.00 24 25 Processing cost 26 Purchase cost 27 Total cost $0.00 $0.00 $0.00

Click on excel "solver" tool present under "menu" bar. Put the following values in "solver" window: Figure 3: Solver screenshot X х Solver Parameters Set Objective: $B$27 To: Max Min Value Of: 0 By Changing Variable Cells: $B$17:$D$20

Subject to the Constraints: $B$21:$D$21 = $B$23:$D$23 $E$17:$E$20 <= $F$10:$F$13 Add Change Delete Reset All Load/Save | Make Unconstrained variables Non-Negative Select a Solving Method: GRG Nonlinear Options

Solving 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. Help Solve Close

Click on “Solve" option. Then click on "OK" option. Doing this would give the following results: А B с D E F 1 Recycling Yield 2 Newsprint Packaging Print Stock 3 Newspaper 85% 80% 4 Mixed Paper 90% 80% 70% 5 White Office Paper 90% 85% 80% 6 Cardboard 80% 70% 7

7 Processing Costs per Ton 8 9 Purchase Tons Cost per Ton Available Newsprint Packaging Print Stock $6.50 $11.00 $15 600 Newspaper 10 11 Mixed Paper $9.75 $12.25 $9.50 $16 500 $4.75 $7.75 $8.50 $19 300 White Office Paper 12 13 Cardboard 14 $7.50 $8.50 $17 400

14 15 16 17 Newspaper 18 Mixed Paper 19 White Office Paper 20 Cardboard 21 Total 22 Recycling Yield Newsprint Packaging 588.24 11.76 0.00 71.43 0.00 300.00 0.00 397.78 500.00 600.00 Print Stock 0.00 428.57 0.00 0.00 300.00 Total 600.00 500.00 300.00 397.78 23 500.00 600.00 300.00 24

24 25 Processing cost 26 Purchase cost 27 Total cost $14,605.49 $29,462.24 $44,067.74 Hence, it is concluded that minimum cost required for producing the required amount of newsprint pulp, packaging paper pulp, and stock quality pulp is $44,067.74.

b. The optimal solution is given below: x = 588.24, X2 = 11.76, -0.00 *2 = 0.00, X 22 = 71.43, = 428.57 *32 = 300.00, = 0.00, X42 = 397.78, The optimal value obtained using the optimal solution is $44,067.74. 131 = 0.00, 23 = 33 = 0.00 *43 = 0.00