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IE 323: 

IE 323 Blending problems

Basic issue in blending: 

2 Basic issue in blending Given quantities of (continuous) inputs Blend these to get outputs Generally, quantity constraints on output Quality (octane, min content, etc.) constraints Costs/constraints inside blending process Among all feasible ways, choose one giving best profit (or least cost, etc.)

Blending for Chandler Oil Basic statement of problem : 

3 Blending for Chandler Oil Basic statement of problem Blend crude oils in order to obtain end products (gasoline, heating oil) Obtain end products with acceptable quality levels Unlimited demand Maximize revenue minus advertising costs

Blending for Chandler Oil Decision variables: 

8 Blending for Chandler Oil Decision variables Barrels of each type of crude oil to be used for each end product Note we need a 2-way classification here Just taking barrels of each type used will not work (can’t track where they go)

Blending for Chandler Oil Objective: 

4 Blending for Chandler Oil Objective Maximize “total profit,” which includes: Revenue from selling end products Advertising costs required to sell end products Advertising cost assumed linear in amount sold This is not actually profit, but profit contribution More realistic (= complex) problem would have operating costs for processes, etc.

Blending for Chandler Oil Constraints: 

5 Blending for Chandler Oil Constraints Can’t use more crude oil than is available Must produce end products with acceptable quality levels Quality requirements stated as averages Modeling issue here in using LP

Average quality issue: 

6 Average quality issue Crude oils: 1 (Q = 10), 2 (Q = 5). Blend x11 of Type 1, x21 of Type 2, need average Q >= 8 Blend’s average Q: (10 x11 + 5 x21)/(x11 + x21) >= 8 Not a linear constraint! Instead, write 10 x11 + 5 x21 >= 8(x11 + x21) This is linear; works even if x11 = x21 = 0 Could think of these as “quality points”

Blending for Chandler Oil Inputs: 

7 Blending for Chandler Oil Inputs Availability of each type of crude oil (barrels) Required quality level for each end product Selling price per barrel of each end product Advertising cost per barrel of each end product

Spreadsheet model (See “Steps 1,2” sheet in Excel): 

9 Spreadsheet model (See “Steps 1,2” sheet in Excel) Step 1: Enter all inputs Step 2: Enter any values in the changing cells Don’t forget numbers of barrels used must be nonnegative

Spreadsheet model (See “Steps 3,4” sheet in Excel): 

10 Spreadsheet model (See “Steps 3,4” sheet in Excel) Step 3: Calculate the total of each input used with the SUM function Step 4: Calculate the total of each end product produced with the SUM function

Spreadsheet model (See “Steps 5,6” sheet in Excel): 

11 Spreadsheet model (See “Steps 5,6” sheet in Excel) Step 5: Calculate the profit contribution per barrel of each output by subtracting the unit advertising cost from the unit selling price Step 6: Calculate the total profit contribution with the SUMPRODUCT function

Spreadsheet model (See “Steps 7,8” sheet in Excel): 

12 Spreadsheet model (See “Steps 7,8” sheet in Excel) Step 7: Calculate the “quality points” achieved for each end product with the SUMPRODUCT function Step 8: Calculate the required “quality points” for each end product by multiplying the required quality per barrel by the number of barrels

Spreadsheet model (See “Optimal” sheet in Excel): 

13 Spreadsheet model (See “Optimal” sheet in Excel) Click on Tools/Solver Fill in the Solver dialog box See next slide Don’t forget to check the Assume Linear Model box under Options Click on Solve

Solver Dialog Box: 

14 Solver Dialog Box

Optimal solution properties (“Optimal” sheet): 

15 Optimal solution properties (“Optimal” sheet) All available crude oil is used Quality constraints are met exactly - no slack Much more of the less profitable end product (heating oil) is produced! Result of the quality constraints

Blending for Chandler Oil Sensitivity information: 

16 Blending for Chandler Oil Sensitivity information Reduced costs are all zero (no decision variable is at a bound) Value if we could get more crude oil: $29.70 (Type 1) and $17.45 (Type 2) Quality constraints don’t tell us much: Example: 10x + 5y >= 8x + 8y Rewrite: 2x - 3y >= 0 Shadow price applies to 0 in RHS, not to 8!

Blending for Chandler Oil Modeling issues: 

17 Blending for Chandler Oil Modeling issues We can do multi-step blending similarly E.g. two-step process, 2 crudes --> 3 intermediate products --> 5 gasolines; different yield ratios pij : amount of product j from crude i prj: total amount of product j (just sum) qjk: amount of gasoline k from product j Use the qjk for quality constraints

Blending for Chandler Oil Summary: 

18 Blending for Chandler Oil Summary Blending idea is simple but execution can get tricky Nonlinear quality constraints made linear by simple multiplication Careful choice of decision variables, just as in post office scheduling example Split variables to track quality levels