Sunco Blending Problem Page 84 Winston

Sunco Blending Problem: (with Application of WinQSB for LP) How to Prepare a Report for a Manager

Source of the problem: page 86, W. L. Winston, "Operations Research, Application and Algorithms", 4th Edition, Thomson Learning, 2004, ISBN: 0-534-38058-1

By: Dr. Parisay

This problem is modeled as a Linear Programming problem. To solve this model we can use Simplex Method. I used WinQSB software that has Simplex Method to solve this problem. This problem has 4 solutions (four alternatives). The complete/detail files are in Class Notes web page.

1 One of the solutions to the problem (Solution 1):

Decision Solution Unit Cost or Total Reduced Basis Allowable Allowable Variable Value Profit c(j) Contribution Cost Status Min c(j) Max. c(j)

1 X11 2,000.00 21 42,000.00 0 basic -M 21 2 X12 2,200.00 11 24,200.00 0 basic 11 11 3 X13 800 1 800 0 basic 1 52.125 4 X21 1,000.00 31 31,000.00 0 basic 31 93.7 5 X22 4,000.00 21 84,000.00 0 basic 21 21 6 X23 0 11 0 0 at bound -M 11 7 X31 0 41 0 0 basic -5.35 41 8 X32 3,300.00 31 102,300.00 0 basic 31 31 9 X33 200 21 4,200.00 0 basic 21 225.5 10 A1 0 -1 0 -209 at bound -M 208 11 A2 750 -1 -750 0 basic -105.5 M 12 A3 0 -1 0 -409 at bound -M 408

Objective Function (Max.) = 287,750.00 (Note: Alternate Solution Exists!!)

Allowable Left Hand Right Hand Slack or Shadow Allowable Max. Constraint Side Direction Side Surplus Price Min. RHS RHS

1 C1 3,000.00 = 3,000.00 0 -20.8 2,500.00 3,400.00 2 C2 2,000.00 = 2,000.00 0 0.1 -M 9,500.00 3 C3 1,000.00 = 1,000.00 0 -40.8 500 1,250.00 4 C4 5,000.00 <= 5,000.00 0 57.25 4,800.00 5,200.00 5 C5 5,000.00 <= 5,000.00 0 20.9 1,000.00 5,400.00 6 C6 3,500.00 <= 5,000.00 1,500.00 0 3,500.00 M 7 C7 13,500.00 <= 14,000.00 500 0 13,500.00 M 8 C8 0 >= 0 0 0 0 800 9 C9 800 >= 0 800 0 -M 800 10 C10 5,200.00 >= 0 5,200.00 0 -M 5,200.00 11 C11 0 <= 0 0 3,090.00 0 5 12 C12 0 <= 0 0 3,090.00 -33 5 13 C13 0 <= 0 0 3,090.00 -5 5

2 Summary of the above solution (a tool for report writing): Below is the summary of this solution to the problem (Solution 1) arranged as a table (a tool for report writing):

Sulfur Octane Total Production GAS Optimal value of decision sales plan variables $830,00 barrel/day 0 total = 13500 1% 10 210000 3000 1 X11 X21 X31 A1 (max (min 10) = = = 0 = 0 1%) 2000 1000 2% 8.08 570000 9500 2 X12 X22 X32 A2 (max (min 10) = = = = 2%) 2200 4000 3300 75 0 1% 11.2 50000 1000 3 X13 X23 X33 A3 (max (min 10) = = 0 = = 0 1%) * 800 200 OIL 1 2 3 Total purchase $487,500 2250 1750 8750 00 00 0 Required oil, barrel/day 5000 5000 3500 * Sample calculation: Octane of Gas 3 = (800*12 + 200*8)/(800 + 200) = 11.2

- for each type of gas: 1 $ ad/day will increase demand by 10 barrel/day - $4 to transform 1 barrel oil into one barrel gas - Max gas to produce 14000 barrel /day

Decision variables: Ai = dollars spent daily on advertising gas i (i = 1, 2, 3) Xij = barrel of crude oil i used daily to produce gas j (i = 1, 2, 3; j = 1, 2, 3)

Max profit = Total sales - Total cost = 830000 – (487500 + 750 + 4*13500) = 287,750

General guideline to prepare report for managers: a) Just imagine what information different managers may look for. - What a top manager may look for? Total profit (for shareholders), - What a sales manager may look for? Total production of each gas and total sales (for sales planning), quality specification (octane and sulfur) of final product (for customer satisfaction) - What a financial manger may look for? Total requirements of each oil and total purchases (procurement planning, budgeting planning), budget for advertising and on which gas,

3 - What a production manager may look for? Amount of each of one the oils, to be used for each one of the gases. b) Answer to variety of “What if” questions (sensitivity analysis). For example, if we have some extra budget, which gas should be produced more? Which oil should be bought more?

We can add some explanation on how some information (rhs or coefficient of OF) can change within a limit and not have changes (or major changes) in optimal solution, therefore highlight some flexibility. c) Consider possible new limitations or scenarios. For example, what if we have limited budget, such as total budget for purchasing oil is $400,000. d) Examine possible alternate solutions. Discuss the importance of multi optimal (alternate) solution to provide flexibility for decision making.

The followings are my demonstration of the above guidelines for writing report, in the case of Sunco problem: a) Just imagine what information different managers may look for.

* Main items in the report to the top manager: What a top manager may look for? Total profit (for shareholders)

Total maximum profit is $287,750. (from WinQSB output)

* Main items in the report to the sales manager: What a sales manager may look for? Total production of each gas and total sales (for sales planning), quality specification (octane and sulfur) of final product (for customer satisfaction)

The information for gases being produced is in the table below. (from WinQSB output and performing some calculations) Gas Barrels $ sales per Total $ sales % sulfur Octane produced barrel per gas (required) (required) 1 3000 70 210000 1 (at most 1) 10 (at least 10) 2 9500 60 570000 2 (at most 2) 8.08 (at least 8) 3 1000 50 50000 1 (at most 1) 11.2 (at least 6) Total 13500 830000

4 * Main items in the report to the financial manager: What a financial manger may look for? Total requirements of each oil and total purchases (procurement planning, budgeting planning), budget for advertising on each gas

The information on budget for each one of the oils is in the table below. (from WinQSB output and performing some calculations) Oil Barrels $ cost per Total $ cost Advertising purchased barrel per oil 1 5000 45 225000 0 2 5000 35 175000 750 3 3500 25 87500 0 Total 13500 487500 750

Total transformation cost: $4*(13500 barrels) = $54000 Total production cost = cost of oil + cost of ad + cost of transformation =487500 + 750 + 54000 = $542250

* Main items in the report to the inventory manager: What an inventory manger may look for? Total requirements of each oil and the expected quality of each one (for quality control upon arrival), frequency and amount of order

The information for each one of the oils is in the table below. (from WinQSB output and performing some calculations) Oil Barrels purchased % sulfur content Octane content 1 5000 0.5 12 2 5000 2 6 3 3500 3 3 Total 13500

5 * Main items in the report to the production manager: What a production manager may look for? Amount of each one of the oils to be used for each one of the gases, quality required for each gas

The information for each one of the gases is in the table below. (from WinQSB output and performing some calculations) Gas Barrels % sulfur Octane Oil 1 Oil 2 Oil 3 produced (required) (required) used used used 1 3000 1 (at most 1) 10 (at least 10) 2000 1000 0 2 9500 2 (at most 2) 8.08 (at least 8) 2200 4000 3300 3 1000 1 (at most 1) 11.2 (at least 6) 800 0 200 Total 13500 5000 5000 3500

b) Answer to variety of “What if” questions (sensitivity analysis). This can be as extra items in the report to the production, inventory, financial, and sales managers:

* Perform sensitivity analysis and draw important sensitivity graphs using WinQSB. Example 1: If we have some extra budget, which gas should be produced more? Notice that Gas 1 and 3 have negative shadow price, so do not add to them!

Example 2: If we have some extra budget, which oil should be bought more? Notice that there is no point buying more Oil 3 as shadow price is zero. We prefer to buy more Oil 1 than Oil 2 because its shadow price is more positive.

* Provide some explanation on how some of the information (limit on resources, rhs, or coefficients of objective function) can change within a limit and not have changes (or major changes) in optimal solution, therefore highlight some flexibility. Be careful about interrelationship between your variables, such as example below. Example 3: (Based on Solution 1) Assume we are interested to increase the total profit by increasing profit from production of any gas, while there is no change in amount of oils used for production (current solution). Looking at X11, X12, X13 (Oil 1 used in each gas) we will notice that only the profit from X13 (its coefficient is 1) can be increased. The rest are at their limit value (maximum allowable 21 and 11). Considering that profit is (sales – cost) we should either reduce the cost of Oil 1 or increase the sales price of Gas 3. We cannot reduce the cost of Oil 1 as it will increase the coefficient of X11 and X12, which is undesirable. Therefore, we are left with increasing the sales price of Gas 3. If we increase the sales price of Gas 3 then the profit for X23 (at its max 11) and X33 (at its min 21) will also increase. But, we cannot increase profit from X23 as it is in its maximum allowable limit. Conclusion: forget about making more profit from reducing cost of Oil 1 or increasing sales price of Gas 3!!!!!

6 c) Consider possible new limitations or scenarios. Example 4: What if we have limited budget for purchasing oil, such as $400,000. This is much below the budget required ($487500). We need to solve the problem again after adding one more constraint, using WinQSB. d) Examine possible alternate solutions. Discuss the importance of multi optimal (alternate) solution to provide flexibility for decision making. Example 5: We have four optimal solutions in this problem. I have summarized these solutions in the following table (an effective communication tool). Variable Variable Solution Solution Solution Solution 1 2 3 4 Oil 1 used in Gas 1 X11 2000 2088.9 2000 2222.2 Oil 1 used in Gas 2 X12 2200 2111.1 2333.3 2111.1 Oil 1 used in Gas 3 X13 800 800 666.7 666.7 Oil 2 used in Gas 1 X21 1000 777.8 1000 444.5 Oil 2 used in Gas 2 X22 4000 4222.2 3666.7 4222.2 Oil 2 used in Gas 3 X23 0 0 333.3 333.3 Oil 3 used in Gas 1 X31 0 1333.3 0 333.3 Oil 3 used in Gas 2 X32 3300 3166.7 3500 3166.7 Oil 3 used in Gas 3 X33 200 200 0 0 Ad budget for Gas 1 A1 0 0 0 0 Ad budget for Gas 2 A2 750 750 750 750 Ad budget for Gas 3 A3 0 0 0 0

We can discuss the importance of integer solutions. In some cases, similar to here, that the solutions are large values (such as 666.7 and 2222.2), it is possible to round the non-integer solutions up and down without major violation of constraints or major change in the value of optimal objective function. Therefore, it is not critical to round non-integer values.

Prepared: 1-30-03 updated: 1-7-04