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Examples for Linear programming and Transportation Problems

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Part one
RMS, Inc. produces three raw materials that are blended together (mixed together) to produce two products: a fuel additive and a solvent base. Each kilogram of fuel additive is a mixture of 2/5 kilogram of material 1 and 3/5 kilogram of material 3. A kilogram of solvent base is a mixture of 1/2 kilogram of material 1, 1/5 kilogram of material 2, and 3/10 kilogram of material 3. The profit is $40 for each kilogram of fuel additive produced and $30 for each kilogram of solvent base produced.
RMS's has available the following quantities of raw materials:
Material 1 20 Kilograms
Material 2 5 Kilograms
Material 3 21 Kilograms

What is the linear programming model for this problem? What is the maximum profit?

Part two
Referring to the following table, what is the minimum transportation cost?
Values not in the margins are cost per unit, $.

1 10 14 10 210
2 12 17 20 140
3 11 11 12 150
4 18 8 13 160
DEMAND 220 220 220

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Solution Summary

One example each for product mix problem and transportation problem is formulated and solved by using solver of MS Excel.

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1. When formulating a linear programming problem constraint, strict inequality signs (i.e., less than < or, greater than >) are not allowed.

2. In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be (less-than-or-equal-to) type.

3. The standard form for the computer solution of a linear programming problem requires all variables on the left side, and all numerical values on the right side of the inequality or equality sign.

4. Media selection is an important decision that advertisers have to make. In most media selection decisions, the objective of the decision maker is to minimize cost.

5. In a media selection problem, maximization of audience exposure may not result in maximization of total profit.

6. In an unbalanced transportation model, supply does not equal demand and supply constraints have signs.

7. Due to the unique characteristics of a transportation problem, the solution values of the decision variables are always integer values.

8. In a transportation problem, a demand constraint for a specific destination represents the amount of product demanded by a given destination (customer, retail outlet, store).

9. In maximization linear programming problem profit is maximized in the objective function by subtracting cost from revenue.

10. Let: rj = regular production quantity for period j, oj =overtime production quantity in period j, ii = inventory quantity in period j, and di = demand quantity in period j
Correct formulation of the demand constraint for a multi-period scheduling problem is:
rj + oj + i2 - i1 di

11. When ____________________ command is used with Excel spreadsheets, the objective function value is computed by multiplying all the values in a column containing the objective function coefficient values with all the corresponding values in a column containing the values of the decision variables. After the respective values in the two columns are multiplied, the resulting values are summed to determine the total profit or total cost.

12. In an unbalanced transportation model, supply does not equal demand and supply constraints have _____________ signs.

13. The ______________is the marginal economic value of one additional unit of a resource.

14. In a linear programming model, a resource constraint is a problem constraint with a¬¬¬¬¬¬¬__________________ sign.

15. The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar chips $0.50. For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

16. If Xab = the production of product a in period b, then to indicate that the limit on production of the company's 3 products in period 2 is 400,

17. Danson furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the optimal purchase quantity of each shelf and what is the maximum profit?

18. The dietician for the local hospital is trying to control the calorie intake of the heart surgery patients. Tonight's dinner menu could consist of the following food items: chicken, lasagna, pudding, salad, mashed potatoes and jello. The calories per serving for each of these items are as follows: chicken (600), lasagna (700), pudding (300), salad (200), mashed potatoes with gravy (400) and jello (200). If the maximum calorie intake has to be limited to 1200 calories. What is the dinner menu that would result in the highest calorie in take without going over the total calorie limit of 1200.
a. chicken, mashed potatoes and gravy, jello and salad
b. lasagna, mashed potatoes and gravy, and jello
c. chicken, mashed potatoes and gravy, and pudding
d. lasagna, mashed potatoes and gravy, and salad
e. chicken, mashed potatoes and gravy, and salad

19. If Xab = the production of product a in period b, then to indicate that the limit on production of the company's "3" products in period 2 is 400,
a. X32 400
b. X21 + X22 + X23 400
c. X12 + X22 + X32 400
d. X12 + X22 + X32 400
e. X23 400

20. Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1.
a. x11 + x12 (.35)(x11 + x21)
b. x11 .35 (x11 + x21)
c. x11 .35 (x11 + x12)
d. -.65x11 + .35x21 0
e. .65x11 - .35x21 0

21. Why should decision makers who are primarily concerned with marketing or finance or production know about linear programming?

See attached file for full problem description.

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