Podcast
Questions and Answers
What is the primary goal of the linear programming (LP) model?
What is the primary goal of the linear programming (LP) model?
- To create conditions where decision variables can only be integers.
- To ensure certainty in all parameters.
- To increase the number of decision variables.
- To maximize or minimize the objective function. (correct)
Which assumption implies that the total resources consumed must equal the sum contributed by individual decision variables?
Which assumption implies that the total resources consumed must equal the sum contributed by individual decision variables?
- Certainty
- Divisibility
- Non-negativity
- Additivity (correct)
Which constraint related to decision variables ensures that values cannot be negative?
Which constraint related to decision variables ensures that values cannot be negative?
- Linearity
- Non-negativity (correct)
- Certainty
- Additivity
Which of the following is NOT a characteristic of the LP model?
Which of the following is NOT a characteristic of the LP model?
If each unit of product A requires 3Kg of raw material, what does the assumption of proportionality imply for unit A's production?
If each unit of product A requires 3Kg of raw material, what does the assumption of proportionality imply for unit A's production?
Which of the following meets the criteria for the 'certainty' assumption in the LP model?
Which of the following meets the criteria for the 'certainty' assumption in the LP model?
When formulating an LP model for products A and B, why is the note about units of labor hours and raw materials important?
When formulating an LP model for products A and B, why is the note about units of labor hours and raw materials important?
What is the primary goal of the Product Mix Problem?
What is the primary goal of the Product Mix Problem?
What is the primary goal of the production planning problem stated in the model?
What is the primary goal of the production planning problem stated in the model?
Which statement is true about the non-negativity constraints?
Which statement is true about the non-negativity constraints?
What is the significance of non-negativity constraints in this model?
What is the significance of non-negativity constraints in this model?
What does the optimal solution $(x_1, x_2) = (470.6, 329.4)$ represent?
What does the optimal solution $(x_1, x_2) = (470.6, 329.4)$ represent?
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Study Notes
Linear Programming Model Development
- Objective: Maximize monthly profit by determining an optimal product mix of ties.
- Decision Variables:
- X1 = quantity of all-silk ties produced
- X2 = quantity of polyester ties produced
- X3 = quantity of blend 1 poly-cotton ties produced
- X4 = quantity of blend 2 poly-cotton ties produced
- Profit Calculations:
- Silk tie profit: 6.70−(6.70 - (6.70−(21 * 0.125) = $4.08
- Polyester profit: 3.55−(3.55 - (3.55−(6 * 0.08) = $3.07
- Blend 1 profit: 4.31−((4.31 - ((4.31−((6 * 0.05) + (9∗0.05))=9 * 0.05)) = 9∗0.05))=3.56
- Blend 2 profit: 4.81−((4.81 - ((4.81−((6 * 0.03) + (9∗0.07))=9 * 0.07)) = 9∗0.07))=4.00
- Objective function: Maximize profit = 4.08X1+4.08X1 + 4.08X1+3.07X2 + 3.56X3+3.56X3 + 3.56X3+4.00X4.
Constraints Overview
- Total material availability constraints:
- Silk availability: 0.125X1 ≤ 800
- Polyester availability: 0.08X2 + 0.05X3 + 0.03X4 ≤ 3,000
- Cotton availability: 0.05X3 + 0.07X4 ≤ 1,600
- Contract constraints:
- X1 ≥ 6,000 (Silk ties)
- X2 ≥ 10,000 (Polyester)
- X3 ≥ 13,000 (Blend 1)
- X4 ≥ 6,000 (Blend 2)
- Demand constraints:
- X1 ≤ 7,000
- X2 ≤ 14,000
- X3 ≤ 16,000
- X4 ≤ 8,500
- Non-negativity constraints: X1, X2, X3, X4 ≥ 0
Assumptions in Linear Programming
- Linearity/Proportionality: Objective functions and constraints must be proportional to decision variables.
- Divisibility: Non-integer (fractional) values of decision variables are acceptable.
- Certainty: All parameter values are known and remain constant.
- Additivity: Total effects of decision variables must sum to the overall effect.
- Non-negativity: Decision variables are restricted to values greater than or equal to zero.
Production Planning Model
- Production scenario with constraints on regular and overtime production.
- Monthly demand for four months: 1000, 800, 1200, 900 units.
- Regular time capacity: 800 units/month; Overtime capacity: 200 units/month.
- Regular production cost: 20/unit;Overtimecost:20/unit; Overtime cost: 20/unit;Overtimecost:25/unit; Inventory holding cost: $3/unit/month.
- Decision Variables:
- Rt = Quantity of Regular production in month t
- Ot = Quantity of Overtime production in month t
- It = Inventory carried over to the next month
- Objective Function: Minimize total cost Z = 20(R1+R2+R3+R4) + 25(O1+O2+O3+O4) + 3(I1+I2+I3).
Diet Problem Example
- Objective: Determine the minimum-cost feed mix of corn and soybean meal for daily special feed.
- Decision Variables:
- x1 = pounds of corn in the daily mix
- x2 = pounds of soybean meal in the daily mix
- Constraints:
- Daily feed minimum: x1 + x2 ≥ 800
- Protein requirement: 0.09x1 + 0.6x2 ≥ 0.30(x1 + x2)
- Fiber requirement: 0.02x1 + 0.06x2 ≤ 0.05(x1 + x2)
- Non-negativity: x1 ≥ 0, x2 ≥ 0.
Classification of LP Solutions
- Multiple/Alternate Optimal Solutions: More than one optimal solution can exist, providing flexibility in decision-making.
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