Example Linear Programming Questions PDF

Here’s a set of slightly more challenging and practical multiple-choice questions for your preparation. These questions test your understanding of theoretical and practical applications of linear programming, transportation, and production planning. 1. Identifying a Binding Constraint A company is...

Here’s a set of slightly more challenging and practical multiple-choice questions for your preparation. These questions test your understanding of theoretical and practical applications of linear programming, transportation, and production planning. 1. Identifying a Binding Constraint A company is solving a production problem with the following labor constraint: 0.5x1+0.3x2≤500.0.5x_1 + 0.3x_2 \leq 500. The optimal solution uses 400 labor hours. What is true about this constraint? a. It is binding because the left-hand side is less than the right-hand side.​ b. It is binding because all labor hours are not used.​ c. It is not binding because 400 < 500.​ d. It is not binding because the left-hand side does not reach the right-hand side. Answer: c. It is not binding because 400 < 500. (Binding occurs only if all available labor is fully utilized.) 2. Shadow Price Interpretation A factory has a machine capacity constraint: 2x1+3x2≤600.2x_1 + 3x_2 \leq 600. The shadow price for this constraint is 8. What does this mean? a. Adding one unit of machine capacity increases the total profit by $8.​ b. Adding one unit of machine capacity reduces the total profit by $8.​ c. Increasing machine capacity decreases production by 8 units.​ d. This constraint is non-binding, so the shadow price does not apply. Answer: a. Adding one unit of machine capacity increases the total profit by $8. 3. Transportation Problem Formulation A company has two factories and three warehouses. The supply at Factory 1 is 100 units, and Factory 2 is 200 units. The demand at Warehouse 1, Warehouse 2, and Warehouse 3 is 120, 100, and 80 units, respectively. What would the demand constraint for Warehouse 2 look like? a. x12+x22≥100x_{12} + x_{22} \geq 100.​ b. x12+x22=100x_{12} + x_{22} = 100.​ c. x11+x12+x13≤200x_{11} + x_{12} + x_{13} \leq 200.​ d. x21+x22+x23=120x_{21} + x_{22} + x_{23} = 120. Answer: b. x12+x22=100x_{12} + x_{22} = 100 (total shipments arriving at Warehouse 2 must meet its demand of 100 units). 4. Objective Function for Production Planning A company produces two products, AA and BB, with unit costs of $10 and $15, respectively. Holding inventory costs $2/unit for AA and $3/unit for BB. The company also incurs a $5 cost per unit increase in production and a $4 cost per unit decrease. Which is the correct objective function to minimize total costs? a. Z=10xA+15xB+2sA+3sB+5I+4DZ = 10x_A + 15x_B + 2s_A + 3s_B + 5I + 4D.​ b. Z=10xA+15xB−2sA−3sB+5I+4DZ = 10x_A + 15x_B - 2s_A - 3s_B + 5I + 4D.​ c. Z=10xA+15xB+2xA+3xB+5I+4DZ = 10x_A + 15x_B + 2x_A + 3x_B + 5I + 4D.​ d. Z=10xA+15xB+5I+4DZ = 10x_A + 15x_B + 5I + 4D. Answer: a. Z=10xA+15xB+2sA+3sB+5I+4DZ = 10x_A + 15x_B + 2s_A + 3s_B + 5I + 4D (adds up production costs, inventory holding, and changes in production). 5. Feasibility Check Which of the following solutions is feasible for the constraint: 3x1+5x2≤15, x1,x2≥0?3x_1 + 5x_2 \leq 15, \, x_1, x_2 \geq 0? a. x1=2,x2=2x_1 = 2, x_2 = 2.​ b. x1=1,x2=3x_1 = 1, x_2 = 3.​ c. x1=0,x2=4x_1 = 0, x_2 = 4.​ d. x1=3,x2=2x_1 = 3, x_2 = 2. Answer: b. x1=1,x2=3x_1 = 1, x_2 = 3 (calculates to 3(1)+5(3)=153(1) + 5(3) = 15, which satisfies the constraint). 6. Production Constraints A factory produces two products P1P_1 and P2P_2. Each unit of P1P_1 requires 3 hours of labor, and each unit of P2P_2 requires 2 hours. If the available labor is 600 hours, what is the correct constraint? a. 3x1+2x2≥6003x_1 + 2x_2 \geq 600.​ b. 3x1+2x2≤6003x_1 + 2x_2 \leq 600.​ c. x1+x2≤600x_1 + x_2 \leq 600.​ d. 3x1+2x2=6003x_1 + 2x_2 = 600. Answer: b. 3x1+2x2≤6003x_1 + 2x_2 \leq 600 (labor cannot exceed 600 hours). 7. Demand Fulfillment in Transportation A warehouse receives shipments from two factories with the following constraint: x11+x21≥400.x_{11} + x_{21} \geq 400. What does this constraint ensure? a. At least 400 units are shipped from Factory 1.​ b. At least 400 units are shipped to the warehouse.​ c. A maximum of 400 units can be received by the warehouse.​ d. At most, 400 units are shipped from both factories. Answer: b. At least 400 units are shipped to the warehouse. 8. Transshipment Problem In a transshipment model, the constraint for a warehouse states: xin+xjn=xnk+xnl.x_{in} + x_{jn} = x_{nk} + x_{nl}. What does this constraint enforce? a. Supply at the warehouse equals demand.​ b. Shipments into the warehouse equal shipments out.​ c. Total production equals total demand.​ d. The warehouse cannot receive more than it ships out. Answer: b. Shipments into the warehouse equal shipments out (flow balance). 9. Blending Problem Constraints A blending problem requires at least 30% of material A in a mix. What is the correct constraint if xAx_A is the amount of material A and xBx_B is the amount of material B? a. xA≥0.3(xA+xB)x_A \geq 0.3(x_A + x_B).​ b. xA≤0.3(xA+xB)x_A \leq 0.3(x_A + x_B).​ c. xA≥30x_A \geq 30.​ d. xB≥0.3(xA+xB)x_B \geq 0.3(x_A + x_B). Answer: a. xA≥0.3(xA+xB)x_A \geq 0.3(x_A + x_B) (ensures material A is at least 30% of the total mix). 10. Objective Function in Transportation If shipping costs are $4/unit from Factory 1 to Warehouse A and $6/unit from Factory 2 to Warehouse A, what is the objective function to minimize shipping costs? a. Z=4x11+6x12Z = 4x_{11} + 6x_{12}.​ b. Z=4x11+6x21Z = 4x_{11} + 6x_{21}.​ c. Z=6x11+4x21Z = 6x_{11} + 4x_{21}.​ d. Z=4x21+6x11Z = 4x_{21} + 6x_{11}. Answer: b. Z=4x11+6x21Z = 4x_{11} + 6x_{21} (accounts for correct shipping costs from Factory 1 and Factory 2 to Warehouse A). These questions will challenge your ability to interpret constraints, objective functions, and relationships in LP models. Let me know if you'd like more practice! Additional Notes for Quiz Preparation Based on the questions from your last quiz, here are critical notes and concepts to keep in mind: 1. Decision Variables  ​ Definition: Represent quantities to determine in order to solve the problem (e.g., units to produce, ship, or allocate). ​ Key Characteristic: Decision variables must be measurable and directly tied to the problem's constraints and objective function. 2. Linear Programming Components ​ Objective Function: ○​ Defines what is being maximized or minimized (e.g., cost, profit). ○​ Linear equations like Z=3x1+5x2Z = 3x_1 + 5x_2. ​ Constraints: ○​ Define the limits or requirements in the model (e.g., 2x1+4x2≤1002x_1 + 4x_2 \leq 100). ○​ Types: ​ Resource constraints (e.g., machine capacity, labor hours). ​ Supply/Demand constraints in transportation models. ​ Quality constraints in blending models. ○​ Feasible Region: The solution space where all constraints are satisfied. 3. Shadow Price ​ Definition: Indicates the change in the objective function's value (profit/cost) if the right-hand side (RHS) of a binding constraint increases by one unit. ​ Applies Only to Binding Constraints. 4. Special Cases in LP ​ Unbounded Problem: ○​ Occurs when the feasible region extends indefinitely in the direction of optimization. ​ Infeasible Problem: ○​ No solution satisfies all constraints simultaneously. ​ Alternate Optimal Solutions: ○​ Occurs when the objective function is parallel to a binding constraint, resulting in multiple solutions along a line. 5. Slack Variables ​ Role: Convert inequalities into equalities for ≤ constraints. ​ Significance: Indicates unused resources (e.g., extra machine hours available). 6. Sensitivity Report ​ Binding Constraint: Fully utilized; shadow price > 0. ​ Non-Binding Constraint: Not fully utilized; shadow price = 0. 7. Feasible Region ​ Definition: The area where all constraints overlap. ​ Optimal Solution: Found at an extreme point (corner) of the feasible region. Advanced Questions Based on Lecture Notes and Textbook (Chapters 4, 6, and Notes 3, 4, 5) 1. Objective Function in Production Planning A factory produces two products, P1P_1 and P2P_2, with unit production costs of $20 and $15, respectively. Inventory holding costs are $2/unit for P1P_1 and $3/unit for P2P_2. What is the objective function to minimize total cost? a. Z=20x1+15x2+2s1+3s2Z = 20x_1 + 15x_2 + 2s_1 + 3s_2​ b. Z=20x1+15x2−2s1−3s2Z = 20x_1 + 15x_2 - 2s_1 - 3s_2​ c. Z=20x1+15x2+s1+s2Z = 20x_1 + 15x_2 + s_1 + s_2​ d. Z=20x1+15x2+2s1Z = 20x_1 + 15x_2 + 2s_1 Answer: a. Z=20x1+15x2+2s1+3s2Z = 20x_1 + 15x_2 + 2s_1 + 3s_2. 2. Feasibility of a Solution Which of the following is a feasible solution for the constraint 3x1+4x2≤123x_1 + 4x_2 \leq 12? a. x1=2,x2=1x_1 = 2, x_2 = 1​ b. x1=1,x2=2x_1 = 1, x_2 = 2​ c. x1=3,x2=1x_1 = 3, x_2 = 1​ d. x1=4,x2=0x_1 = 4, x_2 = 0 Answer: b. x1=1,x2=2x_1 = 1, x_2 = 2 (yields 3(1)+4(2)=113(1) + 4(2) = 11, which satisfies the constraint). 3. Transportation Problem A company ships from two warehouses (W1 and W2) to three stores (S1, S2, S3). The supply at W1 is 500 units, and W2 is 300 units. The demand at S1, S2, and S3 is 200, 300, and 300 units, respectively. What is the constraint for supply at W1? a. x11+x12+x13≤300x_{11} + x_{12} + x_{13} \leq 300​ b. x11+x12+x13≥500x_{11} + x_{12} + x_{13} \geq 500​ c. x11+x12+x13≤500x_{11} + x_{12} + x_{13} \leq 500​ d. x11+x21+x31≤500x_{11} + x_{21} + x_{31} \leq 500 Answer: c. x11+x12+x13≤500x_{11} + x_{12} + x_{13} \leq 500. 4. Shadow Price A shadow price of 5 for a machine capacity constraint indicates: a. Increasing machine capacity by one unit decreases total cost by $5.​ b. Increasing machine capacity by one unit increases total cost by $5.​ c. Increasing machine capacity by one unit increases profit by $5.​ d. This constraint is non-binding, so the shadow price does not apply. Answer: c. Increasing machine capacity by one unit increases profit by $5. 5. Alternate Optimal Solutions When do alternate optimal solutions exist in linear programming? a. When the feasible region is unbounded.​ b. When the problem is infeasible.​ c. When the objective function is parallel to a binding constraint.​ d. When slack variables are introduced. Answer: c. When the objective function is parallel to a binding constraint. 6. Transshipment Problem In a transshipment model, what does the constraint xin+xjn=xnk+xnlx_{in} + x_{jn} = x_{nk} + x_{nl} enforce? a. The supply at a node equals demand.​ b. The total flow into a node equals total flow out.​ c. The shipments from all origins equal shipments to all destinations.​ d. The total demand is satisfied by the total supply. Answer: b. The total flow into a node equals total flow out. 7. Infeasibility What causes a linear programming model to become infeasible? a. The objective function is maximized instead of minimized.​ b. There is no feasible region where all constraints overlap.​ c. There are multiple alternate optimal solutions.​ d. The problem is unbounded. Answer: b. There is no feasible region where all constraints overlap. 8. Feasible Region For the constraint 4x1+6x2≥244x_1 + 6x_2 \geq 24, which solution is infeasible? a. x1=3,x2=0x_1 = 3, x_2 = 0.​ b. x1=4,x2=0x_1 = 4, x_2 = 0.​ c. x1=2,x2=3x_1 = 2, x_2 = 3.​ d. x1=0,x2=5x_1 = 0, x_2 = 5. Answer: d. x1=0,x2=5x_1 = 0, x_2 = 5 (results in 4(0)+6(5)=304(0) + 6(5) = 30, which does not meet ≥24\geq 24). These questions should help you practice and prepare for a quiz with both conceptual and practical challenges. Let me know if you’d like even more examples or deeper explanations for specific topics!