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!