Finite Mathematics Quiz: Shadow And Binding Constraints
20 questions · exam conditions
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Shadow And Binding ConstraintsQuestion 1 of 20

A company's linear programming model has four constraints. At the optimal solution, constraints 1 and 3 have positive slack values, constraint 2 is satisfied as an equality with a shadow price of $15, and constraint 4 is satisfied as an equality with a shadow price of $0. If the right-hand side of constraint 4 were increased by 2 units, what would happen to the optimal objective value?

It would increase by $30 because constraint 4 is binding and has been relaxed by 2 units.
It would remain unchanged because constraint 4 has a shadow price of zero despite being binding.
It would decrease because increasing a constraint's right-hand side always makes the problem more restrictive.
It would increase by an unknown amount because the shadow price only applies to infinitesimal changes.
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Finite Mathematics Quiz

Finite Mathematics Quiz: Shadow And Binding Constraints

Practice Shadow And Binding Constraints in Finite Mathematics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Shadow And Binding Constraints, giving you a quick way to practice the rules, question types, and explanations that matter most for Finite Mathematics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A company's linear programming model has four constraints. At the optimal solution, constraints 1 and 3 have positive slack values, constraint 2 is satisfied as an equality with a shadow price of $15, and constraint 4 is satisfied as an equality with a shadow price of $0. If the right-hand side of constraint 4 were increased by 2 units, what would happen to the optimal objective value?

  1. It would increase by $30 because constraint 4 is binding and has been relaxed by 2 units.
  2. It would remain unchanged because constraint 4 has a shadow price of zero despite being binding. (correct answer)
  3. It would decrease because increasing a constraint's right-hand side always makes the problem more restrictive.
  4. It would increase by an unknown amount because the shadow price only applies to infinitesimal changes.
Explanation: Constraint 4 is binding (satisfied as equality) but has a shadow price of zero, meaning additional relaxation of this constraint provides no benefit to the objective function. The optimal value remains unchanged. Choice A incorrectly assumes positive shadow price. Choice C misunderstands that increasing RHS typically relaxes constraints. Choice D is incorrect because shadow prices apply to small discrete changes, not just infinitesimal ones, and here the shadow price is zero regardless.

Question 2

A linear programming solution shows that the optimal point lies at the intersection of constraints A and B, both of which are satisfied as equalities. Constraint C has a slack of 5 units. If the shadow price of constraint A is $12 and the shadow price of constraint B is $8, and we simultaneously increase the right-hand side of constraint A by 1 unit and decrease the right-hand side of constraint B by 1 unit, what is the expected change in the optimal objective value?

  1. An increase of $4, calculated as the difference between the shadow prices of the two constraints.
  2. An increase of $20, calculated as the sum of the absolute values of both shadow prices.
  3. No change, because the simultaneous modifications to binding constraints will cancel each other out.
  4. The change cannot be determined without knowing whether the modifications keep the current basis optimal. (correct answer)
Explanation: When multiple binding constraints are modified simultaneously, the shadow prices cannot simply be added or subtracted because they assume ceteris paribus conditions. The combined effect depends on whether the current basis remains optimal after both changes, which requires checking the range of validity for simultaneous changes. Choice A incorrectly treats shadow prices as additive. Choice B incorrectly sums absolute values. Choice C incorrectly assumes cancellation occurs with any simultaneous changes.

Question 3

A linear program has been solved with three decision variables and five constraints. The optimal tableau shows that constraints 2 and 4 correspond to basic variables in the final solution, while constraints 1, 3, and 5 correspond to non-basic variables. Given this information about the final tableau structure, which constraints are binding at the optimal solution?

  1. Constraints 2 and 4 are binding because they correspond to basic variables in the optimal tableau.
  2. Constraints 1, 3, and 5 are binding because they correspond to non-basic variables with zero values.
  3. Constraints 1, 3, and 5 are binding because their slack variables are non-basic (equal to zero). (correct answer)
  4. All constraints are binding because the optimal solution has been reached at a vertex of the feasible region.
Explanation: When slack variables are non-basic (equal to zero), it means the original constraints are satisfied as equalities, making them binding. Constraints 1, 3, and 5 have non-basic slack variables, so these constraints are binding. Choice A confuses the role of basic/non-basic variables. Choice B incorrectly identifies which constraints are binding. Choice D is wrong because not all constraints need to be binding at an optimal vertex.

Question 4

A company's optimal production plan uses all available skilled labor (800 hours) and raw materials (1200 kg), but only 600 of the 900 available machine hours. The shadow prices are: skilled labor $45/hour, raw materials $12/kg, machine hours $0/hour. Due to a supply chain disruption, raw materials availability will decrease to 1000 kg. Assuming the shadow price remains valid, what is the most accurate description of the impact?

  1. Profit will decrease by $2400, and the company should prioritize finding alternative material suppliers over acquiring additional machine capacity. (correct answer)
  2. Profit will decrease by $2400, but the company might be able to compensate by utilizing some of the 300 hours of unused machine capacity.
  3. Profit will decrease by $2400, and this loss is definitive because raw materials are a binding constraint in the current solution.
  4. The profit impact will be less than $2400 because the disruption might make machine hours binding, which could partially offset the materials shortage.
Explanation: The profit decrease is $12/kg × 200 kg = $2400. Since machine hours currently have zero shadow price (excess capacity exists), acquiring more machine capacity provides no benefit. The company should focus on material suppliers. Choice B incorrectly suggests unused machine capacity can compensate for material shortage - these are different constraint types. Choice C is correct about the $2400 but adds unnecessary absolute language. Choice D incorrectly suggests machine hours becoming binding would help offset material shortage.

Question 5

In a production planning problem, the optimal solution utilizes the entire capacity of Machine A (500 hours) and Machine B (300 hours), while Machine C has 50 hours of unused capacity out of 400 available hours. The shadow price analysis shows: Machine A has a shadow price of $25/hour, Machine B has a shadow price of $40/hour, and Machine C has a shadow price of $0/hour. If the company can rent additional capacity for Machine A at $30/hour or Machine B at $35/hour, which decision makes economic sense?

  1. Rent additional capacity for Machine A because it has the higher absolute capacity utilization rate.
  2. Rent additional capacity for Machine B because its shadow price of $40 exceeds the rental cost of $35. (correct answer)
  3. Rent additional capacity for both machines because both have positive shadow prices indicating binding constraints.
  4. Do not rent additional capacity for either machine because the rental costs exceed the shadow prices for both.
Explanation: Machine B should be rented because its shadow price (40/hour)exceedstherentalcost(40/hour) exceeds the rental cost (35/hour), providing a net benefit of 5/hour.MachineAshouldnotberentedbecauseitsrentalcost(5/hour. Machine A should not be rented because its rental cost (30/hour) exceeds its shadow price ($25/hour). Choice A incorrectly focuses on capacity utilization rather than economic value. Choice C ignores the cost comparison. Choice D incorrectly states that both rental costs exceed shadow prices.

Question 6

In a profit-maximization linear programming problem, the constraint corresponding to the availability of a specific raw material has a shadow price of $0. Which of the following statements must be true?

  1. The raw material is not used at all in the optimal production plan.
  2. The constraint for the raw material is binding at the optimal solution.
  3. There is a surplus of the raw material at the optimal solution. (correct answer)
  4. The cost of acquiring more of the raw material is prohibitively expensive.
Explanation: A shadow price of 00 for a constraint implies that the constraint is non-binding. For a 'less than or equal to' resource constraint, being non-binding means that not all of the available resource is being used at the optimal solution. Therefore, there must be a surplus (or slack) of that raw material.

Question 7

A company maximizes profit P=60x+70yP = 60x + 70y subject to constraints on two resources, R1 and R2:

R1:x+2y≤400R1: x + 2y \le 400 R2:4x+3y≤1200R2: 4x + 3y \le 1200

The optimal solution is found to be x=300,y=0x=300, y=0. The company is considering acquiring more of one of the resources. Which statement accurately describes the situation?

  1. Increasing the availability of resource R1 is the only action that could increase profit.
  2. Increasing the availability of resource R2 is the only action that could increase profit. (correct answer)
  3. Increasing the availability of either resource R1 or R2 could potentially increase profit.
  4. Neither resource is fully utilized, so acquiring more of either will not increase profit.
Explanation: First, check which constraints are binding at the optimal solution (300,0)(300, 0). For R1: 300+2(0)=300300 + 2(0) = 300. Since 300<400300 < 400, this constraint is non-binding. There is a slack of 400−300=100400 - 300 = 100 units of R1. For R2: 4(300)+3(0)=12004(300) + 3(0) = 1200. Since 1200=12001200 = 1200, this constraint is binding. Because R1 is not fully utilized (non-binding), acquiring more of it will not increase profit. Because R2 is fully utilized (binding), it is a bottleneck. Acquiring more of R2 could allow for a change in the production plan and an increase in profit.

Question 8

A craft brewery produces an IPA (II) and a Stout (SS). The process is limited by total brewing capacity (I+S≤1000I + S \le 1000 gallons) and fermentation tank space (I≤800I \le 800, S≤500S \le 500). Marketing estimates maximum sales of 700 gallons of IPA and 400 gallons of Stout. At the profit-maximizing production level, the brewery uses all of its brewing capacity. It also finds that it could have sold more of both types of beer if it had been able to produce them. Which of the following constraints must be non-binding?

  1. The constraint on total brewing capacity (I+S≤1000I + S \le 1000).
  2. The constraint on maximum sales of IPA (I≤700I \le 700). (correct answer)
  3. Both the IPA sales constraint and the Stout sales constraint must be binding.
  4. All production and sales constraints must be binding at the optimal point.
Explanation: The problem states that the brewery 'uses all of its brewing capacity,' which means the constraint I+S≤1000I + S \le 1000 is met as an equality (I+S=1000I + S = 1000), so it is binding. The problem also states the brewery 'could have sold more of both types of beer.' This means that the optimal production quantity for IPA, IoptI_{opt}, is less than the maximum sales of 700, and the optimal quantity for Stout, SoptS_{opt}, is less than 400. Because Iopt<700I_{opt} < 700, the constraint I≤700I \le 700 has slack and is therefore non-binding.

Question 9

In a standard maximization linear programming problem, if the ii-th constraint is non-binding at the optimal solution, what does the complementary slackness theorem imply about the corresponding variable in the dual problem?

  1. The corresponding dual variable must be positive.
  2. The value of the corresponding dual variable is equal to the slack in the primal constraint.
  3. The corresponding dual constraint must also be non-binding.
  4. The corresponding dual variable must be equal to zero. (correct answer)
Explanation: When you encounter questions about complementary slackness in linear programming, you're dealing with a fundamental relationship between primal and dual problems that connects constraint binding status with variable values. The complementary slackness theorem states two key conditions: (1) if a primal constraint has positive slack (is non-binding), then the corresponding dual variable equals zero, and (2) if a dual variable is positive, then the corresponding primal constraint must be binding (have zero slack). Since the problem tells us the ii-th constraint is non-binding at the optimal solution, it has positive slack. By complementary slackness, this immediately means the corresponding dual variable must equal zero. Looking at why the other answers miss the mark: Choice A contradicts the complementary slackness theorem directly—a non-binding primal constraint cannot correspond to a positive dual variable. Choice B confuses the relationship between slack and dual variables; while both relate to the same constraint, the dual variable value doesn't equal the slack amount. Choice C misunderstands the primal-dual relationship; the binding status of dual constraints relates to whether primal variables are positive or zero, not to the binding status of primal constraints. Remember this pattern: non-binding primal constraint ↔ zero dual variable and positive dual variable ↔ binding primal constraint. These complementary relationships are symmetric and absolute—there are no exceptions. When studying linear programming, practice identifying these connections by working through small examples where you can verify both the primal and dual solutions.

Question 10

A pet food company is creating a mix that must contain at least 15 grams of fiber per serving. The final, cost-minimizing mixture is found to contain exactly 15 grams of fiber. What is the most accurate conclusion regarding the fiber requirement constraint?

  1. The constraint is binding, implying that cost optimization would have preferred a lower fiber content. (correct answer)
  2. The constraint is binding, indicating its shadow price is zero.
  3. The constraint is non-binding because the company successfully met the minimum requirement.
  4. The constraint is non-binding, so adding more fiber will not increase the cost.
Explanation: When you encounter linear programming problems involving constraints, understanding the difference between binding and non-binding constraints is crucial for interpreting optimal solutions. A constraint is binding when the optimal solution uses up the entire allowable amount - it sits exactly on the constraint boundary. Here, the cost-minimizing mix contains exactly 15 grams of fiber, which means the solution lies precisely on the constraint line of "at least 15 grams." This constraint is actively limiting the company's choices. Since this is a cost-minimization problem and the solution lands exactly at the minimum fiber requirement, we can infer that the optimization process would have preferred even less fiber if possible. The constraint is forcing the company to include more fiber than would be ideal from a pure cost perspective. Choice A correctly identifies this binding constraint and its economic implication. Choice B incorrectly states the shadow price is zero - when a constraint is binding in optimization problems, it typically has a positive shadow price representing the cost of tightening that constraint. Choice C misunderstands terminology: meeting a requirement exactly makes the constraint binding, not non-binding. A non-binding constraint would mean the optimal solution naturally exceeded 15 grams without the constraint forcing it. Choice D also incorrectly labels this as non-binding and makes a false claim about cost effects. Study tip: Remember that "binding" means "actively constraining" - if your optimal solution sits exactly on a constraint boundary, that constraint is binding and affecting your optimization.

Question 11

In a linear programming problem, the optimal solution occurs at the point where three constraints intersect: 2x+3y=602x + 3y = 60, x+4y=50x + 4y = 50, and 5x+y=455x + y = 45. However, the feasible region is actually bounded by only the first two constraints, while the third constraint passes through the optimal point but doesn't limit the feasible region. Which statement best describes the constraint 5x+y=455x + y = 45?

  1. It is a binding constraint because it passes through the optimal solution point.
  2. It is a shadow constraint because it intersects the optimal point without being part of the active constraint set.
  3. It is a redundant constraint because it doesn't affect the shape of the feasible region at the optimum. (correct answer)
  4. It is a non-binding constraint because the optimal solution satisfies it as an equality by coincidence.
Explanation: A constraint is redundant when it passes through the optimal point but doesn't actually bound the feasible region - removing it wouldn't change the feasible region or optimal solution. Choice A is incorrect because binding constraints must actually limit the feasible region. Choice B misuses 'shadow constraint' - this isn't standard terminology in linear programming. Choice D is incorrect because non-binding constraints are satisfied as strict inequalities, not equalities, at the optimal solution.

Question 12

A manufacturing process is subject to constraints on labor hours and raw materials. The optimal solution to the profit-maximization problem results in a surplus of 100 kg of raw materials but uses all available labor hours. Which of the following actions is LEAST likely to increase the company's maximum possible profit?

  1. Securing 20 additional hours of labor.
  2. Negotiating a lower purchase price for the raw materials.
  3. Implementing a new technique that reduces the labor needed per product.
  4. Purchasing an additional 50 kg of raw materials. (correct answer)
Explanation: This question tests your understanding of linear programming and resource constraints in optimization problems. When a manufacturing process has surplus materials but uses all labor hours, labor is the "binding constraint" while raw materials are non-binding. The correct answer is D because purchasing additional raw materials won't increase profit when you already have a 100 kg surplus. Adding more of a resource you can't fully utilize is wasteful and provides no benefit to the optimization solution. Let's examine why the other options would likely increase profit: A) Securing additional labor hours would increase profit because labor is the binding constraint. Since all current labor hours are being used, more labor means you can produce more units and generate additional profit. B) Negotiating lower raw material prices reduces costs, which directly increases profit margins on each unit produced. Even though you have surplus materials, you're still purchasing them, so lower prices improve your bottom line. C) Implementing techniques that reduce labor requirements per product effectively increases your labor capacity. If each product needs less labor, you can produce more units with the same labor hours, increasing total profit. Remember this key principle: in linear programming, only changes to binding constraints (resources that are fully utilized) or cost reductions will improve the optimal solution. Adding more of a non-binding resource that you already have in surplus provides no additional value. Always identify which constraints are active versus which have slack when analyzing optimization problems.

Question 13

A factory's production plan is determined by a profit-maximization linear program. The constraint on assembly line time is binding with a shadow price of $80 per hour. The sensitivity analysis report indicates this shadow price is valid for an increase of up to 20 hours. If the factory manager can acquire an additional 30 hours of assembly line time, what is the guaranteed increase in maximum profit?

  1. The maximum profit will increase by exactly $2400.
  2. The maximum profit will increase by exactly $1600.
  3. The maximum profit is guaranteed to increase by at least $1600. (correct answer)
  4. The maximum profit will not change because the increase is outside the allowable range.
Explanation: The shadow price of $80 per hour is only valid for the first 20 additional hours. For this range, the profit will increase by 20 \times \80 = $1600$. The effect of the remaining 10 hours (from 20 to 30) cannot be determined from the given information. Beyond the 20-hour range, the basis of the optimal solution changes, and a new, lower shadow price would apply. Therefore, we can only guarantee an increase of at least $1600.

Question 14

A company manufactures two products, X and Y. The production process is subject to constraints on labor and materials. The linear programming model for maximizing profit includes the constraint for labor hours: 4X+2Y≤3004X + 2Y \le 300. The optimal solution is to produce X=50X=50 and Y=50Y=50. What can be concluded about the labor hours constraint at this optimal solution?

  1. The constraint is binding, as all available labor hours are being utilized. (correct answer)
  2. The constraint is non-binding, as there is a surplus of available labor hours.
  3. The constraint is binding, which implies its shadow price must be zero.
  4. The constraint is non-binding, because the production levels for X and Y are equal.
Explanation: To determine if the constraint is binding, substitute the optimal values of XX and YY into the constraint equation. The total labor hours used are 4(50)+2(50)=200+100=3004(50) + 2(50) = 200 + 100 = 300. Since the hours used (300) are equal to the hours available (300), the constraint is satisfied as an equality. This means the constraint is binding, and the resource is fully utilized.

Question 15

A manufacturer is solving a linear programming problem to maximize profit. The constraint for available machine time on a specialized cutting tool is found to be binding, with a shadow price of $45. What is the most accurate interpretation of this shadow price?

  1. Each hour of cutting tool time generates $45 in revenue for the company.
  2. The cost to run the cutting tool is $45 per hour at the optimal production level.
  3. Acquiring one additional hour of cutting tool time would increase maximum profit by $45, within a certain range. (correct answer)
  4. Reducing the use of the cutting tool by one hour would increase maximum profit by $45.
Explanation: The shadow price of a binding constraint in a maximization problem indicates the amount by which the objective function's optimal value will increase for a one-unit increase in the right-hand side of that constraint. This interpretation is only valid as long as the current basis remains optimal, which corresponds to a specific range of change. Therefore, an additional hour of machine time would increase the maximum possible profit by $45, assuming the change is within the allowable range.

Question 16

A portfolio manager uses a linear program to maximize returns subject to three constraints: (1) Total investment cannot exceed $500,000. (2) At least $100,000 must be in bonds. (3) The amount in stocks cannot exceed twice the amount in bonds.

The optimal solution invests $150,000 in bonds and $300,000 in stocks. The remaining capital is uninvested. What can be concluded about the shadow prices of the constraints?

  1. The shadow price for the total investment constraint is non-zero.
  2. The shadow prices for the bond minimum and the stock-to-bond ratio are both non-zero.
  3. All three shadow prices are zero because the solution is optimal.
  4. The shadow price for the stock-to-bond ratio constraint is non-zero. (correct answer)
Explanation: When you encounter linear programming problems, understanding shadow prices is crucial. A shadow price tells you how much the objective function would improve if you relaxed a constraint by one unit. Importantly, a constraint has a non-zero shadow price only when it's binding (fully utilized) at the optimal solution. Let's examine which constraints are binding in this solution. With $150,000 in bonds and $300,000 in stocks, the total investment is $450,000, which is $50,000 less than the $500,000 limit—so constraint (1) is not binding. The bond minimum requires at least $100,000, but we have $150,000 invested—so constraint (2) is not binding either. However, constraint (3) limits stocks to twice the bond amount: with $150,000 in bonds, the maximum allowed in stocks is exactly $300,000, which matches our solution. This constraint is binding. Since only the stock-to-bond ratio constraint is binding, only it has a non-zero shadow price. Answer D correctly identifies this. Answer A is wrong because the total investment constraint has slack ($50,000 uninvested), so its shadow price is zero. Answer B incorrectly claims the bond minimum constraint is binding—it's not, since we exceed the minimum by $50,000. Answer C misunderstands shadow prices entirely; optimal solutions often have non-zero shadow prices for binding constraints. Remember this pattern: shadow prices are non-zero only for binding constraints. Always check which constraints are fully utilized at the optimal solution to determine where shadow prices exist.

Question 17

A diet is being formulated to minimize cost while meeting nutritional requirements. One constraint requires the diet to provide at least 90 units of vitamin C: 5x+15y≥905x + 15y \ge 90. At the optimal solution, this constraint is binding, and its associated shadow price is $0.25. What is the correct interpretation?

  1. If the vitamin C requirement were increased by one unit, the minimum cost of the diet would decrease by $0.25.
  2. If the vitamin C requirement were increased by one unit, the minimum cost of the diet would increase by $0.25. (correct answer)
  3. The diet provides more vitamin C than required, so there is a surplus.
  4. Each unit of vitamin C in the final diet mix adds $0.25 to the total cost.
Explanation: The shadow price measures the change in the optimal objective function value for a one-unit increase in the constraint's right-hand side. Here, the objective is to minimize cost. The constraint is a minimum requirement (≥\ge). Increasing the requirement from 90 to 91 makes the constraint harder to satisfy, which can only increase or hold constant the minimum cost. A shadow price of $0.25 indicates that this increase of one unit in the requirement will cause the minimum cost to increase by $0.25.

Question 18

A linear programming problem has been solved, and sensitivity analysis reveals that the shadow price of constraint 3 remains valid for right-hand side values between 45 and 65, while the current right-hand side value is 55. The shadow price is $18 per unit. If a manager proposes changing the right-hand side of constraint 3 from 55 to 70, which analysis is most appropriate?

  1. The change would improve the objective by $270, calculated as $18 × (70 - 55), since we're within the validity range.
  2. The change would improve the objective by $270, but this assumes the shadow price remains constant beyond its validity range.
  3. The change would improve the objective by $180, calculated as $18 × (65 - 55), since we can only use the shadow price up to the range limit.
  4. The current shadow price is invalid for this change, so a complete re-optimization is needed to determine the impact. (correct answer)
Explanation: When you encounter sensitivity analysis questions in linear programming, the key concept is understanding the validity range of shadow prices. A shadow price tells you how much the objective function changes per unit increase in a constraint's right-hand side, but this rate is only guaranteed to hold within a specific range. In this problem, the shadow price of $18 per unit is only valid for right-hand side values between 45 and 65. The proposed change moves from 55 to 70, which goes beyond the upper limit of 65. Once you exceed the validity range, the shadow price becomes unreliable and may change significantly due to shifts in the optimal solution structure. Option A incorrectly applies the shadow price to the entire change (55 to 70) and wrongly claims this falls within the validity range. Option B makes the same calculation error but at least acknowledges that the shadow price assumption may not hold beyond the range—however, it still inappropriately uses the invalid shadow price. Option C attempts to limit the calculation to the valid range (55 to 65), but this approach is flawed because it assumes you can partially apply shadow prices and ignore changes beyond the range, which isn't how linear programming works. Option D correctly recognizes that when your proposed change exceeds the validity range, the current shadow price becomes unreliable, requiring complete re-optimization to determine the true impact. Strategy tip: Always check whether proposed changes fall within the shadow price validity range. If any part of the change goes outside these bounds, you need re-optimization—partial calculations using shadow prices won't give accurate results.

Question 19

A manufacturing company produces widgets and gadgets. The production is subject to three constraints: labor hours, material costs, and machine time. After solving the linear programming problem to maximize profit, the optimal solution uses all available labor hours and all available machine time, but only 80% of the available material budget. If the company could obtain one additional hour of labor at no cost, what can be concluded about the effect on the optimal profit?

  1. The optimal profit would definitely increase because labor is a binding constraint in the current solution.
  2. The optimal profit would remain unchanged because the material constraint is not binding in the current solution.
  3. The optimal profit might increase, but only if the shadow price of labor is positive at the current solution. (correct answer)
  4. The optimal profit would increase by exactly the shadow price of the labor constraint multiplied by the current labor allocation.
Explanation: Since labor hours are fully utilized (binding constraint), additional labor could potentially increase profit, but only if the shadow price is positive. The shadow price represents the marginal value of one additional unit of the constraint. Choice A is incorrect because being binding doesn't guarantee profit increase - the shadow price could be zero. Choice B is wrong because non-binding material constraint doesn't affect labor's potential impact. Choice D misunderstands shadow price - it's the rate of profit change per additional unit, not related to current allocation.

Question 20

Consider a transportation problem where the optimal solution ships goods along certain routes while leaving other routes unused. Routes A→1 and B→2 are used in the optimal solution, while routes A→2 and B→1 have zero shipments. In the context of binding constraints, which statement correctly describes the relationship between supply/demand constraints and the used routes?

  1. Supply constraints for A and B are binding because these origins are used in the optimal shipping pattern.
  2. The demand constraint for destination 1 is binding because route A→1 is used with positive flow.
  3. All supply and demand constraints are binding because transportation problems require all supply to be shipped and all demand to be met. (correct answer)
  4. Only the constraints corresponding to unused routes A→2 and B→1 are binding in the optimal solution.
Explanation: In standard transportation problems, all supply must be shipped and all demand must be satisfied, making all supply and demand constraints binding (satisfied as equalities) regardless of which routes are used. The route usage determines which route variables are basic/non-basic, not which supply/demand constraints are binding. Choices A and B incorrectly link constraint binding status to route usage. Choice D confuses route variables with supply/demand constraints.