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Chapter 11: Problem 14

Describe the three steps in solving a linear programming problem.

Short Answer

Expert verified
The three steps in solving a linear programming problem are as follows: Step 1: Formulate the Linear Programming Problem, which involves identifying the objective function, decision variables, and linear constraints. The objective function represents what we aim to maximize or minimize, the decision variables are the unknowns that influence the objective function, and the linear constraints represent the restrictions of the problem. Step 2: Graph the Feasible Region by plotting the linear constraints on a coordinate plane. The feasible region is the area where all constraints are satisfied simultaneously. This region holds all potential solutions. Step 3: Identify the Optimal Solution within the feasible region. For maximization problems, find the point(s) in the region where the objective function has the highest value. For minimization problems, seek the point(s) where the objective function has the lowest value. These points are the optimal solutions. Even though, sometimes, there can exist multiple optimal solutions or no solution at all based on the specifics of the feasible region and the problem.

Step by step solution

01

Formulate the Linear Programming Problem

To solve a linear programming problem, the first step is to formulate the problem by identifying the objective function, decision variables, and the linear constraints. The objective function represents the quantity (such as profit or cost) that needs to be maximized or minimized. The decision variables are the unknowns that we need to find that will influence the objective function, and the linear constraints represent the limitations of the problem.
02

Graph the Feasible Region

Once the problem is formulated, plot the feasible region on a coordinate plane by graphing the linear constraints as equations. The feasible region is the area where all the constraints are satisfied simultaneously, often in the form of a polygon. It contains all the possible solutions to the linear programming problem.
03

Identify the Optimal Solution

The last step is to identify the optimal solution within the feasible region. If the problem is to maximize the objective function, find the vertex or vertices in the feasible region where the objective function has the highest value. If the problem is to minimize the objective function, find the vertex or vertices in the feasible region where the objective function has the lowest value. These points represent the optimal solution of the linear programming problem. Sometimes, there may be multiple optimal solutions or no solution at all, depending on the feasible region and the nature of the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Objective Function
The objective function in linear programming is a key concept that directs how the entire problem is approached. It represents the goal of the problem, typically involving maximizing or minimizing a specific quantity, such as profit, cost, or resource usage.
This function is constructed using the decision variables, which are essentially the factors that can be controlled or adjusted. For example, if a business aims to maximize profit through the sale of products, the objective function would be an expression involving the number of units for each product and their respective profits.
Writing the objective function correctly is essential, as any error can lead to aiming for the wrong goal. It is generally expressed like this:
\( Z = c_1x_1 + c_2x_2 + \ldots + c_nx_n \),
where \( Z \) is the quantity to be optimized, \( c_1, c_2, \dots, c_n \) are the coefficients that represent the contribution of each decision variable, and \( x_1, x_2, \dots, x_n \) are the decision variables themselves.
Decision Variables
Decision variables are the unknowns in a linear programming problem. They are the pieces of the puzzle that you aim to determine. These variables represent the items or actions that affect the outcome of the objective function.
In practice, decision variables must be chosen carefully, as they define what can be changed in the problem. For instance, if you're looking at a factory scenario, decision variables could be the number of units produced, the hours of labor, or the usage of raw materials. These quantities are expressed as \( x_1, x_2, \ldots, x_n \) in mathematical models.
Properly identifying decision variables ensures that you correctly model the situation and solve the problem effectively. They are crucial because they lay the foundation for defining both the objective function and the constraints.
Feasible Region
The feasible region in linear programming is the set of all possible points that satisfy the problem's constraints. It's essentially the solution space where an answer to the problem can exist. Visualized typically on a graph, the feasible region is the polygonal shape where all the constraints overlap.
Each corner or vertex of this region represents a potential solution, with one of these likely being the optimal solution. It is bounded by the lines created by the constraints, indicating which areas are permissible based on the problem's limitations.
It's crucial to correctly plot this region, as it guarantees that the solutions considered are viable and meet all problem requirements. If the feasible region is identified correctly, finding a reliable solution becomes much easier.
Optimal Solution
The optimal solution in linear programming is the point within or on the edge of the feasible region where the objective function achieves its best value. Depending on whether the goal is to maximize or minimize the function, the optimal solution will be the maximum or minimum value of the objective function.
In many cases, this is found at the vertices of the feasible region, which is typically a polygon. You evaluate the objective function at each vertex to determine which point gives the most favorable outcome.
Sometimes, there might be multiple vertices where the optimal value is the same, indicating multiple optimal solutions. In other instances, if the constraints are too restrictive or improperly defined, there might be no feasible region, and thus no solution at all. Ensuring accurate identification of this point ensures the problem-solving process is effective and reliable.

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Most popular questions from this chapter

Ace Cleaning Service is considering expanding into one or more new market areas. Which costs are relevant to Ace's decision on whether to expand?

Define relevant costs. Why are historical costs irrelevant?

(CMA, adapted) The Reward One Company manufactures windows. Its manufacturing plant has the capacity to produce 12,000 windows each month. Current production and sales are 10,000 windows per month. The company normally charges \(\$ 250\) per window. cost information for the current activity level is as follows: Reward One has just received a special one-time-only order for 2,000 windows at \(\$ 225\) per window. Accepting the special order would not affect the company's regular business or its fixed costs. Reward One makes windows for its existing customers in batch sizes of 100 windows \((100 \text { batches } \times 100 \text { windows per batch }=10,000\) windows). The special order requires Reward 0ne to make the windows in 25 batches of 80 windows. 1\. Should Reward One accept this special order? Show your calculations. 2\. Suppose plant capacity were only 11,000 windows instead of 12,000 windows each month. The special order must either be taken in full or be rejected completely. Should Reward One accept the special order? Show your calculations. 3\. As in requirement 1, assume that monthly capacity is 12,000 windows. Reward 0ne is concerned that if it accepts the special order, its existing customers will immediately demand a price discount of \(\$ 20\) in the month in which the special order is being filled. They would argue that Reward One's capacity costs are now being spread over more units and that existing customers should get the benefit of these lower costs. Should Reward One accept the special order under these conditions? Show your calculations.

"Cost written off as depreciation on equipment already purchased is always irrelevant." Do you agree? Why?

(CPA) Choose the best answer. 1\. The Cozy Company manufactures slippers and sells them at \(\$ 10\) a pair. Variable manufacturing cost is \(\$ 5.75\) a pair, and allocated fixed manufacturing cost is \(\$ 1.75\) a pair. It has enough idle capacity available to accept a one-time-only special order of 25,000 pairs of slippers at \(\$ 7.50\) a pair. Cozy will not incur any marketing costs as a result of the special order. What would the effect on operating income be if the special order could be accepted without affecting normal sales: (a) \(\$ 0,(b) \$ 43,750\) increase, \((c) \$ 143,750\) increase, or (d) \(\$ 187,500\) increase? Show your calculations. 2\. The Manchester Company manufactures Part No. 498 for use in its production line. The manufacturing cost per unit for 10,000 units of Part No. 498 is as follows: The Remnant Company has offered to sell 10,000 units of Part No. 498 to Manchester for \(\$ 71\) per unit. Manchester will make the decision to buy the part from Remnant if there is an overall savings of at least \(\$ 45,000\) for Manchester. If Manchester accepts Remnant's offer, S11 per unit of the fixed overhead allocated would be eliminated. Furthermore, Manchester has determined that the released facilities could be used to save relevant costs in the manufacture of Part No. \(575 .\) For Manchester to achieve an overall savings of \(\$ 45,000\) the amount of relevant costs that would have to be saved by using the released facilities in the manufacture of Part No. 575 would be which of the following: (a) \(\$ 30,000,(\mathrm{b}) \$ 115,000,(\mathrm{c}) \$ 125,000,\) or \((\mathrm{d}) \$ 100,000 ?\) Show your calculations. What other factors might Manchester consider before outsourcing to Remnant?
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