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Chapter 11: Problem 14

Describe the three steps in solving a linear programming problem.

Short Answer

Expert verified
Define decision variables, formulate the objective function, and set up the constraints.

Step by step solution

01

Define the Decision Variables

In a linear programming problem, the first step is to clearly define what your decision variables represent. These are the variables that you will adjust in order to achieve the best outcome in terms of maximizing or minimizing the objective function. For instance, if the problem is to maximize profit from selling two different products, your decision variables could be the quantities of each product to produce.
02

Formulate the Objective Function

Next, create an objective function that represents the goal of the problem, such as maximizing profit or minimizing cost. The function is a linear combination of the decision variables. Using the same example, if product 1 and product 2 generate certain profits per unit, your objective function might be something like: \( Z = 50x + 30y \), where \(x\) and \(y\) are the quantities of products 1 and 2, and \(Z\) is the total profit.
03

Set Up the Constraints

Finally, determine the constraints that need to be satisfied in the problem. These are usually inequalities that represent limitations like resource availability, time, or budget. For instance, if the products require different amounts of a resource that is in limited supply, you would set up constraints like: \( 3x + 2y \leq 100 \), where \(3x + 2y\) represents the total resources used by the production of the two products, and 100 is the maximum available resources.

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Key Concepts

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

Decision Variables
In linear programming, decision variables are the heart of the problem-solving process. These variables are the parameters you will adjust to find the best possible outcome. Decision variables can represent a variety of things. They might be quantities of products to be produced, amounts of resources to be allocated, or hours of time to be scheduled. The key is that they are the variables you control and change during the optimization process.

For example, if you're solving a problem involving two products, your decision variables could be the numbers of each product to manufacture. If you label these as \( x \) and \( y \), you have a starting point for addressing how different scenarios impact results. Always remember, defining decision variables clearly is crucial because they form the basis for both the objective function and the constraints.

When choosing decision variables, ensure they are meaningful and relevant to the problem context. Clear definitions allow for smoother communication and analysis as the problem-solving progresses.
Objective Function
The objective function is the mathematical expression that represents your primary goal in the linear programming problem. Whether you're aiming to maximize profits or minimize costs, the objective function is what guides your decision-making process. It's formulated as a linear equation that involves the decision variables.

In the context of maximizing profit, an objective function might look like: \[ Z = 50x + 30y \] where \( x \) and \( y \) are the quantities of two products, and \( 50 \) and \( 30 \) are their respective profit contributions per unit.

This function tells you how your decision variables affect the overall goal. The coefficients of the decision variables indicate how much each unit contributes to the objective. By changing the values of \( x \) and \( y \), you can observe different profit scenarios and find the best one. The objective function should be simple to understand but reflect the complexities of the real-world situation you're modeling.
Constraints
Constraints in linear programming set the limitations within which the solution must fall. These are boundary conditions that the decision variables must adhere to, often modeled as linear inequalities. Constraints can arise from resource limitations, time restrictions, or budgetary caps.

For example, if you're producing two products, and both require a certain amount of a limited resource, the constraint might look like:\[ 3x + 2y \leq 100 \] Here, \( 3x + 2y \) denotes the total resource consumption by the production, and \( 100 \) is the maximum available resource units.

Constraints are essential as they define the feasible region within which the solution must lie. They help balance between different factors and ensure that the solution is realistic and applicable. Identifying the correct constraints ensures that the solution not only meets the objective function's aim but also adheres to real-world limits. It is important to carefully consider all potential constraints to avoid overlooking critical limitations that could skew the results.

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Most popular questions from this chapter

Sanchez Corporation runs two convenience stores, one in Connecticut and one in Rhode Island. Operating income for each store in 2012 is as follows: The equipment has a zero disposal value. In a senior management meeting, Maria Lopez, the management accountant at Sanchez Corporation, makes the following comment," Sanchez can increase its profitability by closing down the Rhode Island store or by adding another store like it." 1\. By closing down the Rhode Island store, Sanchez can reduce overall corporate overhead costs by \(\$ 44,000 .\) Calculate Sanchez's operating income if it closes the Rhode Island store. Is Maria Lopez's statement about the effect of closing the Rhode Island store correct? Explain. 2\. Calculate Sanchez's operating income if it keeps the Rhode Island store open and opens another store with revenues and costs identical to the Rhode Island store (including a cost of \(\$ 22,000\) to acquire equipment with a one- year useful life and zero disposal value). Opening this store will increase corporate overhead costs by \(\$ 4,000\). Is Maria Lopez's statement about the effect of adding another store like the Rhode Island store correct? Explain.

Bob Moody manages the Knoxville plant of George Manufacturing. He has been approached by a representative of Darda Engineering regarding the possible replacement of a large piece of manufacturing equipment that George uses in its process with a more efficient model. While the representative made some compelling arguments in favor of replacing the 3-year old equipment, Moody is hesitant. Moody is hoping to be promoted next year to manager of the larger Chicago plant, and he knows that the accrual-basis net operating income of the Knoxville plant will be evaluated closely as part of the promotion decision. The following information is available concerning the equipment replacement decision: \(\bullet\)The historic cost of the old machine is \(\$ 300,000\). It has a current book value of \(\$ 120,000,\) two remaining years of useful life, and a market value of \(\$ 72,000\). Annual depreciation expense is \(\$ 60,000\). It is expected to have a salvage value of \(\$ 0\) at the end of its useful life. \(\bullet\)The new equipment will cost \(\$ 180,000\). It will have a two-year useful life and a \(\$ 0\) salvage value. George uses straight-line depreciation on all equipment. \(\bullet\)The new equipment will reduce electricity costs by \(\$ 35,000\) per year, and will reduce direct manufacturing labor costs by \(\$ 30,000\) per year. For simplicity, ignore income taxes and the time value of money. 1\. Assume that Moody's priority is to receive the promotion, and he makes the equipment replacement decision based on next year's accrual-based net operating income. Which alternative would he choose? Show your calculations. 2\. What are the relevant factors in the decision? Which alternative is in the best interest of the company over the next two years? Show your calculations. 3\. At what cost of the new equipment would Moody be willing to purchase it? Explain.

How might the optimal solution of a linear programming problem be determined?

(H. Schaefer) The Wild Boar Corporation is working at full production capacity producing 13,000 units of a unique product, Rosebo. Manufacturing cost per unit for Rosebo is as follows: Manufacturing overhead cost per unit is based on variable cost per unit of \(\$ 4\) and fixed costs of \(\$ 39,000\) (at full capacity of 13,000 units). Marketing cost per unit, all variable, is \(\$ 2,\) and the selling price is \(\$ 26\). A customer, the Miami Company, has asked Wild Boar to produce 3,500 units of Orangebo, a modification of Rosebo. Orangebo would require the same manufacturing processes as Rosebo. Miami has offered to pay Wild Boar \(\$ 20\) for a unit of 0 rangebo and share half of the marketing cost per unit. 1\. What is the opportunity cost to Wild Boar of producing the 3,500 units of Orangebo? (Assume that no overtime is worked. 2\. The Buckeye Corporation has offered to produce 3,500 units of Rosebo for Wolverine so that Wild Boar may accept the Miami offer. That is, if Wild Boar accepts the Buckeye offer, Wild Boar would manufacture 9,500 units of Rosebo and 3,500 units of Orangebo and purchase 3,500 units of Rosebo from Buckeye. Buckeye would charge Wild Boar \(\$ 18\) per unit to manufacture Rosebo. 0 n the basis of financial considerations alone, should Wild Boar accept the Buckeye offer? Show your calculations. 3\. Suppose Wild Boar had been working at less than full capacity, producing 9,500 units of Rosebo at the time the Miami offer was made. Calculate the minimum price Wild Boar should accept for Orangebo under these conditions. (lgnore the previous \(\$ 20\) selling price.)

Define opportunity cost.
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