Question

The Brown Company has two warehouses and three retail outlets. Warehouse number one (which will be denoted by \(\mathrm{W}_{1}\) ) has a capacity of 12 units; warehouse number two \(\left(\mathrm{W}_{2}\right)\) holds 8 units. These warehouses must ship the product to the three outlets, denoted by \(\mathrm{O}_{1}, \mathrm{O}_{2}\), and \(\mathrm{O}_{3} \cdot \mathrm{O}_{1}\) requires 8 units. \(\mathrm{O}_{2}\) requires 7 units, and \(\mathrm{O}_{3}\) requires 5 units. Thus, there is a total storage capacity of 20 units, and also a demand for 20 units. The question is, which warehouse should ship how many units to which outlet? (The objective being, of course, to accomplish this at the least possible cost.) Costs of shipping from either warehouse to any of the outlets are known and are summarized in the following table, which also sets forth the warehouse capacities and the needs of the retail outlets: $$ \begin{array}{|c|c|c|c|c|} \hline & \mathrm{O}_{1} & \mathrm{O}_{2} & \mathrm{O}_{3} & \text { Capacity } \\\ \hline \mathrm{W}_{1} \ldots \ldots \ldots \ldots . . & \$ 3.00 & \$ 5.00 & \$ 3.00 & 12 \\ \hline \mathrm{W}_{2} \ldots \ldots \ldots \ldots . . & 2.00 & 7.00 & 1.00 & 8 \\\ \hline \text { Needs (units) } & 8 & 7 & 5 & \\ \hline \end{array} $$

Short answer

The optimal shipping arrangement to minimize the total cost is: \(x_{11} = 8, x_{12} = 4, x_{13} = 0, x_{21} = 0, x_{22} = 3, x_{23} = 5\). The minimum total cost is $Z = 3(8) + 5(4) + 3(0) + 2(0) + 7(3) + 1(5) = 24 + 20 + 21 + 5 = 70 $. So, \(\mathrm{W}_{1}\) should ship 8 units to \(\mathrm{O}_{1}\) and 4 units to \(\mathrm{O}_{2}\), while \(\mathrm{W}_{2}\) should ship 3 units to \(\mathrm{O}_{2}\) and 5 units to \(\mathrm{O}_{3}\).

Step-by-step solution

Define the objective function

We want to minimize the overall shipping cost. Let \(x_{ij}\) be the number of units shipped from warehouse \(i\) to outlet \(j\) for \(i = 1,2\) and \(j = 1,2,3\). Then, the objective function to minimize is the total shipping cost, given by: \(Z = 3x_{11} + 5x_{12} + 3x_{13} + 2x_{21} + 7x_{22} + 1x_{23}\)

Define the constraints

We must also consider the constraints on capacities and needs of the warehouses and outlets, respectively. Warehouse capacity constraints: 1. \(x_{11} + x_{12} + x_{13} \le 12\) (for \(\mathrm{W_{1}}\)) 2. \(x_{21} + x_{22} + x_{23} \le 8\) (for \(\mathrm{W_{2}}\)) Outlet demand constraints: 3. \(x_{11} + x_{21} \ge 8\) (for \(\mathrm{O_{1}}\)) 4. \(x_{12} + x_{22} \ge 7\) (for \(\mathrm{O_{2}}\)) 5. \(x_{13} + x_{23} \ge 5\) (for \(\mathrm{O_{3}}\)) And the non-negativity constraints: 6. \(x_{ij} \ge 0\) for all \(i = 1,2\) and \(j = 1,2,3\)

Solve the linear programming problem

At this point, we have formulated this problem as a linear programming problem with an objective function (step 1), constraints (step 2), and non-negativity requirements (step 6). Since there are 6 unknowns (\(x_{11}, x_{12}, x_{13}, x_{21}, x_{22}, x_{23}\)) and 6 constraints, we can use linear programming techniques, such as the Simplex method or graphing, to solve this problem and determine the optimal shipping arrangement that minimizes the total cost. Note that sometimes, the final solution might not be a whole number, in which case it is necessary to perform further analysis to find the optimal integer solution.

Key concepts

Transportation Problem

In mathematics and logistics, a transportation problem is a type of linear programming issue where the aim is to determine the optimal way to transport goods from multiple sources (such as warehouses) to multiple destinations (like retail outlets) while minimizing the transportation cost.
This problem is fundamental in supply chain management; it helps decision-makers allocate resources effectively to meet specific demands without exceeding capacity constraints.
For the Brown Company, the transportation problem involves two warehouses with fixed capacities, and three retail outlets each with specific needs. To solve the transportation problem, you identify the amount each warehouse should ship to each outlet to ensure all demands are met without any warehouse exceeding its storage capable shipment.
The relationships and constraints involved make the transportation problem a unique subclass of linear programming problems. Understanding this problem helps businesses improve their logistics and distribution strategy, ensuring both cost-effectiveness and efficiency.

Optimization

Optimization in the context of the transportation problem is about finding the best solution or result, typically the one that requires the least resources or incurs the least cost.
It involves developing mathematical models that represent the problem scenario, including the objective function and constraints, and then finding the most efficient way to satisfy those conditions.
In our exercise, the objective is to minimize the overall shipping cost. This is accomplished through setting up an objective function: a mathematical equation representing the total shipping cost.

• The objective function includes variables like costs per unit shipped from different warehouses to different outlets.
• Constraints ensure that the solution meets all capacity and demand requirements.

Optimization techniques, like the Simplex Method or intuitive allocation methods, are applied to find the shipping plan that allows the Brown Company to fulfill all outlet demands while keeping costs low.

Cost Minimization

Cost minimization is the central goal of solving the transportation problem.
In logistics, minimizing costs entails determining how to supply goods to various destinations most cost-effectively.
The problem is boiled down to balancing the cost of shipping goods from two warehouses to three retail outlets with the given transport costs. Costs vary based on different routes taken by shipments.

• We need to calculate the total cost for each potential route and find a combination where these costs are as low as possible.
• The challenge is ensuring no warehouse oversteps its capacity while all retail outlets receive the units they need.

Formulating an equation or model of the problem helps in recognizing these patterns and constraints, providing a clear pathway to achieve the minimal cost solution that meets all requirements.

Simplex Method

The Simplex Method is a popular algorithm used for solving linear programming problems like the transportation problem outlined here.
It iteratively tests the vertices of the feasible region defined by constraints, checking for the optimal solution along the way.
In our scenario, the Simplex Method becomes invaluable in navigating through different permutations of shipping quantities from warehouses to outlets until the least cost configuration is uncovered.

• Starting from a vertex or potential solution, usually the basic feasible solution, the method evaluates adjacent vertices iteratively.
• Decisions are based on the cost coefficient associated with each shipping route, progressively minimizing the overall cost.

By applying the Simplex Method, effectively manage multiple variables and constraints at play, drawing closer to the ideal shipping arrangement that aligns with the cost minimization strategy inherent to the transportation problem.