Question

A small manufacturing company must determine how many units of their primary product should be produced during each of the next four quarters. Estimated demand for the product is as follows: first quarter, 40 units; second quarter, 60; third quarter, 75; fourth quarter, 25. The company must meet demand on time. At the beginning of the first quarter, the company has an inventory of 10 units. At the beginning of each quarter, the company must decide how many product units should be produced during the current quarter. For simplicity, assume that units manufactured during the quarter can be used to meet demand for the current quarter. During each quarter, the company can produce up to 40 product units at a cost of $400 per unit. By having employees work overtime during a quarter, the company can produce additional product units at a cost of $450 per unit. At the end of each quarter (after production has occurred and the current quarter’s demand has been satisfied), a carrying or holding cost of $20 per unit is incurred.

Formulate a balanced transportation problem to minimize the sum of production and inventory costs during the next four quarters by balancing the demand and supply. Find the optimal solution in R. Properly state the objective function and the constraints.

Answer 1

To minimize costs, one must balance the supply and demand across four quarters, taking into account production and inventory costs, with constraints such as production limits and inventory carryover.

Step-by-step explanation:

The student is tasked with formulating a balanced transportation problem to minimize production and inventory costs over four quarters, given demand levels, production capacities, and cost structures. The objective function will be to minimize the sum of production (regular and overtime costs) and inventory carrying costs. The constraints include satisfying quarterly demand, production capacity limits, and inventory balance from quarter to quarter.

For example, using the given information, if demand for the first quarter is 40 units and the company has an inventory of 10 units, they need to produce at least 30 more units to meet demand, considering production costs of $400 or $450 per unit for regular and overtime production, respectively, and carrying costs of $20 for the ending inventory. However, finding the optimal solution specifically in 'R' would involve using an appropriate package or algorithm, such as linear programming with the 'lpSolve' package.