# During the next four months, a customer requires, respectively, 500, 650, 1000, and 700 units of a commodity, and no backlogging is allowed (that is, the customer’s requirements must be met on time). Production costs are $50, $80, $40, and $70 per unit during these months. The storage cost from one month to the next is $20 per unit (assessed on ending inventory). It is estimated that each unit on hand at the end of Month 4 can be sold for $60. Assume there is no beginning inventory: ● Determine how to minimize the net cost incurred in meeting the demands for the next four months. ● Use SolverTable to see what happens to the decision variables and the total cost when the initial inventory varies from 0 to 1000 in 100-unit increments. ● How much lower would the total cost be if the company started with 100 units in inventory, rather than none? ● Would this same cost decrease occur for every 100 -unit increase in initial inventory?

To minimize the net cost incurred in meeting the demands for the next four months, we need to determine the optimal production and inventory levels. We can approach this problem using mathematical optimization techniques, specifically linear programming.

Let’s define our decision variables:

– Xi: The number of units produced in Month i (where i varies from 1 to 4).

– Yi: The number of units kept in inventory at the end of Month i (where i varies from 1 to 4).

Now, let’s set up our objective function and constraints:

Objective function:

Minimize the net cost incurred, which consists of production costs, storage costs, and the opportunity cost of holding inventory.

Net Cost = Σ(Ci * Xi) + Σ(20 * Yi) + (60 * Y4)

where Ci represents the production cost per unit in Month i.

Constraints:

1. The demand must be met for each month: Xi + Yi = Demandi

where Demandi represents the demand for Month i.

2. Inventory at the end of the fourth month must be zero: Y4 = 0

3. Non-negativity constraints: Xi ≥ 0, Yi ≥ 0

Now, we can set up the mathematical model and solve it using a linear programming solver or Excel Solver. By solving the model, we can obtain the optimal production and inventory levels for each month to minimize the net cost.

To analyze the impact of initial inventory on the total cost, we can use SolverTable in Excel. By varying the initial inventory from 0 to 1000 with 100-unit increments, we can observe how the decision variables (production and inventory levels) and the total cost change.

By setting the initial inventory to 100 units instead of none, we would expect the total cost to decrease. This is because having an initial inventory reduces the need for production in Month 1, resulting in lower production costs. Additionally, it decreases the need for storage and inventory buildup throughout the four months, leading to lower storage costs.

The cost decrease may not be the same for every 100-unit increase in initial inventory. It depends on the demand pattern and the production costs in each month. If the demand for one particular month is very high, a larger initial inventory may have a greater impact in reducing production and storage costs. On the other hand, if the demand for that month is low, the impact may be relatively smaller. Therefore, the cost decrease may vary depending on the specific demand and cost structure of the problem.

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