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Question: The costs of placing an order is $150. It is estimated that 1,000 units will be used in the next 12 months. The carrying cost per unit per month is$2.50.

a. Compute the optimum order size

b. Now suppose that the company could lower the ordering cost to $50; the cost of the effort to accomplish this change is$1,000. Assume the product will be sold only for the next two years. Compute the new optimal order quantity if the lower ordering cost were implemented, and determine if the company should invest $1,000 in the ordering-cost reduction program. Solution: (a) The following is obtained: Hence, the optimal order quantity is $EOQ=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2\times 1000\times 150}{2.5}}=346.41$ The total inventory cost is TC =$866.03.

(b) IF the setup cost was reduced to $50, we would get In this case, the optimal order quantity is EOQ = 200, and the total inventory cost is$500. The savings in cost for 2 years operation are (866.03 – 500)*2 =$732.06, which is less than$1000 (the cost of reducing the setup cost), and hence, the ordering-cost reduction program should not be pursued.

Question: A logistics specialist for Wiethoff Inc. must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are:

What are the total monthly transportation costs for the optimal solution?

Solution: We have 9 variables $${{x}_{ij}}$$, where $${{x}_{ij}}$$ represents the amount that is sent from factoryto assembly plant. For convenience of notation, we name the factories A, B and C as 1, 2 and 3.

The problem is written as

\begin{align} & \text{Minimize }\sum\limits_{i,j=1}^{3}{{{C}_{ij}}{{X}_{ij}}} \\ & \text{subject to }{{X}_{11}}+{{X}_{12}}+{{X}_{13}}\le 200 \\ & \text{ }{{X}_{21}}+{{X}_{22}}+{{X}_{23}}\le 400 \\ & \text{ }{{X}_{31}}+{{X}_{32}}+{{X}_{33}}\le 200 \\ & \text{ }{{X}_{11}}+{{X}_{21}}+{{X}_{31}}=120 \\ & \text{ }{{X}_{12}}+{{X}_{22}}+{{X}_{32}}=620 \\ & \text{ }{{X}_{13}}+{{X}_{23}}+{{X}_{33}}=60 \\ & \text{ }{{X}_{ij}}\ge 0 \\ \end{align}

We know solve this problem using Excel’s solver. The output is shown below:

 Plants Variables 1 2 3 1 2 3 A 5 9 16 A 0 200 0 Factories B 1 2 6 B 0 400 0 C 2 8 7 C 120 20 60 1 2 3 $${{\mathbf{c}}_{ij}}{{\mathbf{x}}_{ij}}$$ 1 0 1800 0 2 0 800 0 Objective 3420 3 240 160 420 Restrictions Supply Slack 1 0 200 Slack 2 0 400 Slack 3 0 200 Demand Slack 4 0 Slack 5 0 Slack 6 0

This means that 120 cases of parts are sent from factory C to Plant 1. The total monthly transportation costs are \$3,420.