Consider a supply chain network for a product with n supplier, m manufacturer, and p warehouse nodes. The unit shipment costs from supplier i to plant j are represented by cij and plant j to warehouse k are represented by cjk. Assume that the product flow is from suppliers to plants to warehouses. In addition, the same part is sourced from all suppliers and there is a one to one relationship between the sourced part and the final product produced at the manufacturing plants.
Given the capacities at suppliers and plants, and demands at warehouses, formulate a mixed-integer programming model that minimizes the total cost in satisfying demand based on the following requirements:
(i) Assume that suppliers and plants are possible choice nodes with fixed costs f_{i} and f_{j }associated in selecting supplier i and plant j, respectively
(ii) Assume that the total supply is greater than total demand
(iii) The optimization model needs to minimize both transportation/transhipment and fixed costs in satisfying demand. Clearly define the decision variables and express the objective function and constraints in general mathematical form.