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Question $6$

The SG electronics supplies its two retail outlets from its three warehouses. Warehouse A will distribute $45$ shipments next month. Warehouse B has not yet set its shipping schedule for next month but has the capacity to ship any amount up to a maximum of $50$ shipments. Warehouse C can only ship at most $20$ shipments to Retail outlet $1$ while it has no shipments limit to Retail outlet $2.$ Retail outlet $1$ has submitted its order for $75$ shipments for next month. Retail outlet $2$ needs a minimum of $45$ shipments next month but would be happy to receive more. The shipping costs at the three warehouses differ. The shipping cost per shipment from each warehouse to each retail outlet is given below, along with a summary of the other data.

The distribution manager, Alvin Chu, now needs to develop a plan for how many shipments to send from each warehouse to each of the retail outlets next month. His objective is to minimize the total shipping cost.

$\backslash$table$[[,$Unit Shipping Cost,$],[,$Retail Outlet $1,$Retail Outlet $2,$Supply$],[$Warehouse A$,$$$650,$$$400,=45$ shipments$],[$Warehouse B$,$$$800,$$$600,50$ shipments$],[$Warehouse C$,$$$830,$$$680,--]]$

$($a$)$ Identify the individual decisions that Alvin needs to make. For each of these decisions, define a decision variable to represent the decision.

$($b$)$ Write a mathematical expression for the total shipping cost in terms of the decision variables.

$($c$)$ Write a mathematical expression for each of the constraints on what the values of the decision variables can be$.$

$($d$)$ State a complete mathematical model for Alvin's problem.

$($e$)$ What do you think Alvin's shipping plan should be$?$ Explain your reasoning. Then express your shipping plan in terms of the decision variables.