Civil CE152: and Environmental Engineering Systems Analysis
Midterm
Prof. Madanat and Sengupta
Fall 04
Problem 1 (6 points)
A standard problem in logistics is to determine the optimal size of shipments (i.e., the number
of items) to ship from a warehouse to a retailer, assuming constant demand. The manager seeks
the shipment size that provides the best tradeo between the inventory (or holding) cost and the
transport (or movement) cost.
The inventory cost (in $) is given by:
Ci = AV
where A is a variable cost per item in inventory, and V is the shipment size.
The transport cost (in $) is given by:
Ct = B/V
where B is a xed cost.
Solve for the optimal shipment size.
Problem 2 (6 points)
As production manager of a factory your task is to determine the monthly production quantity for
the next T months. The inventory It at the end of month t is:
It = It1 + Pt1 dt1 ,
t = 1, . . . , Pt T
where is the monthly production quantity and dt the monthly demand forecast. Your aim is to
avoid uctuation of the production quantity from month to month. This avoids labor shortages or
under-utilization. Thus the production plan should minimize the cost function:
T 1
T
cinv
Pt+1 Pt
It + cprod
t=1
t=0
Formulate the production planning problem as a linear program. Include all reasonable constraints
in your formulation. Do not include unnecessary constraints. Your factory is unable to hold more
than K units of inventory at the end of a given month. Make sure you state the meaning of any
additional variables you introduce.
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CE152
Midterm
Problem 3 (8 points)
Derive an optimal solution to the following linear program:
min
X1 ,X2 ,Y1 ,Y2
2(X1 + X2 )
subject to
Y1 = X1 2
X1 + Y2 = X2 + 3
X1 0
X2 0
Y1 0
Y2 0
Graphical or analytical derivations are acceptable.
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