Binders Beverage & Landscaping

To help Bill Binder pick the best route to transport his bottles from his plant to his warehouse we can use the shortest path model. The shortest path model finds how to transport items/people from on location to another while minimizing the total distance traveled, time taken or some other measures. In Bill’s case we will minimize the time it takes to transport his bottles from his plant at node 1 to his warehouse at node 10. A road map for Binder’s Beverage with the nodes and distances stated in minutes is depicted below.

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Binder’s Beverage Road Map

The solution of Bill’s problem shows that the shortest path (measured in terms of time) from the plant to the warehouse is 60 minutes and involves traveling through nodes 2, 4, 8 and 9. The Excel Solution from Solver is attached separately. At node one which is the supply node the net flow is -1. At node 10 which is the demand node the net flow is 1. All other nodes are transshipment nodes and hence have a net flow equal to zero. Below you can find a screenshot from Excel.


1. LP Problem Formulation:
In this case study there are four decision variables.

C30 = Pounds of C-30 to put in a fifty bag fertilizer
C92 = Pounds of C-92 to put in a fifty bag fertilizer
D = Pounds of D-21 to put in a fifty bag fertilizer
E = Pounds of E-11 to put in a fifty bag fertilizer

The objective is to minimize the cost of chemical compounds used and can be written as:

Minimize Total Cost = $0.12(C30) + $0.09(C92) + $0.11(D) + $0.04(E)

subject to following constraints:

a) E>= 0.15 (C30+C92+D+E)
b) C92+C30>= 0.45 (C30+C92+D+E)
c) D+C92= 0

2. The Best solution is to use £7.5 of C-30, £15 of C-92 and £27.5 of E-11 and not to include D-21 in the mix at all. This will result in a cost of $3.35 per £50 bag.

Excel Solution is attached as a separate file. Screenshots of solution and formulas are given below.

All constraints are binding with the exception of constraint (a), which indicated that E-11 should at least comprise 15% of the blend. This would mean the lower bound of E-11 to be used was £7.5, where in fact we ended up using £27.5 of E-11. The difference of £20 is shown in the answer report as a slack of 20.

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