When planning procurement strategies in a multi-plant context, it often happens that by aggregating orders across plants order size can be increased sufficiently to qualify for volume discounts not available to each plant individually. When this is the case, and where there are more suppliers than absolutely necessary to supply the total order, a question arises as to the best way to allocate to the total order to the various suppliers. 

If volume discount schedules are provided by each of the suppliers being considered for a given order, then the problem of allocating demand to the various suppliers can be formulated and solved as a mixed-integer programming problem. Binary variables are introduced to flag the price breaks qualified for at each of the suppliers included in the model. Continuous variables are included to represent the amount purchased from each supplier, and the shipping plan from supplier to demand points. 

An EXCEL application has been developed to illustrate this optimization model on a test database involving 5 plants and 10 suppliers. Each supplier has three price-breaks giving four different unit costs for the product, each applicable to a different range of volumes. The unit cost decreases as the volume increases. The user inputs a total demand for the 5 plants combined, upper and lower bounds on demand for the individual plants and upper and lower bounds on amount purchased from each supplier, and then a Solver is called to optimize the objective function subject to the stated constraints, including the discount opportunities for high volume orders. The objective function itself can be one of several kinds. It can include the two basic costs only, cost of items procured and cost of shipping. As an option, a cost for time to delivery can be added with a user supplied cost per item-week factor. As another option, an effective interest rate can be included with which to compute a cost of implied inventory holdings associated with the lead-times from the various suppliers. And finally, one can also include a cost of risk term in the objective, which estimates the cost of replacement goods in the event that one of the selected suppliers does not deliver on time. These are based on subjective probability of default factors for each supplier supplied by the user. In addition, trade-offs between time and cost can be analyzed by varying the cost per unit-week cost factor across a range of values. One obtains the minimum cost solution at one extreme, and the minimum time solution at the other extreme. The various intermediate solutions are mapped out parametrically as the cost per unit-week is gradually increased from small to large values. The various solutions obtained in this way are said to be time-cost Pareto Optimal. 

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