Mathematical formulation

Multi-period, multi-item production planning problem

Let PRODS be the set of products (glass types) and WEEKS = {1,...,NT} the set of time periods. We write DEM_pt for the demand for product p in time period t. We also have CPROD_p and CSTOCK_p the production and storage cost for glass type p. This cost is identical for all time periods, but it would be easy to model a different cost per time period by adding an index for the time period.

TIMEW_p and TIMEM_p denote the worker and machine times respectively required per unit of product p, and correspondingly, SPACE_p the storage area. The initial stock ISTOCK_p is given, as is the desired final stock level FSTOCK_p per product (see data in Table Data for six glass types). We write CAPW and CAPM for the capacities of workers and machines respectively, and CAPS for the capacity of the storage area.

To solve this problem, we need variables produce_pt to represent the production of glass type p in time period t. The variables corresponding to the stock level of every product p at the end of period t are called store_pt. By convention, the initial stock level ISTOCK_p may be considered as the stock level at the end of time period 0 and we use the notation store_p0 to simplify the formulation of the stock balance constraints:

These stock balance constraints state that the quantity store_pt of product that is held in stock at the end of a time period t equals the stock level store_p,t-1 at the end of the preceding period plus the production produce_pt of the time period t minus the demand DEM_pt of this time period.

We wish to have a certain amount of product in stock at the end of the planning period to avoid that stocks run down to zero at the end of the planning horizon. These constraints on the final stock levels are expressed by the constraints:

We now formulate the various capacity constraints for every time period. The following constraints guarantee that the capacity limits on manpower, machine time, and storage space are kept:

The cost function that is to be minimized is the sum of production and storage costs for all products and time periods.

We obtain the complete mathematical model by the non-negativity constraints for the production variables and for the stored quantities to the constraints described above.

Robust optimization problem

Various assumptions behind the mathematical model that we have stated above might be questioned:

1. Constant resource capacity: availability of personnel most likely will not be constant over time (subject to holidays, training, sick leave etc.) and there is also a risk of scheduled (maintenance) or unscheduled (breakdown) machine outages.
2. Exact demand quantities are known: demand forecasts typically are estimates, most often resulting from an analysis of historical values.

Workers' absence

The case of machine outage (formulated as k contingencies) is studied in Sections Production planning under energy supply uncertainty and Robust unit commitment of this whitepaper. Let us therefore here take a look at how we might capture the uncertainty in the availability of personnel.

Let ABSENCE_t be a maximum limit on the absence hours per time period t (the actual absence will take at most this value), and we also know by experience what is the average absence over a longer period of time (expressed as a percentage A of the total worker hours). We introduce uncertains absent_t to represent the actual absence hours per time period. The worker capacity constraints from the original model are then replaced by the following:

where the polyhedral uncertainty set U_absent is characterized by

Demand scenarios

For a robust formulation of the demand, assume that we have got a number of different scenarios—obtained from historical data for comparable periods or possibly resulting from different forecasting methodologies—that describe the space of possible realizations of demand. In the place of the fixed demand quantities DEM_pt we now work with uncertain quantities demand_pt that are determined by the demand scenario data SCENDEM_spt for a given set of scenarios s∈SCEN. The demand s included in the original model formulation via the stock balance constraints.

A naive 'robustification' might attempt to simply replace the fixed demand quantities by the uncertains demand_pt. However, this approach would not lead to the desired result: by introducing a single uncertain quantity in an equality constraint we do not leave any room for different realizations of the uncertain. We therefore now work with the following two sets of inequalities in the place of the stock balance constraints.

The first of these inequalities states that the demand must be satisfied from the production in a period and the difference between stock levels at the beginning and end of the period, which can be more easily seen in this transformed version of the inequality:

The second inequality forces the final stock per period to be at least what remains after satisfying the largest possible demand.