Most inverse optimization models impute unspecified parameters of an objective function to make an observed solution optimal for a given optimization problem with a fixed feasible set. We propose two approaches to impute unspecified left-hand-side constraint coefficients in addition to a cost vector for a given linear optimization problem. The first approach identifies parameters minimizing the duality gap, while the second minimally perturbs prior estimates of the unspecified parameters to satisfy strong duality, if it is possible to satisfy the optimality conditions exactly. We apply these two approaches to the general linear optimization problem. We also use them to impute unspecified parameters of the uncertainty set for robust linear optimization problems under interval and cardinality constrained uncertainty. Each inverse optimization model we propose is nonconvex, but we show that a globally optimal solution can be obtained either in closed form or by solving a linear number of linear or convex optimization problems.

Aim To estimate the cost‐minimizing size and skill mix of a nursing resource team (NRT). Background Nurse absences can be filled by an NRT at lower hourly cost than staffing agencies or nurses working overtime, but an NRT must be appropriately sized to minimize total cost. Methods Using all registered nurse (RN) absences at an academic teaching hospital from 1 October 2014 to 31 March 2018, we developed a generalized additive model (GAM) to forecast the weekly frequency of each of ten types of absence over 52 weeks. We used the forecasts in an optimization model to determine the cost‐minimizing NRT composition. Results The median weekly frequencies for the ten absence types ranged between 12 and 65.5. The root mean squared errors of the GAMs ranged between 4.55 and 9.07 on test data. The NRT dimensioned by the optimization model yields an estimated annual cost reduction of $277,683 (Canadian dollars) (7%). Conclusions The frequency of RN absences in a hospital can be forecasted with high accuracy, and the use of forecasting and optimization to dimension an NRT can substantially reduce the cost of filling RN absences. Implications for Nursing Management This methodology can be adapted by any hospital to optimize nurse staffing.

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