I was wondering if something is missing in my solution. If so: an explanation would be good. Thank you.

The health care department has to hire new doctors. In the department there are m specializations (orthopedics, pediatrics, etc.). For each specialization j a minimum number of new doctors bj are required. There are n candidate doctors. Each doctor i can cover several specializations (at the same time). Si denotes the set of specializations that doctor i covers. The salary cost of each doctor i, if hired, is ci. The problem of the health care department manager is to hire a subset of doctors that guarantees to cover all requirements at minimum cost. Formulate the problem with a linear model. Variant: According to interregional agreements, some specializations of the department may be left uncovered, by activating a contract with nearby regions. For each specialization j the cost of the contract is tj. In case the contract j is activated, there is no need to hire doctors to cover the bj positions. How does the model change, if the objective becomes to minimize the total cost for hiring doctors and activating the contracts? Parameters and sets: Mi number of specializations n : number of doctor? bj : minimum number of new docfors required for specialization j Ci: the solary cost of docfor i Si set of specializations that docfor i covers Variable Sets, Indicate the indices and their range, the meaning of the variables and their nature (binary, integer...): To determune whether docfor i is to be Wred or not we use the binary variabie Xi xi={70ifhiredelsexi{0,7} Objective function: mini=1nxici doctor requirement constraints: i=7nxibi for all j=1,,m counts the number of doctors that covers speciatization j. jSi takes the value I or 0 based on doctor isSi. Other constraints if needed: New variables of the variant (if needed): Confract or not is indicaled by yj. yj={7ifcontractisactivatedforspe0otherwiseyj={0,7} New objective function: mini=1nxici+j=1myjtj Modified constraints (if needed): i=1nxi+yjSibjforalej=7,,mi=1nxiSi(7yj)bjforallj=7,,m