Formulating a mathematical model for a single-sourcing supply chain that incorporates risk constraints involves multiple steps, including defining decision variables, constraints, and an objective function. Here's a structured approach:

1. Define Decision Variables

Let:

• xx = Order quantity from the single supplier
• yy = Inventory level at the buyer's warehouse
• ss = Supplier's production capacity
• c(x)c(x) = Cost function (ordering, holding, and shortage costs)
• rr = Risk factor affecting supply (e.g., probability of supplier failure)
• dd = Demand (may follow a stochastic distribution)
• tt = Lead time

2. Objective Function

A common goal is to minimize total costs while considering risks:

min⁡C=c(x)+H(y)+P(s)+R(x,r)\min \quad C = c(x) + H(y) + P(s) + R(x, r)

where:

• H(y)H(y) = Holding cost of inventory
• P(s)P(s) = Supplier production cost
• R(x,r)R(x, r) = Risk-adjusted cost (e.g., disruption penalties, lost sales)

3. Constraints

To capture risks, we introduce constraints:

a. Demand Satisfaction Constraint

x≥dx \geq d

(Ensures enough supply to meet demand)

b. Capacity Constraint

x≤sx \leq s

(Supplier cannot produce beyond their maximum capacity)

c. Inventory Balance Equation

yt+1=yt+x−dy_{t+1} = y_t + x - d

(Inventory at next period depends on current inventory, orders, and demand)

d. Risk Constraint (Supplier Disruption Probability)

P(supplier fails)≤θP(\text{supplier fails}) \leq \theta

(This sets an upper bound θ\theta on acceptable risk levels)

e. Lead Time Impact

T(x)=f(t,r)T(x) = f(t, r)

(Lead time T(x)T(x) depends on supply risk and disruptions)

4. Simulation & Validation

• Use Monte Carlo simulation to model demand variability and supplier risk.
• Simulate different disruption scenarios and their impact on costs.
• Validate the model using real-world supply chain data or past disruption cases.

5. Sensitivity Analysis

• Vary key parameters like rr, dd, and tt to assess system robustness.
• Identify the threshold where the system fails to meet demand.
• Evaluate the trade-off between cost and reliability.

6. Presenting Solutions

• Risk-aware ordering strategy: Adjust order sizes based on risk exposure.
• Inventory buffering: Maintain safety stock for high-risk situations.
• Alternative sourcing strategy: Identify conditions under which dual sourcing is preferable.

To formulate a supply chain mathematical model for single-sourcing with risk constraints, follow these steps:

1. Define Decision Variables

Let:

• xtx_t = Order quantity from the supplier at time tt.
• ItI_t = Inventory level at the buyer’s warehouse at time tt.
• dtd_t = Demand at time tt (may follow a stochastic distribution).
• ss = Supplier’s maximum production capacity.
• rtr_t = Risk factor affecting supply at time tt (e.g., probability of supplier failure).
• TT = Lead time for replenishment.
• C(xt)C(x_t) = Total cost function including ordering, holding, shortage, and risk costs.

2. Objective Function

A common goal is to minimize total costs while accounting for supply chain risks:

min⁡C=∑t[O(xt)+H(It)+S(dt−It−xt)+R(xt,rt)]\min \quad C = \sum_{t} \left[ O(x_t) + H(I_t) + S(d_t - I_t - x_t) + R(x_t, r_t) \right]

where:

• O(xt)O(x_t) = Ordering cost.
• H(It)H(I_t) = Holding cost for inventory.
• S(dt−It−xt)S(d_t - I_t - x_t) = Shortage or penalty cost if demand is unmet.
• R(xt,rt)R(x_t, r_t) = Risk-adjusted cost, which includes disruption penalties and lost sales.

3. Constraints

To incorporate risks, enforce these constraints:

a. Demand Satisfaction Constraint

It+xt≥dt,∀tI_t + x_t \geq d_t, \quad \forall t

(Ensures that the supply meets demand)

b. Capacity Constraint

xt≤s,∀tx_t \leq s, \quad \forall t

(The supplier cannot produce beyond their maximum capacity)

c. Inventory Balance Equation

It+1=It+xt−dt,∀tI_{t+1} = I_t + x_t - d_t, \quad \forall t

(Inventory at the next period depends on current inventory, orders, and demand)

d. Risk Constraint (Supplier Disruption Probability)

P(supplier fails)≤θP(\text{supplier fails}) \leq \theta

(Defines an upper bound θ\theta for acceptable supply risk)

e. Lead Time Impact

T(xt)=f(t,rt)T(x_t) = f(t, r_t)

(Lead time T(xt)T(x_t) depends on supplier risk and disruptions)

4. Simulation & Validation

• Use Monte Carlo simulation to model stochastic demand and supplier risk.
• Run scenario analysis with different levels of supplier failure probabilities.
• Compare outcomes with historical supply chain data to validate the model.

5. Sensitivity Analysis

• Vary key parameters (rtr_t, dtd_t, ss, and TT) to assess robustness.
• Identify the threshold where the system fails to meet demand.
• Evaluate the cost-risk trade-off for different sourcing strategies.

6. Presenting Solutions

• Risk-aware ordering strategy: Adjust order sizes based on risk exposure.
• Inventory buffering: Maintain safety stock for high-risk situations.
• Contractual risk mitigation: Consider backup suppliers or penalty clauses.

There are probably many similar models in the relevant scientific literature that use mathematical models of the supply chain.

So, you can take a look at one typical such model here on Researchgate from 2017:

Article Supply chain resilience for single and multiple sourcing in ...

which includes practically most of the things/cases/scenarios/analysis you are looking for - in equations (2) to (16) - plus References for the previous 20 years.