Setting Product Availability for Multiple Products Under Capacity Constraints in a Supply Chain

In our discussion up to this point, we have assumed that a firm can set its desired level of product availability, and no constraints interfere with this choice. A common scenario in which this assumption fails occurs when the desired level of product availability results in an order size that exceeds the available capacity at the supplier. When ordering a single product, it is optimal for the buyer to order the minimum of the available capacity and the optimal order quantity. When ordering multiple products, however, the buyer needs to consider the trade-off between ordering more of one product versus another.

Consider a department store that plans to order two styles of sweaters from an Italian supplier. Demand for the high-end sweater is forecast to be normally distributed, with a mean of mi = 1,000 and a standard deviation of s₁ = 300. Demand for the mid-range sweater is normally distributed, with a mean of m₂ = 2,000 and a standard deviation of s₂ = 400. The high-end sweater has a retail price of p₁ = $150, a cost c₁ = $50, and a salvage value of s₁ = $35. The mid-range sweater has a retail price of p₂ = $100, a cost c₂ = $40, and a salvage value of s₂ = $25. The following analy­sis is also detailed in the spreadsheet Section13.4. Using Equation 13.1, the optimal level of product availability for the high-end sweater is (150 – 50)/(150 – 35) = 0.87 and that for the mid­range sweater is (100 – 40) /(100 – 25) = 0.80. Thus, without capacity constraints, it is opti-mal for the department store to order 1,337 [ = NORMINV(0.87, 1000, 300)] units of the high-end sweater and 2,337 [ = NORMINV(0.80, 2000, 400)] units of the mid-range sweater. If the supplier has a capacity constraint of 3,000 units, though, the desired ordering policy is not feasible, and the department store must decrease the size of its order by a total of at least 674 units. The preceding analysis is provided in the worksheet no constraint. Where should this decrease come from? Should the decrease be divided evenly between the two products?

First let us consider the simplistic approach of decreasing the order size of each product by 337 units to get an order of 1,000 high-end sweaters and 2,000 mid-range sweaters (use work­sheet capacity constraint). This order size meets the capacity constraint, and the expected profit is $194,268 (using Equation 13.3). To check whether this order size is optimal, we can think in terms of how capacity is allocated to the two styles. Let us assume that we have decided to allo­cate 1,000 units to the high-end sweater and 1,999 units to the mid-range sweater. That leaves only the last unit of capacity to be allocated. Which sweater should this unit be assigned to? It is reasonable to make this decision based on the expected marginal contribution to profits if this unit of capacity is allocated to each of the two styles. The last unit of capacity should be allocated to the sweater with the higher expected marginal contribution. Recall that F_i(Q_i) is the probabil­ity that demand for product i is Qi or less. Let MC_i(Q_i) be the marginal contribution of a sweater of type i if quantity Qi is ordered. The expected marginal contribution is evaluated similar to that in Table 13-2 and is obtained as follows:

Expected marginal contribution for high-end sweater = MC₁ (1,000)

Expected marginal contribution for mid-range sweater = MC₂( 1,999)

Clearly, it is better to allocate the last unit of capacity to the high-end sweater rather than the mid-range sweater. In fact, changing the order size to 1,001 high-end sweaters and 1,999 mid-range sweaters increases the expected profits by almost $20. One can now decrease the order size for the mid-range sweater to 1,998 and ask how the last unit of capacity should be allocated. Repeating this procedure indicates that the order size for the high-end sweaters should be increased to at least 1,002. In fact, the order size for the high-end sweater should be increased until the expected marginal contribution for the high-end sweater is the same as that for the mid­range sweater. At that point, it no longer makes sense to move capacity from one type of sweater to another. The optimal allocation of capacity turns out to be 1,089 high-end sweaters and 1,911 mid-range sweaters. The expected profits for this order size are $195,152. Observe that at opti­mality, the high-end sweater is allocated a relatively high share of the available capacity because its margin relative to the cost of overstocking is higher than that of the mid-range sweater.

The idea of allocating the available capacity to the product with the highest expected mar­ginal contribution can be converted into a solution procedure. Let each product i have a mean demand of μi and a standard deviation of σi. Product i has a retail price of pi, a cost c, and a salvage value of si. If quantity Qi is allocated to product i, the expected marginal contribution is obtained as

The following procedure allocates each unit of capacity to the product with the highest expected marginal contribution. Let B be the total available capacity.

1. Set quantity Q_i = 0 for all products
2. Compute the expected marginal contribution MC_i(Q_i) for each product i using Equation 13.8.
3. If no expected marginal contribution is positive, stop. Otherwise, let j be the product with the highest expected marginal contribution. Increase Qj by one unit.

4. If the total quantity across all products is less than B, return to step 2. Otherwise, the capac­ity constraint has been met and the current quantities are optimal.

Partial results from the application of the procedure described above to the department store data are shown in Table 13-5. A more detailed version of Table 13-5 is contained in the worksheet Capacity allocation.

The order quantities under capacity constraints can also be obtained by solving an optimi­zation problem (see worksheet Optimization). Let nfQ) be the expected profit obtained using Equation 13.3 from ordering Qi units of product i. The appropriate order quantities can be obtained by solving the following optimization problem:

Key Point

When ordering multiple products under a limited supply capacity, the allocation of capacity to products should be based on their expected marginal contribution to profits. This approach allocates a relatively higher fraction of capacity to products that have a high margin relative to their cost of overstocking.

Source: Chopra Sunil, Meindl Peter (2014), Supply Chain Management: Strategy, Planning, and Operation, Pearson; 6th edition.